A392371 Numbers k such that if the least prime index of k is x, then all subsequent prime indices are 2x.

1, 2, 3, 5, 6, 7, 11, 13, 17, 18, 19, 21, 23, 29, 31, 37, 41, 43, 47, 53, 54, 59, 61, 65, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 133, 137, 139, 147, 149, 151, 157, 162, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229

Offset

1,2

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.

Also numbers whose trimmed 0-based partial alternating sums of prime indices are all equal. Here, the k-based partial alternating sums of a finite sequence q are given by pas(q,k)_j = (-1)^j * k + Sum_{i=1..j} (-1)^(i+j) * q_i. This is a signed version of the partial sums transformation, inverse to the "first sums" transformation.

Links

Table of n, a(n) for n=1..59.

Formula

Consists of 1 and all numbers of the form prime(x) * prime(2x)^y for some x > 0, y >= 0.

Example

The terms together with their prime indices begin:
    1: {}
    2: {1}
    3: {2}
    5: {3}
    6: {1,2}
    7: {4}
   11: {5}
   13: {6}
   17: {7}
   18: {1,2,2}
   19: {8}
   21: {2,4}
   23: {9}
   29: {10}
   31: {11}
   37: {12}
   41: {13}
   43: {14}
   47: {15}
   53: {16}
   54: {1,2,2,2}
   59: {17}

Mathematica

prix[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
pas[y_, k_]:=Table[(-1)^j*k+Sum[(-1)^(i+j)*y[[i]], {i, j}], {j, 0, Length[y]}];
Select[Range[300], SameQ@@Rest[pas[prix[#], 0]]&]

Crossrefs

For reversed prime indices see A000040, A008578.

Partitions of this type are counted by A001227.

These are positions of constant rows in A391981 or A392374, sums A346699.

Also positions of constant rows in reverse of A391982 or A392711, sums A346697.

For non-trimmed distinct we have A392695, count A392701.

For distinct instead of constant we have A392697, count A392704.

The reverse for distinct is A392698, count A392700.

A055396 gives least prime index, greatest A061395.

A056239 adds up prime indices, row sums of A112798.

A344606 counts alternating permutations of prime indices.

A390307 lists first sums of prime indices, see A390362, A390448, A390449.

Cf. A000097, A001222, A005117, A316524, A344616, A345958, A346703, A392707.

Sequence in context: A330597 A352336 A283599 * A096530 A299157 A325327

Adjacent sequences: A392368 A392369 A392370 * A392372 A392373 A392374

Keyword

nonn

Author

Gus Wiseman, Jan 27 2026

Status

approved