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A180507 Numbers k such that k^2 + 1 = p*q, p and q prime with p == q (mod k). 1
3, 8, 12, 144, 1020, 8040, 13860, 34840, 729180, 1728240, 3232060, 17576520, 39279240, 85184880, 117649980, 778689840, 884737920, 1225045140, 1771563420, 3723878100, 3869896140, 4574299320, 7762395960, 12487172640, 14348911860, 14886940920, 21484957560, 24137574780 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

q - p = k with k = 3, 8, 144.

The next terms with q - p = k are F(432) = 85738...5984 and F(570) where F(n) is the n-th Fibonacci number. All such terms are in A001906; the next such term, if one exists, has more than 25000 decimal digits. - Charles R Greathouse IV, Jan 21 2011

LINKS

Jinyuan Wang, Table of n, a(n) for n = 1..10000

EXAMPLE

a(3) = 12 because 12^2 + 1 = 5*29 and 29 - 5 = 2*12;

a(8) = 34840 because 34840^2 + 1 = 4289 * 283009 and 283009 - 4289 = 278720 = 8*34840.

MAPLE

with(numtheory):for k from 1 to 40000 do: x:=k^2+1:y:=factorset(x):yy:=bigomega(x):if

yy=2 and irem(y[2], k) =y[1] then printf(`%d, `, k):else fi:od:

PROG

(PARI) w(m, r) = Vec(x*(1-x)/(1-(m^2+2)*x+x^2) + O(x^r));

isok(s, t) = isprime(s) && isprime(s+t);

lista(nn) = {my(g, k, m=1, r, u=w(1, nn), v=List([])); for(i=2, r=#u, g=k=(u[i]+sqrtint(5*u[i]^2-4))/2; if(isok(u[i], k), listput(v, k))); while(r>2, u=w(m++, r); for(i=2, #u, k=(m*u[i]+sqrtint((m^2+4)*u[i]^2-4))/2; if(k<g, if(isok(u[i], m*k), listput(v, k)), r=i; break))); Set(v); } \\ Jinyuan Wang, Mar 29 2020

CROSSREFS

Subset of A085722.

Cf. A001906, A002496, A005574, A027862, A134406, A134407.

Sequence in context: A032142 A032046 A247396 * A218889 A131213 A320260

Adjacent sequences: A180504 A180505 A180506 * A180508 A180509 A180510

KEYWORD

nonn

AUTHOR

Michel Lagneau, Jan 20 2011

EXTENSIONS

More terms from Charles R Greathouse IV, Jan 24 2011

Missing terms inserted and more terms from Jinyuan Wang, Mar 30 2020

STATUS

approved

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Last modified February 3 03:08 EST 2023. Contains 360024 sequences. (Running on oeis4.)