A242012 a(n) is the number of positive integers k <= n for which gpf(k^2 + 1) = gpf(n^2 + 1), where gpf is the greatest prime divisor.

1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 1, 1, 2, 1, 1, 1, 2, 3, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 1, 4, 1, 1, 3, 1, 3, 1, 1, 2, 5, 1, 1, 2, 1, 1, 1, 1, 2, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 4, 1, 3, 3, 1, 2, 2, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1

Offset

1,3

Comments

a(n) = 1 if n is a term in A005574 (numbers n such that n^2 + 1 is prime).

a(n) = 1 if gpf(k^2 + 1) <> gpf(n^2 + 1) for every positive integer k < n.

Links

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

Example

a(3) = 2 because the greatest prime divisor of 3^2 + 1 is 5 and n=3 is the 2nd positive value of n at which gpf(n^2 + 1) = 5; the 1st is n=2: gpf(2^2 + 1) = 5.
a(313) = 7 because the greatest prime divisor of 313^2 + 1 is 101, and n=313 is the 7th positive value of n at which this occurs:
   10^2 + 1 = 101;
   91^2 + 1 = 2 * 41 * 101;
  111^2 + 1 = 2 * 61 * 101;
  192^2 + 1 = 5 * 73 * 101;
  212^2 + 1 = 5 * 89 * 101;
  293^2 + 1 = 2 * 5^2 * 17 * 101;
  313^2 + 1 = 2 * 5 * 97 * 101.

Maple

with(numtheory):nn:=200:T:=array(1..nn):k:=0:
for m from 1 to nn do:
x:=factorset(m^2+1):n1:=nops(x):p:=x[n1]:k:=k+1:T[k]:=p:
od:
  for n from 1 to 150 do:
  q:=T[n]:ii:=0:
    for i from 1 to n do:
      if T[i]=q then ii:=ii+1:
      else
      fi:
    od:
    printf(`%d, `, ii):
  od:
# Simpler version:
N:= 1000:  # to get a(n) for n <= N
T:= Array(1..N):
for n from 1 to N do
T[n]:= max(numtheory:-factorset(n^2+1));
  A[n]:= numboccur(T, T[n]);
od:
seq(A[n], n=1..N); # Robert Israel, Aug 12 2014

Prog

(PARI) a(n) = my(gn = vecmax(factor(n^2+1)[, 1])); sum(k=1, n, vecmax(factor(k^2+1)[, 1]) == gn); \\ Michel Marcus, Sep 10 2017

Crossrefs

Cf. A002496, A005574, A014442.

Sequence in context: A072776 A077481 A278113 * A359165 A086290 A136568

Adjacent sequences: A242009 A242010 A242011 * A242013 A242014 A242015

Keyword

nonn

Author

Michel Lagneau, Aug 11 2014

Extensions

Edited by Jon E. Schoenfield, Sep 10 2017

Status

approved