A172367 -id:A172367 - cp's OEIS frontend

A209202 Values of the difference d for 3 primes in geometric-arithmetic progression with the minimal sequence {33^j + jd}, j = 0 to 2.

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, 308, 328, 350, 358, 380, 392, 400, 430, 440, 470, 478, 482, 512, 532, 538, 548, 562, 568, 598, 632, 638, 650, 700, 710, 730

Offset: 1

Sameen Ahmed Khan, Mar 06 2012

nonn

A geometric-arithmetic progression of primes is a set of k primes (denoted by GAP-k) of the form p r^j + j d for fixed p, r and d and consecutive j. Symbolically, for r = 1, this sequence simplifies to the familiar primes in arithmetic progression (denoted by AP-k). The computations were done without any assumptions on the form of d. Primality requires d to be even and coprime to 3.

			d = 8 then {3*3^j + j*d}, j = 0 to 2, is  {3, 17, 43}, which is 3 primes in geometric-arithmetic progression.

Sameen Ahmed Khan, Table of n, a(n) for n = 1..10000
Sameen Ahmed Khan, Primes in Geometric-Arithmetic Progression, arXiv:1203.2083v1 [math.NT], (Mar 09 2012).

Cf. A172367, A209203, A209204, A209205, A209206, A209207, A209208, A209209, A209210.

p = 3; gapset3d = {}; Do[If[PrimeQ[{p, p*p + d, p*p^2 + 2*d}] == {True, True, True}, AppendTo[gapset3d, d]], {d, 0, 1000, 2}]; gapset3d

A209203 Values of the difference d for 4 primes in geometric-arithmetic progression with the minimal sequence {55^j + jd}, j = 0 to 3.

Original entry on oeis.org

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, 454, 462, 496, 538, 552, 562, 582, 588, 684, 714, 736, 744, 798, 804, 814, 832, 882, 894, 912, 946, 972, 994, 1006

Offset: 1

Sameen Ahmed Khan, Mar 06 2012

nonn

Numbers n such that n+25, 2n+125, and 3n+625 are prime.
A geometric-arithmetic progression of primes is a set of k primes (denoted by GAP-k) of the form p r^j + j d for fixed p, r and d and consecutive j.  Symbolically, for r = 1, this sequence simplifies to the familiar primes in arithmetic progression (denoted by AP-k). The computations were done without any assumptions on the form of d. Primality requires d to be even and coprime to 5.
This sequence is infinite on Dickson's conjecture. [Charles R Greathouse IV, Mar 12 2012]

			d = 12  then {5*5^j + j*d}, j = 0 to 3, is {5, 37, 149, 661}, which is 4 primes in geometric-arithmetic progression.

Sameen Ahmed Khan, Table of n, a(n) for n = 1..10000
Sameen Ahmed Khan, Primes in Geometric-Arithmetic Progression, arXiv:1203.2083v1 [math.NT], (Mar 09 2012).

Cf. A172367, A209202, A209204, A209205, A209206, A209207, A209208, A209209, A209210.

p = 5; gapset4d = {}; Do[If[PrimeQ[{p, p*p + d, p*p^2 + 2*d, p*p^3 + 3*d}] == {True, True, True, True}, AppendTo[gapset4d, d]], {d, 0, 1000, 2}]; gapset4d
Select[Range[2,1100,2],And@@PrimeQ[{#+25,2#+125,3#+625}]&] (* Harvey P. Dale, Jan 06 2013 *)

forstep(n=2,1e3,[2,2,2,4],if(isprime(n+25)&&isprime(2*n+125)&&isprime(3*n+625),print1(n", "))) \\ Charles R Greathouse IV, Mar 12 2012

A209204 Values of the difference d for 5 primes in geometric-arithmetic progression with the minimal sequence {55^j + jd}, j = 0 to 4.

Original entry on oeis.org

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, 2658, 2688, 3036, 3162, 3444, 3906, 4524, 5154, 5406, 5544, 5766, 5796, 6018, 6456, 6594, 6636, 6936, 7272, 7938, 8736

Offset: 1

Sameen Ahmed Khan, Mar 06 2012

nonn

A geometric-arithmetic progression of primes is a set of k primes (denoted by GAP-k) of the form p r^j + j d for fixed p, r and d and consecutive j. Symbolically, for r = 1, this sequence simplifies to the familiar primes in arithmetic progression (denoted by AP-k). The computations were done without any assumptions on the form of d. Primality requires d to be multiple of 3# = 6 and coprime to 5.

