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 A357707 Numbers whose prime indices have equal number of parts congruent to each of 1 and 3 (mod 4). 1
 1, 3, 7, 9, 10, 13, 19, 21, 27, 29, 30, 34, 37, 39, 43, 49, 53, 55, 57, 61, 62, 63, 70, 71, 79, 81, 87, 89, 90, 91, 94, 100, 101, 102, 107, 111, 113, 115, 117, 129, 130, 131, 133, 134, 139, 147, 151, 159, 163, 165, 166, 169, 171, 173, 181, 183, 186, 187, 189 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS 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..59. EXAMPLE The terms together with their prime indices begin: 1: {} 3: {2} 7: {4} 9: {2,2} 10: {1,3} 13: {6} 19: {8} 21: {2,4} 27: {2,2,2} 29: {10} 30: {1,2,3} MATHEMATICA primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]]; Select[Range[100], Count[primeMS[#], _?(Mod[#, 4]==1&)]==Count[primeMS[#], _?(Mod[#, 4]==3&)]&] CROSSREFS These partitions are counted by A035544. Includes A066207 = products of primes of even index. The conjugate partitions are ranked by A357636, reverse A357632. The conjugate reverse version is A357640 (aerated). A056239 adds up prime indices, row sums of A112798. A316524 gives alternating sum of prime indices, reverse A344616. A344651 counts partitions by alternating sum, ordered A097805. A357705 counts reversed partitions by skew-alternating sum, half A357704. Cf. A035363, A035550, A035594, A053251, A298311, A357486, A357623, A357638. Sequence in context: A284526 A344413 A339618 * A294571 A083214 A091210 Adjacent sequences: A357704 A357705 A357706 * A357708 A357709 A357710 KEYWORD nonn AUTHOR Gus Wiseman, Oct 12 2022 STATUS approved

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Last modified August 8 12:20 EDT 2024. Contains 375021 sequences. (Running on oeis4.)