# Integer Sequences for the difference for Primes in Arithmetic Progression with the minimal start Sequence {p + j*d}, j = 0 to k-1

1. Sameen Ahmed Khan,
Sequence A206037: 2, 4, 8, 10, 14, 20, 28, 34, 38, 40, 50, 64, 68, 80, 94, 98, 104, 110, 124, 134, 154, 164, 178, 188, 190, 208, 220, 230, 238, 248, ...,
Values of the difference d for 3 primes in arithmetic progression with the minimal start sequence {3 + j*d}, j = 0 to 2.,
in N. J. A. Sloane (Editor), The On-Line Encyclopedia of Integer Sequences,
published electronically at
http://oeis.org/A206037.
(Friday the 03 February 2012).

2. Sameen Ahmed Khan,
Sequence A206038: 6, 12, 18, 42, 48, 54, 84, 96, 126, 132, 252, 348, 396, 426, 438, 474, 594, 636, 642, 648, 678, 804, 858, 1176, 1218, 1272, 1302, 1314, 1362, 1428, ...,
Values of the difference d for 4 primes in arithmetic progression with the minimal start sequence {5 + j*d}, j = 0 to 3.,
in N. J. A. Sloane (Editor), The On-Line Encyclopedia of Integer Sequences,
published electronically at
http://oeis.org/A206038.
(Friday the 03 February 2012).

3. Sameen Ahmed Khan,
Sequence A206039: 6, 12, 42, 48, 96, 126, 252, 426, 474, 594, 636, 804, 1218, 1314, 1428, 1566, 1728, 1896, 2106, 2574, 2694, 2898, 3162, 3366, 4332, 4368, 4716, 4914, 4926, ...,
Values of the difference d for 5 primes in arithmetic progression with the minimal start sequence {5 + j*d}, j = 0 to 4.,
in N. J. A. Sloane (Editor), The On-Line Encyclopedia of Integer Sequences,
published electronically at
http://oeis.org/A206039.
(Friday the 03 February 2012).

4. Sameen Ahmed Khan,
Sequence A206040: 30, 150, 930, 2760, 3450, 4980, 9150, 14190, 19380, 20040, 21240, 28080, 33930, 57660, 59070, 63600, 69120, 76710, 80340, 81450, 97380, 100920, 105960, ...,
Values of the difference d for 6 primes in arithmetic progression with the minimal start sequence {7 + j*d}, j = 0 to 5.,
in N. J. A. Sloane (Editor), The On-Line Encyclopedia of Integer Sequences,
published electronically at
http://oeis.org/A206040.
(Friday the 03 February 2012).

5. Sameen Ahmed Khan,
Sequence A206041: 150, 2760, 3450, 9150, 14190, 20040, 21240, 63600, 76710, 117420, 122340, 134250, 184470, 184620, 189690, 237060, 274830, 312000, 337530, 379410, ...,
Values of the difference d for 7 primes in arithmetic progression with the minimal start sequence {7 + j*d}, j = 0 to 6.,
in N. J. A. Sloane (Editor), The On-Line Encyclopedia of Integer Sequences,
published electronically at
http://oeis.org/A206041.
(Friday the 03 February 2012).

6. Sameen Ahmed Khan,
Sequence A206042: 1210230, 2523780, 4788210, 10527720, 12943770, 19815600, 22935780, 28348950, 28688100, 32671170, 43443330, 47330640, 51767520, 54130440, ...,
Values of the difference d for 8 primes in arithmetic progression with the minimal start sequence {11 + j*d}, j = 0 to 7.,
in N. J. A. Sloane (Editor), The On-Line Encyclopedia of Integer Sequences,
published electronically at
http://oeis.org/A206042.
(Friday the 03 February 2012).

7. Sameen Ahmed Khan,
Sequence A206043: 32671170, 54130440, 59806740, 145727400, 224494620, 246632190, 280723800, 301125300, 356845020, 440379870, 486229380, 601904940, 676987920, ...,
Values of the difference d for 9 primes in arithmetic progression with the minimal start sequence {11 + j*d}, j = 0 to 8.,
in N. J. A. Sloane (Editor), The On-Line Encyclopedia of Integer Sequences,
published electronically at
http://oeis.org/A206043.
(Friday the 03 February 2012).

8. Sameen Ahmed Khan,
Sequence A206044: 224494620, 246632190, 301125300, 1536160080, 1760583300, 4012387260, 4911773580, 7158806130, 8155368060, 15049362300, 15908029410, ...,
Values of the difference d for 10 primes in arithmetic progression with the minimal start sequence {11 + j*d}, j = 0 to 9.,
in N. J. A. Sloane (Editor), The On-Line Encyclopedia of Integer Sequences,
published electronically at
http://oeis.org/A206044.
(Friday the 03 February 2012).

