A292374 a(1) = 1, a(2n) = 0, and for odd numbers n > 1, a(n) = a(A064989(n)) + [n == 1 (mod 4)].

1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 2, 0, 0, 0, 3, 0, 3, 0, 1, 0, 3, 0, 2, 0, 0, 0, 4, 0, 4, 0, 1, 0, 0, 0, 5, 0, 0, 0, 6, 0, 6, 0, 1, 0, 6, 0, 3, 0, 0, 0, 7, 0, 1, 0, 1, 0, 7, 0, 8, 0, 0, 0, 2, 0, 8, 0, 1, 0, 8, 0, 9, 0, 0, 0, 1, 0, 9, 0, 1, 0, 9, 0, 1, 0, 0, 0, 10, 0, 1, 0, 1, 0, 0, 0, 11, 0, 0, 0, 12, 0, 12, 0, 1

Offset: 1

Author

Antti Karttunen, Sep 17 2017

Keywords

• nonn

Comments

For odd numbers > 1, iterate the map x -> A064989(x), which shifts every prime in the prime factorization of n one index step towards smaller primes. a(n) counts the numbers of the form 4k+1 encountered until the first number which is even has been reached. This count includes also n itself if it is of the form 4k+1 (A016813), thus a(1) = 1.

In other words, locate the position where n is in square array A246278 and moving up by that column, count all numbers of the form 4k+1 until an even number at the top of the column is reached.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..16384
• Index entries for sequences computed from indices in prime factorization

Crossrefs

Cf. A016813, A055396, A064989, A246271, A246278, A292375.

Cf. also A038802 (odd bisection of a(n) + A292376(n)).

Programs

Mathematica

a[1] = 1; a[n_] := a[n] = If[EvenQ@ n, 0, a[Times @@ Power[Which[# == 1, 1, # == 2, 1, True, NextPrime[#, -1]] & /@ First@ #, Last@ #] &@ Transpose@ FactorInteger@ n] + Boole[Mod[n, 4] == 1]]; Array[a, 105] (* Michael De Vlieger, Sep 17 2017 *)

PARI

A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A292374(n) = if(1==n,n,if(!(n%2),0,(if(1==(n%4),1,0)+A292374(A064989(n)))));

Scheme

;; With memoization-macro definec.
(definec (A292374 n) (cond ((even? n) 0) ((= 1 n) 1) (else (+ (if (= 1 (modulo n 4)) 1 0) (A292374 (A064989 n))))))

Formula

a(1) = 1, a(2n) = 0, and for odd numbers n > 1, a(n) = a(A064989(n)) + [n == 1 (mod 4)].

a(n) <= A292375(n).

For n >= 2, a(n) + A292376(n) + 1 = A055396(n).