A357631 - OEIS

%I #12 Oct 10 2023 16:22:30

%S 1,12,16,30,63,70,81,108,154,165,192,256,273,286,300,325,442,480,561,

%T 588,595,625,646,700,741,750,874,931,972,1008,1045,1080,1120,1173,

%U 1296,1334,1452,1470,1495,1540,1653,1728,1771,1798,2028,2139,2294,2401,2430

%N Numbers k such that the half-alternating sum of the prime indices of k is 0.

%C We define the half-alternating sum of a sequence (A, B, C, D, E, F, G, ...) to be A + B - C - D + E + F - G - ...

%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 If k is a term, then so is m^4 * k for any m >= 1. - _Robert Israel_, Oct 10 2023

%H Robert Israel, <a href="/A357631/b357631.txt">Table of n, a(n) for n = 1..10000</a>

%e The terms together with their prime indices begin:

%e     1: {}

%e    12: {1,1,2}

%e    16: {1,1,1,1}

%e    30: {1,2,3}

%e    63: {2,2,4}

%e    70: {1,3,4}

%e    81: {2,2,2,2}

%e   108: {1,1,2,2,2}

%e   154: {1,4,5}

%e   165: {2,3,5}

%e   192: {1,1,1,1,1,1,2}

%e   256: {1,1,1,1,1,1,1,1}

%e   273: {2,4,6}

%e   286: {1,5,6}

%e   300: {1,1,2,3,3}

%p f:= proc(n) local F,Q,i;

%p F:= sort(ifactors(n)[2],(s,t) -> s[1]<t[1]);

%p F:= map(t -> numtheory:-pi(t[1])$t[2],F);

%p Q:= [-1,1,1,-1];

%p add(Q[i mod 4 + 1]*F[i],i=1..nops(F))

%p end proc:

%p select(f=0, [$1..10000]); # _Robert Israel_, Oct 10 2023

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

%t halfats[f_]:=Sum[f[[i]]*(-1)^(1+Ceiling[i/2]),{i,Length[f]}];

%t Select[Range[1000],halfats[primeMS[#]]==0&]

%Y The version for original alternating sum is A000290.

%Y The version for standard compositions is A357625, reverse A357626.

%Y Positions of zeros in A357629, reverse A357633.

%Y The skew-alternating form is A357632, reverse A357636.

%Y The reverse version is A357635.

%Y These partitions are counted by A357639, skew A357640.

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

%Y A316524 gives alternating sum of prime indices, reverse A344616.

%Y A351005 = alternately equal and unequal partitions, compositions A357643.

%Y A351006 = alternately unequal and equal partitions, compositions A357644.

%Y A357641 counts comps w/ half-alt sum 0, even A357642.

%Y Cf. A003963, A053251, A055932, A357189, A357485-A357488, A357621-A357624, A357630, A357634, A357637.

%K nonn

%O 1,2

%A _Gus Wiseman_, Oct 09 2022