9. Sameen Ahmed Khan,
Sequence A206045: 1536160080, 4911773580, 25104552900, 77375139660, 83516678490, 100070721660, 150365447400, 300035001630, 318652145070, 369822103350, ...,
Values of the difference d for 11 primes in arithmetic progression with the minimal start sequence {11 + j*d}, j = 0 to 10.,
in N. J. A. Sloane (Editor), The On-Line Encyclopedia of Integer Sequences,
published electronically at
http://oeis.org/A206045.
(Friday the 03 February 2012).

# Integer Sequences for the difference for Primes in Geometric-Arithmetic Progression with the minimal start, and minimal ratio Sequence {p*pj + j*d}, j = 0 to k-1

A geometric-arithmetic progression of primes is a set of k primes (denoted by GAP-k) of the form p1*r j + j*d for fixed p1, r and d and consecutive j, from j = 0 to k - 1. i.e, {p1, p1*r + d, p1*r 2 + 2 d, p1* r 3 + 3 d, ...}. For example 3, 17, 79 is a 3-term geometric-arithmetic progression (i.e, a GAP-3) with a = p1 = 3, r = 5 and d = 2. A GAP-k is said to be minimal if the minimal start p1 and the minimal ratio r are equal, i.e, p1 = r = p, where p is the smallest prime ≥ k. Such GAPs have the form p*p j + j*d. Minimal GAPs with different differences, d do exist. For example, the minimal GAP-5 (p1 = r = 5) has the possible differences, 84, 114, 138, 168, ... (see the Sequence A209204) and the minimal GAP-6 (p1 = r = 7) has the possible differences, 144, 1494, 1740, 2040, .... (see the Sequence A209205). The following article gives the conditions under which, a GAP-k is a set of k primes in geometric-arithmetic progression.

10. Sameen Ahmed Khan,
Sequence A209202: 2, 8, 10, 20, 22, 28, 38, 50, 52, 62, 70, 92, 98, 100, 118, 122, 128, 140, 142, 170, 202, 218, 220, 230, 232, 248, 260, 268, 272, 302, ...,
Values of the difference d for the geometric-arithmetic progression {3*3j + j*d}, j = 0 to j = 2 to be a set of 3 primes,
in N. J. A. Sloane (Editor), The On-Line Encyclopedia of Integer Sequences,
published electronically at http://oeis.org/A209202.
(Tuesday the 06 March 2012).

11. Sameen Ahmed Khan,
Sequence A209203: 6, 12, 16, 28, 34, 36, 54, 76, 78, 84, 114, 124, 132, 138, 142, 148, 154, 166, 168, 208, 226, 258, 268, 288, 324, 348, 376, 414, 436, 442, ...,
Values of the difference d for the geometric-arithmetic progression {5*5j + j*d}, j = 0 to j = 3 to be a set of 4 primes,
in N. J. A. Sloane (Editor), The On-Line Encyclopedia of Integer Sequences,
published electronically at http://oeis.org/A209203.
(Tuesday the 06 March 2012).

12. Sameen Ahmed Khan,
Sequence A209204: 84, 114, 138, 168, 258, 324, 348, 462, 552, 588, 684, 714, 744, 798, 882, 894, 972, 1176, 1602, 1734, 2196, 2256, 2442, 2478, 2568, 2646, ...,
Values of the difference d for the geometric-arithmetic progression {5*5j + j*d}, j = 0 to j = 4 to be a set of 5 primes,
in N. J. A. Sloane (Editor), The On-Line Encyclopedia of Integer Sequences,
published electronically at http://oeis.org/A209204.
(Tuesday the 06 March 2012).

13. Sameen Ahmed Khan,
Sequence A209205: 144, 1494, 1740, 2040, 3324, 4044, 6420, 12804, 13260, 13464 13620, 15444, 25824, 31524, 31674, 31680, 32124, 33720, 38064, 40410, ...,
Values of the difference d for the geometric-arithmetic progression {7*7j + j*d}, j = 0 to j = 5 to be a set of 6 primes,
in N. J. A. Sloane (Editor), The On-Line Encyclopedia of Integer Sequences,
published electronically at http://oeis.org/A209205.
(Tuesday the 06 March 2012).

14. Sameen Ahmed Khan,
Sequence A209206: 3324, 13260, 38064, 46260, 51810, 54510, 58914, 76050, 81510, 82434, 109800, 119340, 120714, 132390, 141480, 154254, 167904, 169734, 185040, ...,
Values of the difference d for the geometric-arithmetic progression {7*7j + j*d }, j = 0 to j = 6 to be a set of 7 primes,
in N. J. A. Sloane (Editor), The On-Line Encyclopedia of Integer Sequences,
published electronically at http://oeis.org/A209206.
(Tuesday the 06 March 2012).

