A373892 - OEIS

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

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OFFSET

0,2

LINKS

Table of n, a(n) for n=0..53.

EXAMPLE

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.

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 *)

PROG

(Magma) f:=func<n|n ne 1 and not IsPrime(n) and Gcd(n, DivisorSigma(1, n)) eq 1>; 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;

CROSSREFS

Cf. A003624.

Sequence in context: A244276 A161448 A031096 * A303194 A188819 A123605

Adjacent sequences: A373889 A373890 A373891 * A373893 A373894 A373895

KEYWORD

nonn

AUTHOR

Marius A. Burtea, Jul 12 2024

STATUS

approved