Subsequence of A209203. - Zak Seidov, Jul 06 2013

			d = 114 then {5*5^j + j*d}, j = 0 to 4, is {5, 139, 353, 967, 3581}, which is 5 primes in geometric-arithmetic progression.

Sameen Ahmed Khan, Table of n, a(n) for n = 1..10000
Sameen Ahmed Khan, Primes in Geometric-Arithmetic Progression, arXiv:1203.2083v1 [math.NT], (Mar 09 2012).

Cf. A172367, A209202, A209203, A209205, A209206, A209207, A209208, A209209, A209210.

p = 5; gapset5d  = {}; Do[If[PrimeQ[{p, p*p + d, p*p^2 + 2*d, p*p^3 + 3*d, p*p^4 + 4*d}] == {True, True, True, True, True}, AppendTo[gapset5d, d]], {d, 2, 10000, 2}]; gapset5d

A209205 Values of the difference d for 6 primes in geometric-arithmetic progression with the minimal sequence {77^j + jd}, j = 0 to 5.

Original entry on oeis.org

144, 1494, 1740, 2040, 3324, 4044, 6420, 12804, 13260, 13464, 13620, 15444, 25824, 31524, 31674, 31680, 32124, 33720, 38064, 40410, 44634, 45804, 46260, 51810, 54510, 56100, 58914, 60810, 68004, 69114, 70794, 74574, 76050, 77694, 80580, 81510, 82434, 89244

Offset: 1

Sameen Ahmed Khan, Mar 06 2012

nonn

A geometric-arithmetic progression of primes is a set of k primes (denoted by GAP-k) of the form p r^j + j d for fixed p, r and d and consecutive j. Symbolically, for r = 1, this sequence simplifies to the familiar primes in arithmetic progression (denoted by AP-k). The computations were done without any assumptions on the form of d. Primality requires d to be multiple of 3# = 6 and coprime to 7.

			d = 1494 then {7*7^j + j*d}, j = 0 to 5, is {7, 1543, 3331, 6883, 22783, 125119}, which is 6 primes in geometric-arithmetic progression.

Sameen Ahmed Khan, Table of n, a(n) for n = 1..10000
Sameen Ahmed Khan, Primes in Geometric-Arithmetic Progression, arXiv:1203.2083v1 [math.NT], (Mar 09 2012).

Cf. A172367, A209202, A209203, A209204, A209206, A209207, A209208, A209209, A209210.

p = 7; gapset6d = {}; Do[If[PrimeQ[{p, p*p + d, p*p^2 + 2*d, p*p^3 + 3*d, p*p^4 + 4*d, p*p^5 + 5*d}] == {True, True, True, True, True, True}, AppendTo[gapset6d, d]], {d, 0, 100000, 2}]; gapset6d

A209206 Values of the difference d for 7 primes in geometric-arithmetic progression with the minimal sequence {77^j + jd}, j = 0 to 6.

Original entry on oeis.org

3324, 13260, 38064, 46260, 51810, 54510, 58914, 76050, 81510, 82434, 109800, 119340, 120714, 132390, 141480, 154254, 167904, 169734, 185040, 209214, 252864, 253110, 256080, 278514, 291930, 292314, 337104, 341694, 379944, 392964, 404730, 406074, 412050

Offset: 1

Sameen Ahmed Khan, Mar 06 2012

nonn

A geometric-arithmetic progression of primes is a set of k primes (denoted by GAP-k) of the form p r^j + j d for fixed p, r and d and consecutive j. Symbolically, for r = 1, this sequence simplifies to the familiar primes in arithmetic progression (denoted by AP-k). The computations were done without any assumptions on the form of d. Primality requires d to be to be multiple of 3# = 6 and coprime to 7.

			d = 13260 then {7*7^j + j*d}, j = 0 to 6, is {7, 13309, 26863, 42181, 69847, 183949, 903103}, which is 7 primes in geometric-arithmetic progression.

Sameen Ahmed Khan, Table of n, a(n) for n = 1..5875
Sameen Ahmed Khan, Primes in Geometric-Arithmetic Progression, arXiv:1203.2083v1 [math.NT], (Mar 09 2012).

Cf. A172367, A209202, A209203, A209204, A209205, A209207, A209208, A209209, A209210.

p = 7; gapset7d = {}; Do[If[PrimeQ[{p, p*p + d, p*p^2 + 2*d, p*p^3 + 3*d, p*p^4 + 4*d, p*p^5 + 5*d, p*p^6 + 6*d}] == {True, True, True, True, True, True, True}, AppendTo[gapset7d, d]], {d, 0, 500000, 2}]; gapset7d

A209207 Values of the difference d for 8 primes in geometric-arithmetic progression with the minimal sequence {1111^j + jd}, j = 0 to 7.

