A119347 -id:A119347 - cp's OEIS frontend

A199919 Number of distinct sums of distinct divisors of n when positive and negative divisors are allowed.

3, 7, 9, 15, 9, 25, 9, 31, 27, 37, 9, 57, 9, 49, 49, 63, 9, 79, 9, 85, 65, 49, 9, 121, 27, 49, 81, 113, 9, 145, 9, 127, 81, 49, 69, 183, 9, 49, 81, 181, 9, 193, 9, 169, 157, 49, 9, 249, 27, 187, 81, 197, 9, 241, 69, 241, 81, 49, 9, 337, 9, 49, 209, 255, 81, 289

Offset: 1

Michel Marcus, Dec 22 2012

			a(2)=7 because the signed divisors of 2 are -2, -1, 1 and 2 and their all possible sums are -1, -2, -3, 0, 1, 2, 3.
a(3)=9 because the signed divisors of 3 are -3, -1, 1 and 3 and their all possible sums are -1, -2, -3, -4, 0, 1, 2, 3, 4.

David A. Corneth, Table of n, a(n) for n = 1..10000
Bernard Jacobson, Sums of distinct divisors and sums of distinct units, Proc. Amer. Math. Soc. 15 (1964), 179-183
David A. Corneth, PARI program

Cf. A005153, A119347.

dsdd[n_]:=Module[{divs=Divisors[n]},Length[Union[Total/@Subsets[ Join[ divs,-divs],2Length[divs]]]]]; Array[dsdd,70] (* Harvey P. Dale, Jan 19 2015 *)

A199919(n) = { my(ds=concat(apply(x -> -x,divisors(n)),divisors(n)),m=Map(),s,u=0); for(i=0,(2^#ds)-1,s = sumbybits(ds,i); if(!mapisdefined(m,s), mapput(m,s,s); u++)); (u); }; \\ Slow!
sumbybits(v,b) = { my(s=0,i=1); while(b>0,s += (b%2)*v[i]; i++; b >>= 1); (s); }; \\ Antti Karttunen, May 19 2021

See PARI-link \\ David A. Corneth, May 20 2021

a(A005153(n)) = 2*sigma(A005153(n)) + 1. - David A. Corneth, May 19 2021

a(p) = 9 for odd primes p. - Antti Karttunen, May 19 2021

A342401 Numbers k such that the number of distinct sums of distinct unitary divisors of k sets a new record.

Original entry on oeis.org

1, 2, 6, 10, 30, 42, 60, 66, 78, 90, 110, 130, 170, 190, 210, 330, 390, 462, 510, 546, 570, 690, 798, 858, 870, 930, 1050, 1110, 1218, 1230, 1290, 1410, 1470, 1554, 1590, 1722, 1770, 1830, 1974, 2010, 2130, 2190, 2310, 2730, 3570, 3990, 4290, 4830, 5610, 6006

Offset: 1

Amiram Eldar, Mar 10 2021

nonn

The corresponding record values are 1, 3, 12, 15, 72, 96, 108, 144, 168, 172, 183, 207, 231, 255, 576, 864, ... (see the link for more values).

The analogous sequence of records with all the divisors (A119347) is conjecturally the highly abundant numbers (A002093), excluding 3 and 10.

			The first 6 terms of A342400 are 1, 3, 3, 3, 3 and 12. The record values, 1, 3 and 12, occur are 1, 2 and 6, the first 3 terms of this sequence.

Amiram Eldar, Table of n, a(n) for n = 1..99
Amiram Eldar, Table of n, a(n), A342400(a(n)) for n = 1..99

Cf. A002093, A119347, A342400.

f[n_] := Module[{d = Select[Divisors[n], CoprimeQ[#, n/#] &], x, s, m, c}, m = Length[d]; s = Plus @@ d; c = Rest @ CoefficientList[Series[Product[1 + x^d[[i]], {i, 1, m}], {x, 0, s}], x]; Count[c, _?(# > 0 &)]]; fmax = -1; seq = {}; Do[f1 = f[n]; If[f1 > fmax, fmax = f1; AppendTo[seq, n]], {n, 1, 10^3}]; seq

A385904 a(n) is the number of nonempty subsets of the divisors of n that sum to a perfect square.

