The OEIS is supported by the many generous donors to the OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A357631 Numbers k such that the half-alternating sum of the prime indices of k is 0. 23
 1, 12, 16, 30, 63, 70, 81, 108, 154, 165, 192, 256, 273, 286, 300, 325, 442, 480, 561, 588, 595, 625, 646, 700, 741, 750, 874, 931, 972, 1008, 1045, 1080, 1120, 1173, 1296, 1334, 1452, 1470, 1495, 1540, 1653, 1728, 1771, 1798, 2028, 2139, 2294, 2401, 2430 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS 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 - ... 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. If k is a term, then so is m^4 * k for any m >= 1. - Robert Israel, Oct 10 2023 LINKS Robert Israel, Table of n, a(n) for n = 1..10000 EXAMPLE The terms together with their prime indices begin: 1: {} 12: {1,1,2} 16: {1,1,1,1} 30: {1,2,3} 63: {2,2,4} 70: {1,3,4} 81: {2,2,2,2} 108: {1,1,2,2,2} 154: {1,4,5} 165: {2,3,5} 192: {1,1,1,1,1,1,2} 256: {1,1,1,1,1,1,1,1} 273: {2,4,6} 286: {1,5,6} 300: {1,1,2,3,3} MAPLE f:= proc(n) local F, Q, i; F:= sort(ifactors(n)[2], (s, t) -> s[1] numtheory:-pi(t[1])\$t[2], F); Q:= [-1, 1, 1, -1]; add(Q[i mod 4 + 1]*F[i], i=1..nops(F)) end proc: select(f=0, [\$1..10000]); # Robert Israel, Oct 10 2023 MATHEMATICA primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]]; halfats[f_]:=Sum[f[[i]]*(-1)^(1+Ceiling[i/2]), {i, Length[f]}]; Select[Range[1000], halfats[primeMS[#]]==0&] CROSSREFS The version for original alternating sum is A000290. The version for standard compositions is A357625, reverse A357626. Positions of zeros in A357629, reverse A357633. The skew-alternating form is A357632, reverse A357636. The reverse version is A357635. These partitions are counted by A357639, skew A357640. A056239 adds up prime indices, row sums of A112798. A316524 gives alternating sum of prime indices, reverse A344616. A351005 = alternately equal and unequal partitions, compositions A357643. A351006 = alternately unequal and equal partitions, compositions A357644. A357641 counts comps w/ half-alt sum 0, even A357642. Cf. A003963, A053251, A055932, A357189, A357485-A357488, A357621-A357624, A357630, A357634, A357637. Sequence in context: A192609 A157678 A334560 * A109240 A020740 A344021 Adjacent sequences: A357628 A357629 A357630 * A357632 A357633 A357634 KEYWORD nonn AUTHOR Gus Wiseman, Oct 09 2022 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified August 8 12:42 EDT 2024. Contains 375021 sequences. (Running on oeis4.)