A280829 -id:A280829 - cp's OEIS frontend

A337853 a(n) is the number of partitions of n as the sum of two Niven numbers.

0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 5, 5, 5, 4, 4, 3, 3, 3, 3, 3, 4, 3, 4, 4, 4, 4, 5, 4, 5, 4, 4, 4, 3, 2, 4, 3, 3, 4, 3, 3, 5, 3, 4, 5, 4, 4, 7, 4, 5, 6, 5, 3, 7, 4, 4, 6, 4, 2, 7, 3, 4, 5, 4, 3, 7, 3, 4, 5, 4, 3, 8, 3, 4, 6, 3, 3, 6, 2, 5, 6, 5, 3, 8, 4, 4, 6

Offset: 0

Marius A. Burtea, Sep 26 2020

a(n) >= 1 for n >= 2 ?.

For n <= 200000, a(n) = 1 only for n = 2, 3, 299, (2 = 1 + 1, 3 = 1 + 2, 299 = 1 + 288) and a(n) = 2 only for n in {4, 5, 35, 59, 79, 95, 97, 149, 169, 179, 389}.

			0 and 1 cannot be decomposed as the sum of two Niven numbers, so a(0) = a(1) = 0.
4 = 1 + 3 = 2 + 2 and 1, 2, 3 are in A005349, so a(4) = 2.
15 = 3 + 12 = 5 + 10 = 6 + 9 = 7 + 8 and 3, 5, 6, 7, 8, 9, 10, 12 are in A005349, so a(15) = 4.

Rémy Sigrist, Table of n, a(n) for n = 0..10000

Cf. A005349, A116357, A172151, A172398, A180149, A280829, A319468, A302479, A319395, A337854.

niven:=func; [#RestrictedPartitions(n,2,{k: k in [1..n-1] | niven(k)}): n in [0..100]];

m = 100; nivens = Select[Range[m], Divisible[#, Plus @@ IntegerDigits[#]] &]; a[n_] := Length[IntegerPartitions[n, {2}, nivens]]; Array[a, m, 0] (* Amiram Eldar, Sep 27 2020 *)

A280832 Sum of the parts in the partitions of n into two squarefree semiprimes.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 12, 0, 0, 0, 16, 0, 0, 0, 40, 21, 0, 0, 24, 25, 0, 27, 56, 29, 30, 31, 64, 0, 0, 35, 108, 37, 0, 39, 80, 82, 42, 86, 132, 90, 0, 94, 192, 147, 50, 0, 156, 106, 108, 110, 168, 114, 0, 118, 240, 305, 0, 63, 128, 195, 132, 201, 340, 138, 140

Offset: 1

Wesley Ivan Hurt, Jan 08 2017

			a(20) = 40; there are two partitions of n into two squarefree semiprimes: (14,6) and (10,10). The sum of the parts in these partitions is 40.

Indranil Ghosh, Table of n, a(n) for n = 1..1000
Index entries for sequences related to partitions

Cf. A280710, A280829.

with(numtheory): A280832:=n->n*add(floor(bigomega(i)*mobius(i)^2/2)*floor(2*mobius(i)^2/bigomega(i))*floor(bigomega(n-i)*mobius(i)^2/2)*floor(2*mobius(n-i)^2/bigomega(n-i)), i=2..floor(n/2)): seq(A280832(n), n=1..100);

Table[n Sum[Floor[PrimeOmega[i] MoebiusMu[i]^2/2] Floor[2 MoebiusMu[i]^2 / PrimeOmega[i]] Floor[PrimeOmega[n - i] MoebiusMu[i]^2 / 2] Floor[2 MoebiusMu[n - i]^2 / PrimeOmega[n - i]], {i, 2, Floor[n/2]}], {n, 1, 70}] (* Indranil Ghosh, Mar 09 2017, translated from Maple code *)
spp[n_]:=Total[Flatten[Select[IntegerPartitions[n,{2}],AllTrue[#,SquareFreeQ] && PrimeOmega[ #]=={2,2}&]]]; Array[spp,70] (* Harvey P. Dale, Jun 12 2022 *)

for(n=1, 70, print1(n * sum(i=2, floor(n/2), floor(bigomega(i) * moebius(i)^2 / 2) * floor(2*moebius(i)^2 / bigomega(i)) * floor(bigomega(n - i)* moebius(i)^2 / 2) * floor(2*moebius(n - i)^2 / bigomega(n - i))),", "))  \\ Indranil Ghosh, Mar 09 2017, translated from Maple code

a(n) = n * Sum_{k=1..floor(n/2)} c(k) * c(n-k), where c = A280710. - Wesley Ivan Hurt, Aug 31 2025

cp's OEIS Frontend

A337853 a(n) is the number of partitions of n as the sum of two Niven numbers.

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