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

Author

Michael Engling, Jul 20 2011

Keywords

• nonn

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)

Crossrefs

Cf. A193280.

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

Programs

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 *)

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(dAntti Karttunen, Nov 29 2024

PARI

A193279(n) = { my(c=[0]); fordiv(n,d, if(dA119347) - Antti Karttunen, Nov 29 2024