A180252 -id:A180252 - cp's OEIS frontend

A181436 Numbers k such that the prime divisors of k^2 + 1 are of the form q^2 + 1.

1, 2, 3, 4, 6, 7, 10, 13, 14, 16, 20, 24, 26, 36, 38, 40, 43, 54, 56, 66, 68, 74, 84, 90, 94, 110, 116, 117, 120, 124, 126, 130, 134, 146, 150, 156, 160, 170, 176, 180, 183, 184, 204, 206, 210, 224, 230, 236, 240, 250, 256, 260, 264, 270, 280, 284, 293, 300, 306, 314, 326, 327

Offset: 1

Michel Lagneau, Jan 29 2011

nonn

			183 is in the sequence because 183^2 + 1 = 2*5*17*197 and 2 = 1^2 + 1, 5 = 2^2+1, 17 = 4^2+1 and 197 = 14^2 + 1.

Ivan Neretin, Table of n, a(n) for n = 1..1000

Cf. A180252, A002144, A002522, A005574.

with(numtheory):nn:=1000:for n from 1 to nn do: x:=n^2+1:y:=factorset(x):ny:=nops(y):id:=0:for
  q from 1 to ny do: z:=y[q]-1:zz:=sqrt(z):if zz=floor(zz) then id:=id+1:else  fi:od:if id=ny then printf(`%d, `,n):else fi:od:

Select[Range@330, And @@ IntegerQ /@ Sqrt[FactorInteger[#^2 + 1][[All, 1]] - 1] &] (* Ivan Neretin, Aug 31 2016 *)

isok(n) = {fn = factor(n^2+1)[,1]; for (k=1, #fn, if (!issquare(fn[k]-1), return (0));); 1;} \\ Michel Marcus, Sep 01 2016

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

Original entry on oeis.org

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 *)

A216784 a(n) is the number of distinct prime divisors of n^2 + 1 of the form m^2 + 1.

Original entry on oeis.org

1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 0, 1, 1, 2, 2, 1, 1, 1, 1, 3, 0, 1, 0, 3, 1, 1, 0, 1, 1, 2, 1, 2, 1, 2, 1, 1, 0, 1, 1, 2, 1, 1, 1, 2, 2, 1, 0, 1, 2, 2, 1, 1, 0, 2, 1, 1, 1, 2, 1, 2, 1, 1, 0, 2

Offset: 1

Michel Lagneau, Oct 15 2012

nonn

a(m) = 0 for m = A217276(n).

			a(13) = 3 because 13^2+1 = 170 = 2*5*17 with 3 divisors of the form m^2+1 such that 2 = 1^2+1, 5=2^2+1 and 17 = 4^2+1.
a(34) = 0 because 34^2+1 = 1157 = 13*89 and the prime divisors 13, 89 are not of the form m^2+1.

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

Cf. A217276, A181436, A180252.

with(numtheory):for n from 1 to 100 do:x:=n^2+1:y:=factorset(x):n1:=nops(y):i:=0:for k from 1 to n1 do:z:=sqrt(y[k]-1):if z=floor(z) then i:=i+1:else fi:od: printf(`%d, `,i):od:
# second Maple program:
a:= n-> nops(select(x-> issqr(x-1), ifactors(n^2+1)[2][..., 1])):
seq(a(n), n=1..87);  # Alois P. Heinz, Jul 24 2025

a[n_] := Length @ Select[FactorInteger[n^2 + 1][[;;,1]], IntegerQ @ Sqrt[# - 1] &]; Array[a, 100] (* Amiram Eldar, Sep 11 2019 *)

A217279 Numbers of the form n^2 + 1 without prime divisors of the form a^2 + 1.

Original entry on oeis.org

1157, 1937, 2117, 2501, 3601, 4901, 5777, 7397, 9217, 10001, 10817, 12997, 18497, 20737, 26897, 34597, 36101, 37637, 38417, 45797, 48401, 51077, 59537, 60517, 64517, 70757, 75077, 81797, 86437, 87617, 92417, 99857, 102401, 104977, 108901, 111557, 119717

Offset: 1

Michel Lagneau, Sep 29 2012

nonn

The corresponding n are in A217276.

a(n) == 1, 17, 37, 57, 77, 97 mod 100.

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

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

Cf. A002522, A005574, A217276, A181436, A180252.

isA217279 := proc(n)
    if issqr(n-1) then
        for d in numtheory[factorset](n) do
            if issqr(d-1) then
                return false;
            end if;
        end do:
        return true ;
    else
        false;
    end if;
end proc:
for n from 1 to 300 do
    if isA217279(n^2+1) then
        printf("%d ",n^2+1) ;
    end if;
end do: # R. J. Mathar, Oct 01 2012

Select[1 + Range[400]^2, Not[PrimeQ[#]] && Intersection[Divisors[#], 1 + Range[Sqrt[# - 1] - 1]^2] == {} &] (* Alonso del Arte, Sep 29 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

A181436 Numbers k such that the prime divisors of k^2 + 1 are of the form q^2 + 1.

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

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A216784 a(n) is the number of distinct prime divisors of n^2 + 1 of the form m^2 + 1.

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A217279 Numbers of the form n^2 + 1 without prime divisors 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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