A349315 -id:A349315 - cp's OEIS frontend

A348674 Number of distinct values that can be produced by splitting n and adding the parts.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4

Offset: 0

Alois P. Heinz, Oct 29 2021

Differs from A055642 first at n=120: a(120) = 4 != 3 = A055642(120).

The number of split positions can vary from 0 to length(n)-1.

			a(0) = 1: 0.
a(10) = 2: 1 = 1+0, 10.
a(100) = 3: 1 = 1+0+0, 10 = 10+0, 100.
a(120) = 4: 3 = 1+2+0, 12 = 12+0, 21 = 1+20, 120.
a(2493690) = 62 = |{33, 51, 60, 69, 78, 87, 96, 105, 114, 123, 132, 141, 150, 159, 168, 177, 186, 195, 213, 267, 294, 321, 348, 375, 384, 402, 420, 510, 564, 591, 618, 708, 726, 744, 789, 807, 942, 951, 969, 1032, 1050, 1185, 2508, 2562, 2589, 3183, 3705, 3723, 3741, 3939, 4947, 5028, 9375, 9393, 24945, 25026, 49371, 93696, 93714, 249369, 493692, 2493690}|.

Alois P. Heinz, Table of n, a(n) for n = 0..20000

Ordinal transform gives A349315.
Where records occur: A349316.
Cf. A000041, A002275, A004216, A007953, A011557, A055642, A089695.

b:= proc(s) option remember; (n-> {parse(s), seq(seq(seq(x+y,
      y=b(s[i+1..n])), x=b(s[1..i])), i=1..n-1)})(length(s))
    end:
a:= n-> nops(b(""||n)):
seq(a(n), n=0..120);

a(n) <= 2^floor(log_10(n)) = 2^A004216(n) for n>0.
a((10^n-1)/9) = a(A002275(n)) <= A000041(n) with equality only for n <= 23.
a(10^n) = a(A011557(n)) = n+1.

A349316 Where records occur in A348674.

Original entry on oeis.org

0, 10, 100, 120, 1010, 1020, 1120, 1230, 10120, 10130, 10230, 10240, 11230, 11240, 12240, 12350, 101130, 101240, 101350, 102240, 102350, 102460, 112240, 112350, 112460, 122360, 122470, 123470, 123580, 1011240, 1011250, 1011350, 1011360, 1012250, 1012360, 1012460

Offset: 1

Alois P. Heinz, Nov 14 2021

Alois P. Heinz, Table of n, a(n) for n = 1..105

Cf. A348674, A349315.

b:= proc(s) option remember; (n-> {parse(s), seq(seq(seq(x+y,
      y=b(s[i+1..n])), x=b(s[1..i])), i=1..n-1)})(length(s))
    end:
a:= proc(n) option remember; local k, t; t:=nops(b(""||(a(n-1))));
      for k from 1+a(n-1) while nops(b(""||k))<=t do od; k
    end: a(1):=0:
seq(a(n), n=1..29);

A349315(a(n)) = 1.

A348674(a(n)) < A348674(a(n+1)).

cp's OEIS Frontend

A348674 Number of distinct values that can be produced by splitting n and adding the parts.

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