A392700 - OEIS

%I #6 Feb 02 2026 10:14:59

%S 1,6,18,21,54,65,133,147,162,319,481,486,731,845,1007,1029,1403,1458,

%T 2059,2449,2527,3293,4141,4374,4601,5311,6943,7203,8201,9211,9251,

%U 10921,10985,12283,13122,13213,15247,16517,17797,19847,22213,24139,25853,28141,29539

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

%C 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.

%C 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.

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

%e The terms together with their prime indices begin:

%e     1: {}

%e     6: {1,2}

%e    18: {1,2,2}

%e    21: {2,4}

%e    54: {1,2,2,2}

%e    65: {3,6}

%e   133: {4,8}

%e   147: {2,4,4}

%e   162: {1,2,2,2,2}

%e   319: {5,10}

%e   481: {6,12}

%e   486: {1,2,2,2,2,2}

%e   731: {7,14}

%e   845: {3,6,6}

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

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

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

%Y See also A391982 or A392711, sums A346697.

%Y Including primes gives A392371, see A325230.

%Y A055396 gives least prime index, greatest A061395.

%Y A056239 adds up prime indices, row sums of A112798.

%Y A344606 counts alternating permutations of prime indices.

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

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

%K nonn

%O 1,2

%A _Gus Wiseman_, Jan 29 2026