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 A357486 Heinz numbers of integer partitions with the same length as alternating sum. 10
 1, 2, 10, 20, 21, 42, 45, 55, 88, 91, 105, 110, 125, 156, 176, 182, 187, 198, 231, 245, 247, 312, 340, 351, 374, 390, 391, 396, 429, 494, 532, 544, 550, 551, 605, 663, 680, 702, 713, 714, 765, 780, 782, 845, 891, 910, 912, 969, 975, 1012, 1064, 1073, 1078 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions. The alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i. LINKS Table of n, a(n) for n=1..53. EXAMPLE The terms together with their prime indices begin: 1: {} 2: {1} 10: {1,3} 20: {1,1,3} 21: {2,4} 42: {1,2,4} 45: {2,2,3} 55: {3,5} 88: {1,1,1,5} 91: {4,6} 105: {2,3,4} 110: {1,3,5} 125: {3,3,3} 156: {1,1,2,6} 176: {1,1,1,1,5} MATHEMATICA primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]]; ats[y_]:=Sum[(-1)^(i-1)*y[[i]], {i, Length[y]}]; Select[Range[100], PrimeOmega[#]==ats[Reverse[primeMS[#]]]&] CROSSREFS For product instead of length we have new, counted by A004526. The version for compositions is A357184, counted by A357182. For absolute value we have A357486, counted by A357487. These partitions are counted by A357189. A000041 counts partitions, strict A000009. A000712 up to 0's counts partitions, sum = twice alt sum, rank A349159. A001055 counts partitions with product equal to sum, ranked by A301987. A006330 up to 0's counts partitions, sum = twice rev-alt sum, rank A349160. A025047 counts alternating compositions. A357136 counts compositions by alternating sum. Cf. A051159, A131044, A262046. Sequence in context: A346810 A347024 A009342 * A350508 A306105 A038103 Adjacent sequences: A357483 A357484 A357485 * A357487 A357488 A357489 KEYWORD nonn AUTHOR Gus Wiseman, Oct 01 2022 STATUS approved

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Last modified September 23 02:06 EDT 2023. Contains 365532 sequences. (Running on oeis4.)