

A218564


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


3



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
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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..(Nt)/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