Original entry on oeis.org

62610, 165270, 420300, 505980, 669780, 903030, 932400, 1004250, 1052610, 1093080, 1230270, 1231020, 1248120, 1433250, 1571430, 1742040, 1908480, 2668290, 2885220, 3367590, 3416520, 3760290, 3813630, 3965250, 3995340, 4137450, 4334610, 5443620, 5939250

Offset: 1

Sameen Ahmed Khan, Mar 06 2012

nonn

A geometric-arithmetic progression of primes is a set of k primes (denoted by GAP-k) of the form p r^j + j d for fixed p, r and d and consecutive j. Symbolically, for r = 1, this sequence simplifies to the familiar primes in arithmetic progression (denoted by AP-k). The computations were done without any assumptions on the form of d. Primality requires d to be multiple of 5# = 30 and coprime to 11.

			d = 165270 then {11*11^j + j*d}, j = 0 to 8, is {11, 165391, 331871, 510451, 822131, 2597911, 20478791, 215515771}, which is 8 primes in geometric-arithmetic progression.

Sameen Ahmed Khan, Table of n, a(n) for n = 1..3233
Sameen Ahmed Khan, Primes in Geometric-Arithmetic Progression, arXiv:1203.2083v1 [math.NT], (Mar 09 2012).

Cf. A172367, A209202, A209203, A209204, A209205, A209206, A209208, A209209, A209210.

p = 11; gapset8d = {}; Do[If[PrimeQ[{p, p*p + d, p*p^2 + 2*d, p*p^3 + 3*d, p*p^4 + 4*d, p*p^5 + 5*d, p*p^6 + 6*d, p*p^7 + 7*d}] == {True, True, True, True, True, True, True, True}, AppendTo[gapset8d, d]], {d, 0, 10^7, 2}]

A209208 Values of the difference d for 9 primes in geometric-arithmetic progression with the minimal sequence {1111^j + jd}, j = 0 to 8.

Original entry on oeis.org

903030, 1004250, 3760290, 7296450, 7763520, 17988210, 28962390, 29956950, 33316320, 37265160, 39013800, 39768150, 43920480, 50110620, 54651480, 56388810, 74306610, 74679810, 75911850, 89115210, 92619690, 98518800, 108718080, 116535300, 116958450, 117671820

Offset: 1

Sameen Ahmed Khan, Mar 06 2012

nonn

A geometric-arithmetic progression of primes is a set of k primes (denoted by GAP-k) of the form p r^j + j d for fixed p, r and d and consecutive j. Symbolically, for r = 1, this sequence simplifies to the familiar primes in arithmetic progression (denoted by AP-k). The computations were done without any assumptions on the form of d. Primality requires d to be multiple of 5# = 30 and coprime to 11.

			d = 1004250 then {11*11^j + j*d}, j = 0 to 8, is {11, 1004371, 2009831, 3027391, 4178051, 6792811, 25512671, 221388631, 2365981691}, which is 9 primes in geometric-arithmetic progression.

Sameen Ahmed Khan, Table of n, a(n) for n = 1..539
Sameen Ahmed Khan, Primes in Geometric-Arithmetic Progression, arXiv:1203.2083v1 [math.NT], (Mar 09 2012).

Cf. A172367, A209202, A209203, A209204, A209205, A209206, A209207, A209209, A209210.

p = 11; gapset9d = {}; Do[If[PrimeQ[{p, p*p + d, p*p^2 + 2*d, p*p^3 + 3*d, p*p^4 + 4*d, p*p^5 + 5*d, p*p^6 + 6*d, p*p^7 + 7*d, p*p^8 + 8*d}] == {True, True, True, True, True, True, True, True, True}, AppendTo[gapset9d, d]], {d, 0, 10^8, 2}]

A209209 Values of the difference d for 10 primes in geometric-arithmetic progression with the minimal sequence {1111^j + jd}, j = 0 to 9.

