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 A357851 Numbers k such that the half-alternating sum of the prime indices of k is 1. 1
 2, 8, 18, 32, 45, 50, 72, 98, 105, 128, 162, 180, 200, 231, 242, 275, 288, 338, 392, 420, 429, 450, 455, 512, 578, 648, 663, 720, 722, 800, 833, 882, 924, 935, 968, 969, 1050, 1058, 1100, 1125, 1152, 1235, 1250, 1311, 1352, 1458, 1463, 1568, 1680, 1682, 1716 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 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. LINKS Table of n, a(n) for n=1..51. EXAMPLE The terms together with their prime indices begin: 2: {1} 8: {1,1,1} 18: {1,2,2} 32: {1,1,1,1,1} 45: {2,2,3} 50: {1,3,3} 72: {1,1,1,2,2} 98: {1,4,4} 105: {2,3,4} 128: {1,1,1,1,1,1,1} 162: {1,2,2,2,2} 180: {1,1,2,2,3} 200: {1,1,1,3,3} 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[#]]==1&] CROSSREFS The version for k = 0 is A357631, standard compositions A357625-A357626. The version for original alternating sum is A001105. Positions of ones in A357629, reverse A357633. The skew version for k = 0 is A357632, reverse A357636. Partitions with these Heinz numbers are counted by A035444, skew A035544. The reverse version is A357635, k = 0 version A000583. 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-length A357642. Cf. A003963, A053251, A055932, A345958, A357621-A357624, A357630, A357634, A357637, A357639, A357640. Sequence in context: A293296 A055044 A356209 * A067051 A074629 A209303 Adjacent sequences: A357848 A357849 A357850 * A357852 A357853 A357854 KEYWORD nonn AUTHOR Gus Wiseman, Oct 28 2022 STATUS approved

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Last modified September 26 04:29 EDT 2023. Contains 365653 sequences. (Running on oeis4.)