A392707 - OEIS

%I #10 Feb 09 2026 17:27:14

%S 1,2,3,5,7,10,11,13,14,17,19,22,23,26,29,31,33,34,37,38,39,41,43,46,

%T 47,51,53,57,58,59,61,62,67,69,71,73,74,79,82,83,85,86,87,89,93,94,95,

%U 97,101,103,106,107,109,110,111,113,115,118,122,123,127,129,130

%N Numbers whose prime indices have strictly increasing 0-based partial alternating sums.

%C First differs from A325460 in having 110.

%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 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.

%C Also numbers whose prime indices are the first sums of some finite strictly increasing sequence of nonnegative integers beginning with 0, where the first sums of a nonempty sequence (a, b, c, d, ...) are (a+b, b+c, c+d, ...).

%C Conjecture: Reversed integer partitions of this type (with strictly increasing 0-based partial alternating sums) are counted by A053251.

%e The prime indices of 10 are (1,3), with 0-based partial alternating sums (0,1,2), which are strictly increasing, so 10 is in the sequence.

%e The prime indices of 130 are (1,3,6), with 0-based partial alternating sums (0,1,2,4), which are strictly increasing, so 130 is in the sequence.

%e The terms together with their prime indices begin:

%e    1: {}

%e    2: {1}

%e    3: {2}

%e    5: {3}

%e    7: {4}

%e   10: {1,3}

%e   11: {5}

%e   13: {6}

%e   14: {1,4}

%e   17: {7}

%e   19: {8}

%e   22: {1,5}

%e   23: {9}

%e   26: {1,6}

%e Sequences of this type grouped by sum begin:

%e   (1)  (2)  (3)  (4)    (5)    (6)    (7)    (8)    (9)      (10)

%e                  (1,3)  (1,4)  (1,5)  (1,6)  (1,7)  (1,8)    (1,9)

%e                                       (2,5)  (2,6)  (2,7)    (2,8)

%e                                                     (1,3,5)  (3,7)

%e                                                              (1,3,6)

%t prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];

%t pas[y_,k_]:=Table[(-1)^j*k+Sum[(-1)^(i+j)*y[[i]],{i,j}],{j,0,Length[y]}];

%t Select[Range[100],Less@@pas[prix[#],0]&]

%Y Partitions of this type appear to be counted by A053251.

%Y The first term with omega > k is A066205(k).

%Y These are positions of strictly increasing rows in A391981, sums A346699.

%Y For weakly instead of strictly increasing we have A392694, apparent count A000009.

%Y For distinct instead of strictly increasing we have A392695, counted by A392701.

%Y For distinct and trimmed we have A392697, counted by A392704.

%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 A392374 lists 0-based partial alternating sums of prime indices, sums A346699.

%Y A392708 ranks comps with distinct 0-based partial alternating sums, count A392703.

%Y Cf. A001222, A005117, A316524, A344616, A345958, A346697, A390445, A390446, A390990, A391982, A392371, A392711.

%K nonn

%O 1,2

%A _Gus Wiseman_, Feb 05 2026