Original entry on oeis.org

903030, 17988210, 28962390, 39768150, 74306610, 89115210, 116535300, 173227980, 186013380, 237952050, 359613030, 386317920, 392253990, 443687580, 499153200, 548024610, 591655080, 652133160, 665780640, 705583830, 758828310, 910046550, 920546160, 921847290

Offset: 1

Sameen Ahmed Khan, Mar 06 2012

nonn

A geometric-arithmetic progression of primes is a set of k primes (denoted by GAP-k) of the form p r^j + j d for fixed p, r and d and consecutive j. Symbolically, for r = 1, this sequence simplifies to the familiar primes in arithmetic progression (denoted by AP-k). The computations were done without any assumptions on the form of d. Primality requires d to be multiple of 5# = 30 and coprime to 11.

			d = 17988210 then {11*11^j + j*d}, j = 0 to 9, is {11, 17988331, 35977751, 53979271, 72113891, 91712611, 127416431, 340276351, 2501853371, 26099318491}, which is 10 primes in geometric-arithmetic progression.

Sameen Ahmed Khan, Table of n, a(n) for n = 1..96
Sameen Ahmed Khan, Primes in Geometric-Arithmetic Progression, arXiv:1203.2083v1 [math.NT], (Mar 09 2012).

Cf. A172367, A209202, A209203, A209204, A209205, A209206, A209207, A209208, A209210.

p = 11; gapset10d = {}; Do[If[PrimeQ[{p, p*p + d, p*p^2 + 2*d, p*p^3 + 3*d, p*p^4 + 4*d, p*p^5 + 5*d, p*p^6 + 6*d, p*p^7 + 7*d, p*p^8 + 8*d, p*p^9 + 9*d}] == {True, True, True, True, True, True, True, True, True, True}, AppendTo[gapset10d, d]], {d, 0, 10^8, 2}]

A209210 Values of the difference d for 11 primes in geometric-arithmetic progression with the minimal sequence {1111^j + jd}, j = 0 to 10.

Original entry on oeis.org

443687580, 591655080, 1313813550, 2868131100, 3525848580, 3598823970, 4453413120, 6075076800, 6644124480, 7429693770, 9399746580, 11801410530, 12450590250

Offset: 1

Sameen Ahmed Khan, Mar 06 2012

nonn

A geometric-arithmetic progression of primes is a set of k primes (denoted by GAP-k) of the form p r^j + j d for fixed p, r and d and consecutive j. Symbolically, for r = 1, this sequence simplifies to the familiar primes in arithmetic progression (denoted by AP-k). The computations were done without any assumptions on the form of d. Primality requires d to be multiple of 5# = 30 and coprime to 11.

			d = 591655080 then {11*11^j + j*d}, j = 0 to 10, is {11, 591655201, 1183311491, 1774979881, 2366781371, 2960046961, 3569417651, 4355944441, 7091188331, 31262320321, 291228221411}, which is 11 primes in geometric-arithmetic progression.

Sameen Ahmed Khan, Primes in Geometric-Arithmetic Progression, arXiv:1203.2083v1 [math.NT], (Mar 09 2012).

Cf. A172367, A209202, A209203, A209204, A209205, A209206, A209207, A209208, A209209.

p = 11; gapset11d = {}; Do[If[PrimeQ[{p, p*p + d, p*p^2 + 2*d, p*p^3 + 3*d, p*p^4 + 4*d, p*p^5 + 5*d, p*p^6 + 6*d, p*p^7 + 7*d, p*p^8 + 8*d, p*p^9 + 9*d, p*p^10 + 10*d}] == {True, True, True, True, True, True, True, True, True, True, True}, AppendTo[gapset11d, d]], {d, 0, 10^8, 2}]

A246519 Primes p such that 4+p, 4+p^2, 4+p^3 and 4+p^5 are all prime.

Original entry on oeis.org

7, 5503, 21013, 301123, 303613, 420037, 469363, 679153, 771427, 991957, 999667, 1524763, 1707367, 2030653, 2333083, 2540563, 2552713, 2710933, 3009967, 3378103, 3441817, 3592213, 4419937, 4704613, 4840723, 5177797, 5691547, 6227587, 6275887, 6395677, 6595597, 6597163

Offset: 1

Zak Seidov, Aug 28 2014

nonn

For even k > 2, 4 + n^k is prime only for n = 1.
From Derek Orr, Aug 28 2014 (edited by Danny Rorabaugh, Apr 19 2015): (Start)
4+p^4 is composite for all primes p. For p = 2, 4+p^4 = 20 is composite. To prove it for odd primes, consider S(n) = 4+(2*n+1)^4. S(n) == 0 (mod 5) unless n == 2 (mod 5). If n == 2 (mod 5), then 2*n+1 == 0 (mod 5), which is only prime for n = 2; this gives p = 5 and 4+5^4 = 629 is composite. For other odd primes p, 4+p^4 is greater than 5 and divisible by 5.
4+p^(4*m) is also composite for any prime p and integer m > 0. For each m, the proof is the same as above.
(End)
All terms are == {3,7} (mod 10). - Zak Seidov, Aug 29 2014

