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

%I A337690 #43 Jan 01 2024 08:00:40
%S A337690 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,
%T A337690 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,
%U A337690 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
%N A337690 a(n) is the number of primitive nondeficient numbers (A006039) dividing n.
%C A337690 As a simple consequence of the definition of a primitive nondeficient number, a(n) is nonzero if and only if n is nondeficient.
%H A337690 Antti Karttunen, <a href="/A337690/b337690.txt">Table of n, a(n) for n = 1..65537</a>
%H A337690 <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>.
%F A337690 a(n) = Sum_{d|n} A341619(d) = Sum_{d|n} [1==A341620(d)]. - Corrected by _Antti Karttunen_, Feb 21 2021
%F A337690 a(A005100(n)) = 0.
%F A337690 a(A006039(n)) = 1.
%F A337690 a(A023196(n)) >= 1.
%F A337690 a(A337479(n)) = A337539(n).
%F A337690 a(n) <= A341620(n). - _Antti Karttunen_, Feb 22 2021
%F A337690 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
%e A337690 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.
%e A337690 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.
%o A337690 (PARI)
%o A337690 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
%o A337690 A337690(n) = sumdiv(n, d, A341619(d));
%Y A337690 A006039 (or equivalently, its characteristic function, A341619) is used to define this sequence.
%Y A337690 See A000203 and A023196 for definitions of deficient and nondeficient.
%Y A337690 Sequences with similar definitions: A080224, A294927, A337539, A341620.
%Y A337690 Cf. A302110, A341604.
%Y A337690 Positions of 0's: A005100.
%Y A337690 Positions of numbers >= k: A023196 (k=1), A337688 (k=2), A337689 (k=3).
%Y A337690 Positions of first appearances are given in A337691.
%Y A337690 Differs from its derived sequence A341618 for the first time at n=120, where a(120)=2, while A341618(120)=1.
%K A337690 nonn
%O A337690 1,60
%A A337690 _Antti Karttunen_ and _Peter Munn_, Sep 15 2020
%E A337690 Data section extended to 120 terms by _Antti Karttunen_, Feb 21 2021