A373892 a(n) is the smallest number that can be partitioned in exactly n ways as the sum of two Duffinian numbers (A003624).

1, 8, 25, 43, 84, 71, 102, 160, 150, 219, 226, 196, 244, 350, 328, 300, 330, 354, 400, 386, 448, 408, 434, 390, 510, 536, 462, 546, 570, 624, 608, 740, 722, 690, 714, 770, 726, 660, 750, 804, 842, 858, 876, 870, 932, 914, 924, 840, 986, 1038, 966, 1108, 1050, 1056

Offset: 0

Author

Marius A. Burtea, Jul 12 2024

Keywords

• nonn

Examples

			1 cannot be written as the sum of two Duffinian numbers, so a(0) = 1.
The numbers from 2 to 7 cannot be written as the sum of two Duffinian numbers and 8 = 4 + 4 = A003624(1) + A003624(1), so a(1) = 8.
25 = 4 + 21 = 9 + 16 and 4 = A003624(1), 9 = A003624(3), 16 = A003624(4), 21 = A003624(5) and the numbers 9 to 24 cannot be written in two ways as a sum of two Duffinian numbers. Thus a(2) = 25.
		

Crossrefs

Cf. A003624.

Programs

Magma

f:=func; b:=[n: n in [1..2000] |f(n)]; a:=[]; for n in [0..60] do k:=1; while #RestrictedPartitions(k,2,Set(b)) ne n do k:=k+1; end while; Append(~a,k); end for; a;

Mathematica

dufQ[n_] := CompositeQ[n] && CoprimeQ[n, DivisorSigma[1, n]]; f[n_] := Sum[If[dufQ[k] && dufQ[n - k], 1, 0], {k, 1, Floor[n/2]}]; seq[len_, nmax_] := Module[{s = Table[0, {len}], c = 0, n = 1, i}, While[c < len && n < nmax, i = f[n] + 1; If[i <= len && s[[i]] == 0, c++; s[[i]] = n]; n++]; s]; seq[54, 2000] (* Amiram Eldar, Jul 19 2024 *)