

A292374


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


3



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
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OFFSET

1,13


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


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).


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 *)


PROG

(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 memoizationmacro definec)
(definec (A292374 n) (cond ((even? n) 0) ((= 1 n) 1) (else (+ (if (= 1 (modulo n 4)) 1 0) (A292374 (A064989 n))))))


CROSSREFS

Cf. A016813, A055396, A064989, A246271, A246278, A292375.
Cf. also A038802 (odd bisection of a(n) + A292376(n)).
Sequence in context: A284687 A046268 A202425 * A292376 A257685 A327172
Adjacent sequences: A292371 A292372 A292373 * A292375 A292376 A292377


KEYWORD

nonn


AUTHOR

Antti Karttunen, Sep 17 2017


STATUS

approved