Original entry on oeis.org

1, 1, 2, 2, 1, 4, 1, 3, 3, 2, 1, 11, 1, 3, 4, 5, 1, 9, 1, 9, 3, 3, 1, 27, 2, 2, 4, 8, 1, 27, 1, 7, 3, 2, 2, 49, 1, 1, 3, 22, 1, 21, 1, 7, 8, 3, 1, 77, 2, 5, 2, 4, 1, 22, 2, 21, 2, 1, 1, 248, 1, 2, 7, 11, 1, 21, 1, 4, 2, 17, 1, 235, 1, 1, 9, 7, 1, 20, 1, 64, 6, 1

Offset: 1

Felix Huber, Jul 21 2025

nonn

			a(6) = 4 because exactly the 4 nonempty subsets {1}, {1, 3}, {1, 2, 6} and {3, 6} of the divisors of 6 sum to a perfect square: 1 = 1^2, 1 + 3 = 2^2, 1 + 2 + 6 = 3^2.

Felix Huber, Table of n, a(n) for n = 1..10000

Cf. A000290, A005835, A027750, A119347, A119348, A237290, A385645, A385646.

with(NumberTheory):
A385904:=proc(n)
    local b,l,j;
    l:=[(Divisors(n))[]]:
    b:=proc(m,i)
        option remember;
        `if`(m=0,1,`if`(i<1,0,b(m,i-1)+`if`(l[i]>m,0,b(m-l[i],i-1))))
	end;
    add(b(j^2,nops(l)),j=1..floor(sqrt(sigma(n))));
end:
seq(A385904(n),n=1..82);

a[n_]:=Module[{nb = 0, d = Divisors[n]},Length[Select[Subsets[d],IntegerQ[Sqrt[Total[#]]]&]]]-1;Array[a,82] (* James C. McMahon, Jul 27 2025 *)

a(n) = my(nb=0, d=divisors(n)); forsubset(#d, s, nb+=issquare(sum(i=1, #s, d[s[i]]))); nb-1; \\ Michel Marcus, Jul 22 2025

a(p) = 1 for primes p != 3.

A307222 Deficient numbers k at least one of whose divisors is the sum of other distinct divisors of k.

Original entry on oeis.org

45, 63, 99, 105, 110, 117, 130, 135, 154, 165, 170, 182, 189, 195, 225, 231, 238, 255, 266, 273, 285, 286, 297, 315, 322, 345, 351, 357, 374, 385, 399, 405, 418, 429, 441, 455, 459, 475, 483, 494, 495, 506, 513, 525, 561, 567, 585, 595, 598, 609, 621, 627, 646

Offset: 1

Amiram Eldar, Mar 29 2019

nonn

Erdős used the term "integers with the property P" for numbers k such that all the 2^d(k) sums formed from the d(k) divisors of k are distinct and proved that they are all deficient numbers and have a positive density. This sequence lists deficient numbers not having this property.

Differs from A051773 from n >= 12.

			45 is in this sequence since its divisors are 1, 3, 5, 9, 15, 45 whose sum is 78 < 90, and thus it is deficient, yet the divisor 15 is the sum of other divisors of 45: 1 + 5 + 9.

S. J. Benkoski and P. Erdős, On weird and pseudoperfect numbers, Math. Comp., 28 (1974), pp. 617-623. Alternate link; 1975 corrigendum
Paul Erdős, Some extremal problems in combinatorial number theory, Mathematical Essays Dedicated to A. J. Macintyre, Ohio Univ. Press, Athens, Ohio (1970), pp. 123-133.

Cf. A000005, A005100, A051773, A119347.

T[n_, k_] := Module[{d = Divisors[n]}, SeriesCoefficient[Series[Product[1 + x^d[[i]], {i, Length[d]}], {x, 0, k}], k]]; seqQ[n_] := DivisorSigma[1, n] < 2n && Max[T[n, #] & /@ Range[DivisorSigma[1, n]]] > 1; Select[Range[1000], seqQ]

cp's OEIS Frontend

A199919 Number of distinct sums of distinct divisors of n when positive and negative divisors are allowed.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A342401 Numbers k such that the number of distinct sums of distinct unitary divisors of k sets a new record.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A385904 a(n) is the number of nonempty subsets of the divisors of n that sum to a perfect square.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A307222 Deficient numbers k at least one of whose divisors is the sum of other distinct divisors of k.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica