A218564 Numbers n such that n^2 + 1 is divisible by a 5th power.

1068, 2057, 4193, 5182, 7318, 8307, 10443, 11432, 13568, 14557, 16693, 17682, 19818, 20807, 22943, 23932, 26068, 27057, 29193, 30182, 32318, 33307, 35443, 36432, 38568, 39557, 41693, 42682, 44818, 45807, 47943, 48932, 51068, 52057, 54193, 55182, 57318, 58307

Offset

1,1

Comments

For each prime p == 1 (mod 4), there are two values of x (mod p^5) that solve x^2 + 1 == 0 (mod p^5), and then x + k*p^5 is in the sequence for every k. Thus the asymptotic density of this sequence should be 1 - Product_p (1 - 2/p^5), where the product is over all primes p == 1 (mod 4). - Robert Israel, Sep 04 2018

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Example

1068 is in the sequence because 1068^2+1 = 1140625 = 5^6*73;
143044 is in the sequence because 143044^2+1 = 20461585937 = 13^5*55109;
390112 is in the sequence because 390112^2+1 = 152187372545 = 5*13*17^6*97.

Maple

N:= 10^5: # to get all terms <= N
P:= select(isprime, [seq(i, i=5..floor((N^2+1)^(1/5)), 4)]):
g:= proc(x, r, N) local t; t:= rhs(op(x)); seq(t+r*k, k=0..(N-t)/r) end proc:
R:= `union`(seq(map(g, {msolve(n^2+1, p^5)}, p^5, N), p=P)):
sort(convert(R, list)); # Robert Israel, Sep 04 2018

Mathematica

Select[Range[2, 20000], Max[Transpose[FactorInteger[#^2+1]][[2]]]>4&]

Prog

(PARI) isok(n) = vecmax(factor(n^2+1)[, 2]) >= 5; \\ Michel Marcus, Sep 04 2018

Crossrefs

Cf. A002522, A049532, A034939, A218562, A218563.

Sequence in context: A123211 A023078 A056102 * A234880 A218565 A145298

Adjacent sequences: A218561 A218562 A218563 * A218565 A218566 A218567

Keyword

nonn

Author

Michel Lagneau, Nov 02 2012

Status

approved