A070303 -id:A070303 - cp's OEIS frontend

A217276 Numbers n such that no prime divisors of n^2 + 1 are of the form a^2 + 1.

34, 44, 46, 50, 60, 70, 76, 86, 96, 100, 104, 114, 136, 144, 164, 186, 190, 194, 196, 214, 220, 226, 244, 246, 254, 266, 274, 286, 294, 296, 304, 316, 320, 324, 330, 334, 346, 354, 356, 360, 366, 374, 390, 410, 416, 424, 426, 434, 454, 456, 460, 476, 484, 486

Offset: 1

Michel Lagneau, Sep 29 2012

nonn

			34 is in the sequence because 34^2+1 = 1157 = 13*89 and the prime divisors 13, 89 are not of the form a^2+1.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A002522, A005574, A070303, A181436, A180252, A181436.

with(numtheory):for n from 1 to 100 do: x:=factorset(n^2+1):n1:=nops(x):ii:=0:for m from 1 to n1 do:y:=sqrt(x[m]-1):if y=floor(y) then ii:=1:else fi:od:if ii=0 then printf(`%d, `,n):else fi:od:

fQ[n_] := Module[{lst = Transpose[FactorInteger[n^2 + 1]][[1]]}, Length[lst] > 1 && And @@ (Not /@ IntegerQ /@ Sqrt[lst - 1])]; Select[Range[500], fQ] (* T. D. Noe, Oct 01 2012 *)

A368544 The number of divisors of n whose prime factors are all of the form k^2+1.

Original entry on oeis.org

1, 2, 1, 3, 2, 2, 1, 4, 1, 4, 1, 3, 1, 2, 2, 5, 2, 2, 1, 6, 1, 2, 1, 4, 3, 2, 1, 3, 1, 4, 1, 6, 1, 4, 2, 3, 2, 2, 1, 8, 1, 2, 1, 3, 2, 2, 1, 5, 1, 6, 2, 3, 1, 2, 2, 4, 1, 2, 1, 6, 1, 2, 1, 7, 2, 2, 1, 6, 1, 4, 1, 4, 1, 4, 3, 3, 1, 2, 1, 10, 1, 2, 1, 3, 4, 2, 1

Offset: 1

Amiram Eldar, Dec 29 2023

The number of terms of A180252 that divide n.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A002496, A005574, A070303, A180252.

q[n_] := AllTrue[FactorInteger[n][[;; , 1]], IntegerQ[Sqrt[# - 1]] &]; f[p_, e_] := If[q[p], e + 1, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f=factor(n)); prod(i=1, #f~, if(issquare(f[i,1]-1), f[i,2] + 1, 1))};

from math import prod
from sympy import factorint
from sympy.ntheory.primetest import is_square
def A368544(n): return prod(e+1 for p, e in factorint(n).items() if is_square(p-1)) # Chai Wah Wu, Dec 30 2023

Multiplicative with a(p^e) = e+1 if p is of the form k^2+1, and 1 otherwise.
a(n) >= 1, with equality if and only if all the prime factors of n are in A070303.
a(n) <= A000005(n), with equality if and only if n is in A180252.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{k in A005574} (1 + 1/k^2) = 2.80986546... .

cp's OEIS Frontend

A217276 Numbers n such that no prime divisors of n^2 + 1 are of the form a^2 + 1.

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A368544 The number of divisors of n whose prime factors are all of the form k^2+1.

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