A392700 Nonprime numbers k such that if the least prime index of k is x, then all subsequent prime indices are 2x.

1, 6, 18, 21, 54, 65, 133, 147, 162, 319, 481, 486, 731, 845, 1007, 1029, 1403, 1458, 2059, 2449, 2527, 3293, 4141, 4374, 4601, 5311, 6943, 7203, 8201, 9211, 9251, 10921, 10985, 12283, 13122, 13213, 15247, 16517, 17797, 19847, 22213, 24139, 25853, 28141, 29539

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 nonprime 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..45.

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: {}
    6: {1,2}
   18: {1,2,2}
   21: {2,4}
   54: {1,2,2,2}
   65: {3,6}
  133: {4,8}
  147: {2,4,4}
  162: {1,2,2,2,2}
  319: {5,10}
  481: {6,12}
  486: {1,2,2,2,2,2}
  731: {7,14}
  845: {3,6,6}

Mathematica

Select[Range[100], !PrimeQ[#]&&SameQ@@Rest[pas[prix[#], 0]]&]

Crossrefs

Partitions of this type are counted by A069283, nonprime case of A001227.

Positions of non-singleton constant rows in A391981 or A392374, sums A346699.

See also A391982 or A392711, sums A346697.

Including primes gives A392371, see A325230.

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. A000040, A000097, A001222, A005117, A316524, A344616, A345958, A346703, A392697, A392707.

Sequence in context: A022418 A350848 A373413 * A358748 A081318 A275168

Adjacent sequences: A392697 A392698 A392699 * A392701 A392702 A392703

Keyword

nonn

Author

Gus Wiseman, Jan 29 2026

Status

approved