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

1,60

Comments

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

Links

Antti Karttunen, Table of n, a(n) for n = 1..65537

Index entries for sequences related to sigma(n).

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

Example

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.

Prog

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

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.

Sequence in context: A341353 A010105 A341618 * A083916 A083893 A187097

Adjacent sequences: A337687 A337688 A337689 * A337691 A337692 A337693

Keyword

nonn

Author

Antti Karttunen and Peter Munn, Sep 15 2020

Extensions

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

Status

approved