A181436 -id:A181436 - 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 *)

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 *)

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

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Author

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Examples

Links

Crossrefs

Programs

Maple

Mathematica

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica