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a(n) is the number of partitions of n as the sum of two Niven numbers.
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%I #21 Sep 08 2022 08:46:25

%S 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,

%T 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,

%U 4,5,4,3,8,3,4,6,3,3,6,2,5,6,5,3,8,4,4,6

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

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

%C 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}.

%H Rémy Sigrist, <a href="/A337853/b337853.txt">Table of n, a(n) for n = 0..10000</a>

%e 0 and 1 cannot be decomposed as the sum of two Niven numbers, so a(0) = a(1) = 0.

%e 4 = 1 + 3 = 2 + 2 and 1, 2, 3 are in A005349, so a(4) = 2.

%e 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.

%t 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 *)

%o (Magma) niven:=func<n | n mod &+Intseq(n) eq 0>; [#RestrictedPartitions(n,2,{k: k in [1..n-1] | niven(k)}): n in [0..100]];

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

%K nonn,base

%O 0,5

%A _Marius A. Burtea_, Sep 26 2020