A261345 Number of distinct prime divisors among the numbers k^2 + 1 for k in 1 <= k <= n.

1, 2, 2, 3, 4, 5, 5, 5, 6, 7, 8, 9, 9, 10, 11, 12, 12, 12, 13, 14, 14, 15, 16, 17, 18, 19, 20, 21, 22, 22, 22, 22, 23, 24, 25, 26, 27, 27, 28, 29, 29, 30, 30, 31, 32, 32, 32, 33, 34, 34, 35, 36, 37, 38, 38, 39, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 49, 50

Offset: 1

Author

Michel Lagneau, Aug 15 2015

Keywords

• nonn

Comments

Conjecture: n/a(n) <= 1.6.

Størmer-number-counting function: a(n) is the number of terms in A005528 less than or equal to n. - Luc Rousseau, Jun 13 2018

Examples

			For a(5), there are 4 distinct prime divisors that occur in the values 1^2+1 = 2, 2^2+1 = 5, 3^2+1 = 2*5, 4^2+1 = 17, 5^2+1 = 26 = 2*13. Taken together, the distinct prime factors are {2,5,13,17}.
		

Links

• Michel Lagneau, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A002144, A002496, A002522, A089122, A005528.

Programs

Maple

with(numtheory):nn:=100:lst:={}:
for n from 1 to nn do:
  p:=n^2+1:x:=factorset(p):n0:=nops(x):
  A:={op(x),x[n0]}:
  lst:=lst union A :n1:=nops(lst):printf(`%d, `,n1):
od:

Mathematica

Array[Length@ Tally@ First@ Transpose@ Flatten[FactorInteger[#^2 + 1] & /@ Range@ #, 1] &, {69}] (* Michael De Vlieger, Aug 18 2015 *)
Module[{nn=70,fi},fi=Table[FactorInteger[n^2+1][[All,1]],{n,nn}];Table[ Length[ Union[Flatten[Take[fi,m]]]],{m,nn}]] (* Harvey P. Dale, Sep 11 2021 *)

PARI

lista(nn) = {v = []; for (n=1, nn, v = Set(concat(v, factor(n^2+1)[,1]~)); print1(#v, ", "););} \\ Michel Marcus, Aug 16 2015