15. Sameen Ahmed Khan,
Sequence A209207: 62610, 165270, 420300, 505980, 669780, 903030, 932400, 1004250, 1052610, 1093080, 1230270, 1231020, 1248120, ...,
Values of the difference d for the geometric-arithmetic progression {11*11j + j*d}, j = 0 to j = 7 to be a set of 8 primes,
in N. J. A. Sloane (Editor), The On-Line Encyclopedia of Integer Sequences,
published electronically at http://oeis.org/A209207.
(Tuesday the 06 March 2012).

16. Sameen Ahmed Khan,
Sequence A209208: 903030, 1004250, 3760290, 7296450, 7763520, 17988210, 28962390, 29956950, 33316320, 37265160, 39013800, 39768150, 43920480, 50110620, 54651480, 56388810, 74306610, 74679810, 75911850, 89115210, 92619690, 98518800, ...,
Values of the difference d for the geometric-arithmetic progression {11*11j + j*d }, j = 0 to j = 8 to be a set of 9 primes,
in N. J. A. Sloane (Editor), The On-Line Encyclopedia of Integer Sequences,
published electronically at http://oeis.org/A209208.
(Tuesday the 06 March 2012).

17. Sameen Ahmed Khan,
Sequence A209209: 903030, 17988210, 28962390, 39768150, 74306610, 89115210, 116535300, 173227980, 186013380, 237952050, 359613030, 386317920, 392253990, 443687580, 499153200, 548024610, 591655080, ...,
Values of the difference d for the geometric-arithmetic progression {11*11j + j*d }, j = 0 to j = 9 to be a set of 10 primes,
in N. J. A. Sloane (Editor), The On-Line Encyclopedia of Integer Sequences,
published electronically at http://oeis.org/A209209.
(Tuesday the 06 March2012).

18. Sameen Ahmed Khan,
Sequence A209210: 443687580, 591655080, 1313813550, 2868131100, 3525848580, 3598823970, 4453413120, 6075076800, 6644124480, 7429693770, 9399746580, 11801410530, ...,
Values of the difference d for the geometric-arithmetic progression {11*11j + j*d }, j = 0 to j = 10 to be a set of 11 primes,
in N. J. A. Sloane (Editor), The On-Line Encyclopedia of Integer Sequences,
published electronically at http://oeis.org/A209210.
(Tuesday the 06 March 2012).

19. Sameen Ahmed Khan,
Sequence A227280: 81647160420, 170655787050, 211212209880, 227961624450, ...,
Values of the difference d for 12 primes in geometric-arithmetic progression with the minimal sequence {13*13j + j*d}, j = 0 to 11,
in N. J. A. Sloane (Editor), The On-Line Encyclopedia of Integer Sequences,
published electronically at
http://oeis.org/A227280.
(Friday the 05 July 2013).

# Integer Sequences for the First primes of arithmetic progressions of k primes each with common difference k# Minimal Difference Sequence {p1 + j*(k#)}, j = 0 to k-1

The minimal possible difference in an AP-k is conjectured to be k# for all k > 7.
The exceptional cases (for k < = 7) are k = 2, k = 3, k = 5 and k = 7.

For k = 2, we have d = 1 and there is ONLY one AP-2 with this difference: {2, 3}.

For k = 3, we have d = 2 and there is ONLY one AP-3 with this difference: {3, 5, 7}.

For k = 4, we have d = 4# = 6 and AP-4 is {5, 11, 17, 23} and is not unique.
The first primes is the Sequence A023271: 5, 11, 41, 61, 251, 601, 641, 1091, 1481, 1601, 1741, 1861, 2371, ...

For k = 5, we have d = 3# = 6 and there is ONLY one AP-5 with this difference: {5, 11, 17, 23, 29}.

For k = 6, we have d = 6# = 30 and AP-6 is {7, 37, 67, 97, 127, 157} and is not unique.
The first primes is the Sequence A156204: 7, 107, 359, 541, 2221, 6673, 7457, 10103, 25643, 26861, 27337, 35051, 56149, ...

For k = 7, we have d = 5*5# = 150 and there is ONLY one AP-7 with this difference: {7, 157, 307, 457, 607, 757, 907}.

20. Sameen Ahmed Khan,
Sequence A227281: 7, 11, 37, 107, 137, 151, 277, 359, 389, 401, 541, 557, 571, 877, 1033, 1493, 1663, 2221, 2251, 2879, 3271, 6269, 6673, 6703, 7457, 7487, 9431, 10103, 10133, 10567, 11981, 12457, 12973, 14723, 17047, 19387, 24061, 25643, 25673, 26861, 26891, 27337, 27367, ...,
First primes of arithmetic progressions of 5 primes each with the common difference 30,
in N. J. A. Sloane (Editor), The On-Line Encyclopedia of Integer Sequences,
published electronically at
http://oeis.org/A227281.
(Friday the 05 July 2013).

21. Sameen Ahmed Khan,
Sequence A227282: 47, 179, 199, 409, 619, 829, 881, 1091, 1453, 3499, 3709, 3919, 10529, 10627, 10837, 10859, 11069, 11279, 14423, 20771, 22697, 30097, 30307, 31583, 31793, 32363, 41669, 75703, 93281, 95747, 120661, 120737, 120871, 120947, 129287, 140603, 153319, 153529, ...,
First primes of arithmetic progressions of 7 primes each with the common difference 210,
in N. J. A. Sloane (Editor), The On-Line Encyclopedia of Integer Sequences,
published electronically at
http://oeis.org/A227282.
(Friday the 05 July 2013).

22. Sameen Ahmed Khan,
Sequence A227283: 199, 409, 619, 881, 3499, 3709, 10627, 10859, 11069, 30097, 31583, 120661, 120737, 153319, 182537, 471089, 487391, 564973, 565183, 825991, 1010747, 1280623, 1288607, 1288817, 1302281, 1302491, 1395209, 1982599, 2358841, 2359051, 2439571, 3161017, 3600521, ...,
First primes of arithmetic progressions of 8 primes each with the common difference 210,
in N. J. A. Sloane (Editor), The On-Line Encyclopedia of Integer Sequences,
published electronically at
http://oeis.org/A227283.
(Friday the 05 July 2013).

23. Sameen Ahmed Khan,
Sequence A227284: 199, 409, 3499, 10859, 564973, 1288607, 1302281, 2358841, 3600521, 4047803, 17160749, 20751193, 23241473, 44687567, 50655739, 53235151, 87662609, 100174043, 103468003, 110094161, 180885839, 187874017, 192205147, 221712811, 243051733, 243051943, 304570103, ...,
First primes of arithmetic progressions of 9 primes each with the common difference 210,
in N. J. A. Sloane (Editor), The On-Line Encyclopedia of Integer Sequences,
published electronically at
http://oeis.org/A227284.
(Friday the 05 July 2013).

24. Sameen Ahmed Khan,
Sequence A227285: 60858179, 186874511, 291297353, 1445838451, 2943023729, 4597225889, 7024895393, 8620560607, 8656181357, 19033631401, 20711172773, 25366690189, 27187846201, 32022299977, 34351919351, ...,
First primes of arithmetic progressions of 11 primes each with the common difference 2310,
in N. J. A. Sloane (Editor), The On-Line Encyclopedia of Integer Sequences,
published electronically at
http://oeis.org/A227285.
(Friday the 05 July 2013).

25. Sameen Ahmed Khan,
Sequence A227286: 14933623, 2085471361, ...,
First primes of arithmetic progressions of 13 primes each with the common difference 30030,
in N. J. A. Sloane (Editor), The On-Line Encyclopedia of Integer Sequences,
published electronically at
http://oeis.org/A227286.
(Friday the 05 July 2013).

26. Sameen Ahmed Khan,

### List of Integer Sequences for "Primes in Arithmetic Progression" from the http://oeis.org/ (Logo). List of Integer Sequences for "Primes in Geometric-Arithmetic Progression" from the http://oeis.org/ (Logo).http://oeis.org/wiki/User:Sameen_Ahmed_Khan at OEIS Wiki (Logo).

 List of 37+ Writeups from the INSPIRE HEP (Logo), Originally SLAC SPIRES (Logo). List of 21+ Writeups from the LANL E-Print archive (see the Atom Feeds).

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4. Resistor Networks.

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6. Salt Solutions.

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In March 2005, I was appointed as the Regular Correspondent for the International Committee for Future Accelerators (ICFA, Logo) Beam Dynamics Panel Newsletters (Logo), for the region of Middle East & Africa. ICFA, the International Committee for Future Accelerators (Logo), provides a forum to discuss and implement plans for further promoting collaborative accelerator-based science. Its primary purpose is to strengthen collaboration in accelerator-based science, to encourage future projects, and to make recommendations to governments. See the International Committee for Future Accelerators (ICFA, Logo) Beam Dynamics Panel Newsletter (Logo), No. 36 (April 2005).
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