A337854 -id:A337854 - 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 *)

A337862 a(n) is the smallest number that can be partitioned into n ways as the sum of two Moran numbers.

Original entry on oeis.org

0, 36, 63, 174, 198, 306, 312, 399, 1176, 930, 1296, 1989, 1110, 888, 2190, 1896, 2688, 3990, 3630, 3090, 3696, 3810, 8316, 6870, 4710, 12420, 11340, 9180, 13350, 12990, 14070, 14364, 14970, 9900, 15444, 14790, 15012, 18570, 19980, 25164, 23610, 25092, 23790

Offset: 0

Marius A. Burtea, Oct 21 2020

			0 cannot be written as the sum of two Moran numbers because A001101(1) = 18, so 0 is a term and a(0) = 0.
36 = 18 + 18 = A001101(1) + A001101(1), so a(1) = 36.
63 = 18 + 27 = A001101(1) + A001101(5) and 63 = 21 + 42 = A001101(2) + A001101(4), so a(2) = 63.
174 = 18 + 156 = 21 + 153 = 63 + 111 and 18, 21, 63, 111, 153, 156 are in A001101, so a(3) = 174.
198 = 27 + 171 = 42 + 156 = 45 + 153 = 84 + 114 and 27, 42, 45, 84, 153, 156, 171 are in A001101, so a(4) = 198.

Cf. A001101, A337854, A337861.

a:=[]; moran:=func; v:={m:m in [1..40000]|moran(m)}; for n in [0..40] do k:=0; while #RestrictedPartitions(k,2,v) ne n do k:=k+1; end while; Append(~a,k); end for; a;

m = 60000; morans = Select[Range[m], PrimeQ[#/Plus @@ IntegerDigits[#]] &]; mx = 43; s = Table[-1, {mx}]; n = 0; c = 0; While[c < mx && n <= m, If[(i = Length[IntegerPartitions[n, {2}, morans]] + 1) <= mx && s[[i]] == -1, c++; s[[i]] = n]; n++]; s (* Amiram Eldar, Oct 21 2020 *)

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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