A193279 Number of distinct sums of distinct proper divisors of n.

0, 1, 1, 3, 1, 6, 1, 7, 3, 7, 1, 16, 1, 7, 7, 15, 1, 21, 1, 22, 7, 7, 1, 36, 3, 7, 7, 28, 1, 42, 1, 31, 7, 7, 7, 55, 1, 7, 7, 50, 1, 54, 1, 31, 27, 7, 1, 76, 3, 31, 7, 31, 1, 66, 7, 64, 7, 7, 1, 108, 1, 7, 29, 63, 7, 78, 1, 31, 7, 72, 1, 123, 1, 7, 31, 31

Offset

1,4

Comments

a(n)=1 if and only if n is prime.

a(n)=n-1 if n is a power of 2.

a(n)=n if n is an even perfect number (is the converse true?)

Note: the count excludes an empty subset of proper divisors that would give 0 as a sum. - Antti Karttunen, Mar 07 2018

Links

Antti Karttunen, Table of n, a(n) for n = 1..20000 (first 10000 terms from Amiram Eldar)

Maple

with(linalg): a:=proc(n) local dl, t: dl:=convert(numtheory[divisors](n) minus {n}, list): t:=nops(dl): return nops({seq(innerprod(dl, convert(2^t+i, base, 2)[1..t]), i=1..2^t-1)}): end: seq(a(n), n=1..76); # Nathaniel Johnston, Jul 23 2011

Mathematica

a[n_] := Module[{d = Most @ Divisors[n], x}, Count[CoefficientList[Product[1 + x^i, {i, d}], x], _?(# > 0 &)] - 1]; Array[a, 100] (* Amiram Eldar, Jun 13 2020 *)

Prog

(PARI)
\\ Slow and naive:
A193279(n) = if(1==n, 0, my(pds = (divisors(n)[1..(numdiv(n)-1)]), maxsum = vecsum(pds), sums = vector(maxsum), psetsiz = (2^length(pds))-1, k = 0, s); for(i=1, psetsiz, s = vecsum(choosebybits(pds, i)); if(!sums[s], k++; sums[s]++)); (k)); \\ Antti Karttunen, Mar 07 2018
(PARI) A193279(n) = { my(p=1); fordiv(n, d, if(d<n, p *= (1 + 'x^d))); sum(i=1, poldegree(p), (0<polcoeff(p, i))); }; \\ Antti Karttunen, Nov 29 2024
(PARI) A193279(n) = { my(c=[0]); fordiv(n, d, if(d<n, c = Set(concat(c, vector(#c, i, c[i]+d))))); (#c)-1; }; \\ (see A119347) - Antti Karttunen, Nov 29 2024

Crossrefs

Cf. A193280.

Cf. A119347 (allows also n to be included in the sums), A378447 (differences).

Sequence in context: A066446 A069625 A111614 * A378547 A378549 A076889

Adjacent sequences: A193276 A193277 A193278 * A193280 A193281 A193282

Keyword

nonn

Author

Michael Engling, Jul 20 2011

Status

approved