A337853 -id:A337853 - cp's OEIS frontend

A337854 a(n) is the smallest number that can be partitioned in exactly n ways as the sum of two Niven numbers.

0, 2, 4, 6, 8, 10, 51, 48, 72, 108, 126, 90, 138, 144, 120, 198, 162, 210, 216, 315, 240, 234, 306, 252, 372, 270, 546, 360, 342, 444, 414, 468, 420, 642, 450, 522, 540, 924, 612, 600, 666, 630, 888, 930, 756, 840, 882, 936, 972, 1098, 1215, 1026, 1212, 1080

Offset: 0

Marius A. Burtea, Sep 26 2020

			a(0) = 0 because 0 cannot be written as the sum of two Niven numbers.
a(1) = 2 because 2 is uniquely written 2 = 1 + 1, with 1 in A005349.
a(2) = 4 because 4 = 1 + 3 = 2 + 2 and 1, 2, 3 are in A005349.
a(3) = 6 because 6 = 1 + 5 = 2 + 4 = 3 + 3 and 1, 2, 3, 4, 5 are terms in A005349.
a(6) = 51, because 51 = 1 + 50 = 3 + 48 = 6 + 45 = 9 + 42 = 21 + 30 = 24 + 27 and 1, 3, 6, 9, 21, 24, 27, 30, 42, 45, 48, 50 are terms in A005349.

Cf. A005349, A337853.

a:=[]; niven:=func; for n in [0..55] do k:=0; while k le 10000 and #RestrictedPartitions(k,2,{m:m in [1..k-1]| niven(m)}) ne n do k:=k+1; end  while; Append(~a,k); end for; a;

m = 1300; nivens = Select[Range[m], Divisible[#, Plus @@ IntegerDigits[#]] &]; a[n_] := Length[IntegerPartitions[n, {2}, nivens]]; mx = 54; s = Table[-1, {mx}]; c = 0; n = 0; While[c < mx, i = a[n] + 1; If[i <= mx && s[[i]] < 0, c++; s[[i]] = n]; n++]; s (* Amiram Eldar, Sep 27 2020 *)

A337861 Numbers that can be written as the sum of two Moran numbers (see A001101).

Original entry on oeis.org

36, 39, 42, 45, 48, 54, 60, 63, 66, 69, 72, 81, 84, 87, 90, 102, 105, 108, 111, 126, 129, 132, 135, 138, 141, 144, 147, 151, 153, 154, 156, 159, 160, 162, 168, 170, 171, 173, 174, 175, 177, 178, 179, 180, 183, 189, 192, 194, 195, 196, 197, 198, 201, 208, 211

Offset: 1

Marius A. Burtea, Oct 21 2020

			36 = 18 + 18 = A001101(1) + A001101(1), so 36 is a term.
39 = 18 + 21 = A001101(1) + A001101(2), so 39 is a term.
87 = 42 + 45 = A001101(4) + A001101(5), so 87 is a term.

Cf. A001101, A005349, A337853.

moran:=func; [n:n in [1..220] | #RestrictedPartitions(n,2,{k:k in [1..n-1] | moran(k)}) ne 0];

m = 211; morans = Select[Range[m], PrimeQ[#/Plus @@ IntegerDigits[#]] &]; Select[Range[m], Length[IntegerPartitions[#, {2}, morans]] > 0 &] (* Amiram Eldar, Oct 21 2020 *)

cp's OEIS Frontend

A337854 a(n) is the smallest number that can be partitioned in exactly n ways as the sum of two Niven numbers.

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