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A174685 Indices of Sophie Germain pentagonal numbers: indices i of pentagonal numbers P(i) = A000326(i) such that 2*P(i)+1 is also a pentagonal number. 2
0, 75, 244, 86359, 281384, 99658019, 324716700, 115005267375, 374722790224, 132715978892539, 432429775201604, 153154124636722439, 499023585859860600, 176739727114798801875, 575872785652503930604, 203957491936353180641119 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Indices j of pentagonal numbers such that 2*PentagonalNumber(j) + 1 = PentagonalNumber(k) for some integer k. This is the pentagonal number analog of A124174, Sophie Germain triangular numbers tr such that 2*tr+1 is also a triangular number. This is to pentagonal numbers A000326 as Sophie Germain primes A005384 are to primes A000040, and as A124174 is to A000217.

LINKS

Ray Chandler, Table of n, a(n) for n = 1..653 (terms to 1000 digits)

Eric W. Weisstein, Pentagonal Number,

Eric W. Weisstein, Sophie Germain prime,

Index entries for linear recurrences with constant coefficients, signature (1, 1154, -1154, -1, 1).

FORMULA

{j such that there exists a k such that 2*(A000326(j))+1 = A000326(k)}. {j such that there exists a k such that 2*(j*(3*j-1)/2)+1 = k*(3*k-1)/2}.

G.f. x^2*(-75-169*x+435*x^2+x^3) / ( (x-1)*(x^2-34*x+1)*(x^2+34*x+1) ) with a(n) = 1/6 -7*A029547(n) +239*A029547(n-1) +35*A029547(n)*(-1)^n/6 +1183*A029547(n-1)*(-1)^(n-1)/6. - R. J. Mathar, Oct 25 2011

a(1)=0, a(2)=75, a(3)=244, a(4)=86359, a(5)=281384, a(n)= a(n-1)+ 1154*a(n-2)-1154*a(n-3)-a(n-4)+a(n-5). - Harvey P. Dale, Jul 17 2014

EXAMPLE

Using P(n) = A000326(n) we have:

a(1) = 0 because 2*P(0)+1 = 2*0+1 = 1 = P(1).

a(2) = 75 because 2*P(75)+1 = 2*8400+1 = 16801 = P(106).

a(3) = 244 because 2*P(244)+1 = 2*89182+1 = 178365 = P(345).

a(4) = 86359 because 2*P(86359)+1 = 2*11186772142+1 = 22373544285 = P(122130). Similarly, the indices for the larger of each pentagonal number pair are a derived sequence (see A260937): 1, 106, 345, 122130, 397937, 140937722, ... .

MATHEMATICA

LinearRecurrence[{1, 1154, -1154, -1, 1}, {0, 75, 244, 86359, 281384}, 30] (* Harvey P. Dale, Jul 17 2014 *)

PROG

(PARI) /* Jack Brennen, who extended this sequence, found that, other than the trivial pair (0, 1), you need to solve: u^2 - 2*v^2 = 23. And take only those values where u and v are both congruent to 5 mod 6. Then you have P((u+1)/6) = 2*P((v+1)/6)+1. The following PARI/GP will give the nontrivial answers: */

{forstep(i=7, 47, 8,

  print(Vec(lift(Mod((x+1), x^2-2)^i*(4*x-3)+(x+1))/6));

  print(Vec(lift(Mod((x+1), x^2-2)^i*(4*x+3)+(x+1))/6)))}

CROSSREFS

Cf. A000326, A005384, A124174, A260937.

Sequence in context: A201916 A098230 A258056 * A158742 A292313 A158765

Adjacent sequences:  A174682 A174683 A174684 * A174686 A174687 A174688

KEYWORD

nonn

AUTHOR

Jonathan Vos Post, Mar 27 2010

STATUS

approved

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Last modified October 17 18:58 EDT 2019. Contains 328127 sequences. (Running on oeis4.)