A390448 Numbers whose prime indices are the first sums of some weakly increasing sequence of positive integers.

3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99, 101, 103, 105, 107, 109, 111, 113, 115, 117, 119, 123, 127, 129, 131, 133

Offset

1,1

Comments

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

The first sums of a nonempty sequence (a, b, c, d, ...) are (a+b, b+c, c+d, ...).

Links

Table of n, a(n) for n=1..62.

Example

The first sums of (1,1,1,2) are (2,2,3), which is the prime indices of 45, so 45 is in the sequence.
The first sums of (3,3,3) are (6,6), which is the prime indices of 169, so 169 is in the sequence.
The terms together with their prime indices begin:
    3: {2}
    5: {3}
    7: {4}
    9: {2,2}
   11: {5}
   13: {6}
   15: {2,3}
   17: {7}
   19: {8}
   21: {2,4}
   23: {9}
   27: {2,2,2}
   29: {10}
   31: {11}
   33: {2,5}
   35: {3,4}
   37: {12}
   39: {2,6}
   41: {13}
   43: {14}
   45: {2,2,3}

Mathematica

prix[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
uncha[tar_, ini_]:=(cur={ini}; Do[AppendTo[cur, tar[[k]]-If[k==1, ini, cur[[k]]]], {k, 1, Length[tar]}]; cur);
gre[n_]:=n==1||With[{m=prix[n]}, Select[Table[uncha[m, i], {i, 1, Max[m]}], LessEqual@@#&]!={}];
Select[Range[2, 100], gre]

Crossrefs

These are numbers whose prime indices are a row of A390307.

Positions of positive terms in A390444 with alternating zeros.

The complement is A390445, counted by A390447.

Partitions of this type are counted by A390446.

Union of A390449.

A version for compositions is A390676, complement A390677.

A112798 lists prime indices, sum A056239, multiplicities A124010.

A342527 counts compositions with all equal first sums.

A390567 counts compositions with all distinct first sums.

Cf. A001222, A004709, A055396, A215366, A243055, A286470, A325328, A355536, A390362, A390364, A390365.

Sequence in context: A033039 A381822 A322660 * A161957 A082720 A033037

Adjacent sequences: A390445 A390446 A390447 * A390449 A390450 A390451

Keyword

nonn

Author

Gus Wiseman, Nov 11 2025

Status

approved