A337690 a(n) is the number of primitive nondeficient numbers (A006039) dividing n.

0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 2, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 2

Offset: 1

Author

Antti Karttunen and Peter Munn, Sep 15 2020

Keywords

• nonn

Comments

As a simple consequence of the definition of a primitive nondeficient number, a(n) is nonzero if and only if n is nondeficient.

Examples

			The least nondeficient number, therefore the least primitive nondeficient number is 6. So a(1) = a(2) = a(3) = a(4) = a(5) = 0 as all primitive nondeficient numbers are larger, and therefore not divisors; and a(6) = 1, as only 1 primitive nondeficient number divides 6, namely 6 itself.
60 has the following 12 divisors: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60. Of these, only 6 and 20 are in A006039, thus a(60) = 2.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537
• Index entries for sequences related to sigma(n).

Crossrefs

A006039 (or equivalently, its characteristic function, A341619) is used to define this sequence.

See A000203 and A023196 for definitions of deficient and nondeficient.

Sequences with similar definitions: A080224, A294927, A337539, A341620.

Cf. A302110, A341604.

Positions of 0's: A005100.

Positions of numbers >= k: A023196 (k=1), A337688 (k=2), A337689 (k=3).

Positions of first appearances are given in A337691.

Differs from its derived sequence A341618 for the first time at n=120, where a(120)=2, while A341618(120)=1.

Programs

PARI

A341619(n) = if(sigma(n) < (2*n), 0, fordiv(n, d, if((d= 2*d), return(0))); (1)); \\ After code in A071395
A337690(n) = sumdiv(n, d, A341619(d));

Formula

a(n) = Sum_{d|n} A341619(d) = Sum_{d|n} [1==A341620(d)]. - Corrected by Antti Karttunen, Feb 21 2021

a(A005100(n)) = 0.

a(A006039(n)) = 1.

a(A023196(n)) >= 1.

a(A337479(n)) = A337539(n).

a(n) <= A341620(n). - Antti Karttunen, Feb 22 2021

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{n>=1} 1/A006039(n) = 0.3... (see A006039 for a better estimate of this constant). - Amiram Eldar, Jan 01 2024

Extensions

Data section extended to 120 terms by Antti Karttunen, Feb 21 2021