A068067 - OEIS

A068067

Number of integers m, 0 < m <= n, such that n divides m(m+1)/2.

1, 0, 2, 0, 2, 1, 2, 0, 2, 1, 2, 1, 2, 1, 4, 0, 2, 1, 2, 1, 4, 1, 2, 1, 2, 1, 2, 1, 2, 3, 2, 0, 4, 1, 4, 1, 2, 1, 4, 1, 2, 3, 2, 1, 4, 1, 2, 1, 2, 1, 4, 1, 2, 1, 4, 1, 4, 1, 2, 3, 2, 1, 4, 0, 4, 3, 2, 1, 4, 3, 2, 1, 2, 1, 4, 1, 4, 3, 2, 1, 2, 1, 2, 3, 4, 1, 4, 1, 2, 3, 4, 1, 4, 1, 4, 1, 2, 1, 4, 1, 2, 3, 2, 1, 8

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OFFSET

1,3

COMMENTS

Least n with a(n) = 2^k is prime(k+1)#/2 = A002110(A000040(k+1))/2. Least n with a(n) = 2^k-1 != 1 is p(k+1)#.

LINKS

G. C. Greubel, Table of n, a(n) for n = 1..10000

FORMULA

a(n) = 0 iff n = 2^k with k >= 1.

If n is even, a(n) = 2^(omega(n)-1) - 1; if n is odd, a(n) = 2^omega(n). Here omega(n) = A001221(n) is the number of distinct prime divisors of n.

A068068(n) - a(n) = 0 if n is odd, 1 if n is even.

MATHEMATICA