			From _K. D. Bajpai_, Jan 20 2015: (Start)
a(2) = 5503:
4 + 5503 = 5507;
4 + 5503^2 = 30283013;
4 + 5503^3 = 166647398531;
4 + 5503^5 = 5046584669419727747;
all five are prime.
(End)

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A007591, A073573, A125260, A172367.
Primes p such that 4+p^7, 4+p^9 and 4+p^11 are also prime is A253937. - K. D. Bajpai, Jan 20 2015
The subsequence with 4+p^7 also prime is A246562. - Danny Rorabaugh, Apr 19 2015

[p: p in PrimesUpTo(2*10^7) | IsPrime(4+p) and IsPrime(4+p^2) and IsPrime(4+p^3) and IsPrime(4+p^5)]; // Vincenzo Librandi, Apr 19 2015

k=4; Select[Prime[Range[1,500000]], PrimeQ[k+#]&&PrimeQ[k+#^2] &&PrimeQ[k+#^3] &&PrimeQ[k+#^5]&]  (*K. D. Bajpai, Jan 20 2015 *)

for(n=1, 6000000, if(isprime(n) && isprime(4+n) && isprime(4+n^2) && isprime(4+n^3) && isprime(4+n^5), print1(n, ", "))) \\ Colin Barker, Aug 28 2014

p=7; forprime(q=11, 1e8, if(q-p==4 && isprime(4+p^2) && isprime(4+p^3) && isprime(4+p^5), print1(p, ", ")); p=q) \\ Charles R Greathouse IV, Aug 28 2014

from sympy import prime, isprime
A246519_list = [p for p in (prime(n) for n in range(1,10**5)) if all([isprime(4+p**z) for z in (1,2,3,5)])]
# Chai Wah Wu, Sep 08 2014

cp's OEIS Frontend

A209202 Values of the difference d for 3 primes in geometric-arithmetic progression with the minimal sequence {3*3^j + j*d}, j = 0 to 2.

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A209203 Values of the difference d for 4 primes in geometric-arithmetic progression with the minimal sequence {5*5^j + j*d}, j = 0 to 3.

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A209204 Values of the difference d for 5 primes in geometric-arithmetic progression with the minimal sequence {5*5^j + j*d}, j = 0 to 4.

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A209205 Values of the difference d for 6 primes in geometric-arithmetic progression with the minimal sequence {7*7^j + j*d}, j = 0 to 5.

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A209206 Values of the difference d for 7 primes in geometric-arithmetic progression with the minimal sequence {7*7^j + j*d}, j = 0 to 6.

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A209207 Values of the difference d for 8 primes in geometric-arithmetic progression with the minimal sequence {11*11^j + j*d}, j = 0 to 7.

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A209208 Values of the difference d for 9 primes in geometric-arithmetic progression with the minimal sequence {11*11^j + j*d}, j = 0 to 8.

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A209209 Values of the difference d for 10 primes in geometric-arithmetic progression with the minimal sequence {11*11^j + j*d}, j = 0 to 9.

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A209202 Values of the difference d for 3 primes in geometric-arithmetic progression with the minimal sequence {33^j + jd}, j = 0 to 2.

A209203 Values of the difference d for 4 primes in geometric-arithmetic progression with the minimal sequence {55^j + jd}, j = 0 to 3.

A209204 Values of the difference d for 5 primes in geometric-arithmetic progression with the minimal sequence {55^j + jd}, j = 0 to 4.

A209205 Values of the difference d for 6 primes in geometric-arithmetic progression with the minimal sequence {77^j + jd}, j = 0 to 5.

A209206 Values of the difference d for 7 primes in geometric-arithmetic progression with the minimal sequence {77^j + jd}, j = 0 to 6.

A209207 Values of the difference d for 8 primes in geometric-arithmetic progression with the minimal sequence {1111^j + jd}, j = 0 to 7.

A209208 Values of the difference d for 9 primes in geometric-arithmetic progression with the minimal sequence {1111^j + jd}, j = 0 to 8.

A209209 Values of the difference d for 10 primes in geometric-arithmetic progression with the minimal sequence {1111^j + jd}, j = 0 to 9.

A209210 Values of the difference d for 11 primes in geometric-arithmetic progression with the minimal sequence {1111^j + jd}, j = 0 to 10.