A359197 Least number k having n subsets of its divisors whose sum is k+1.

1, 2, 18, 12, 162, 24, 342, 80, 36, 198, 156, 48, 126, 150, 1430, 132, 1110, 1302, 1672, 448, 90, 96, 784, 1190, 1408, 84, 320, 72, 1064, 3100, 16048, 744, 702, 60, 920, 690, 984, 750, 594, 2300, 714, 696, 11024, 192, 11696, 400, 2028, 680, 728, 1548, 10672, 546, 616, 2156, 462, 324, 37888, 510, 4698

Offset: 0

Author

Robert G. Wilson v, Dec 19 2022

Keywords

• nonn
• look

Comments

All deficient numbers k > 1 (A005100) have only one subset, {1, k}, whose sum is k+1.

Examples

			a(0) = 1 since there exists no subset of the divisors of 1 which sum to 2.
a(1) = 2. 2 is the least deficient number greater than 1.
a(2) = 18 since two subsets of its divisors, {1, 18} and {1, 3, 6, 9}, sum to 19, and no smaller number has this property.
a(3) = 12 since three subsets of its divisors, {1, 12}, {3, 4, 6} and {1, 2, 4, 6}, sum to 13, and no smaller number has this property.
a(4) = 162 since {1, 162}, {1, 27, 54, 81}, {1, 9, 18, 54, 81} and {1, 3, 6, 18, 54, 81} sum to 163, and no smaller number has this property.
		

Links

• Robert G. Wilson v, Table of n, a(n) for n = 0..262

Crossrefs

Cf. A005100, A359196.

Programs

Maple

f:= proc(n) local x,d; coeff(expand(mul(1+x^d, d=numtheory:-divisors(n))),x,n+1) end proc:
N:= 60: # for a(0)..a(N)
V:= Array(0..N): count:= 0:
for n from 1 while count < N+1 do
  v:= f(n);
if v <= N and V[v] = 0 then V[v]:= n; count:= count+1 fi
od:
convert(V,list); # Robert Israel, Jan 14 2023

Mathematica

f[n_] := Block[{d = Divisors@ n}, SeriesCoefficient[ Series[ Product[1 + x^d[[i]], {i, Length@ d}], {x, 0, n +1}], n +1]]; j = 1; t[_] := 0; While[ j < 10001, b = f@j; If[ t[b] == 0, t[b] = j]; j++]; t /@ Range[0, 50]