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A250028 a(n) is the number of positive integers k <= n such that lpf(k^2 + 1) = lpf(n^2 + 1), where lpf() is the least prime factor function. 1
1, 1, 2, 1, 3, 1, 4, 2, 5, 1, 6, 3, 7, 1, 8, 1, 9, 4, 10, 1, 11, 5, 12, 1, 13, 1, 14, 6, 15, 2, 16, 7, 17, 1, 18, 1, 19, 8, 20, 1, 21, 9, 22, 2, 23, 1, 24, 10, 25, 1, 26, 11, 27, 1, 28, 1, 29, 12, 30, 3, 31, 13, 32, 3, 33, 1, 34, 14, 35, 4, 36, 15, 37, 1, 38, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

The least prime factor of n^2 + 1 is A089120(n).

a(2j+1) = j+1 because 2 is the least prime factor of the even numbers.

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

a(n) = 1 if lpf(n^2 + 1) appears for the first time (example: a(50) = 1 because lpf(50^2 + 1) = lpf(41*61) = 41.

Property: if p = lpf(n^2 + 1), then p divides (n-p)^2 + 1.

LINKS

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

EXAMPLE

a(3) = 2 because the least prime factor of 3^2 + 1 is 2 and 2 is the 2nd positive integer k for which lpf(k^2 + 1) is 2 (the 1st occurrence is 1^2 + 1 = 2).

a(12) = 3 because the least prime factor of 12^2 + 1 = 5*29 is 5, and 5 is the 3rd occurrence (the 1st and 2nd are 2^2 + 1 = 5 and 8^2 + 1 = 5*13, respectively).

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[1]: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:

MATHEMATICA

With[{s = Array[FactorInteger[#^2 + 1][[1, 1]] &, {76}]}, Reap[Do[Sow@ Count[Take[s, i], k_ /; k == FactorInteger[i^2 + 1][[1, 1]]], {i, Length@ s}]][[-1, 1]]] (* Michael De Vlieger, Sep 12 2017 *)

PROG

(PARI) a(n) = my(gn = vecmin(factor(n^2+1)[, 1])); sum(k=1, n, vecmin(factor(k^2+1)[, 1]) == gn); \\ Michel Marcus, Sep 11 2017

CROSSREFS

Cf. A002496, A005574, A089120, A242012.

Sequence in context: A260739 A130747 A055440 * A101279 A064576 A322390

Adjacent sequences:  A250025 A250026 A250027 * A250029 A250030 A250031

KEYWORD

nonn

AUTHOR

Michel Lagneau, Nov 11 2014

EXTENSIONS

Edited by Jon E. Schoenfield, Sep 11 2017

STATUS

approved

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Last modified May 15 21:44 EDT 2021. Contains 343929 sequences. (Running on oeis4.)