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.

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

1,1

Comments

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.

Links

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).

Example

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.

Mathematica

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}]

Crossrefs

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

Sequence in context: A154547 A357494 A209208 * A104847 A251520 A250542

Adjacent sequences: A209206 A209207 A209208 * A209210 A209211 A209212

Keyword

nonn

Author

Sameen Ahmed Khan, Mar 06 2012

Status

approved