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 A350942 Number of odd parts minus number of even conjugate parts of the integer partition with Heinz number n. 20
 0, 1, 0, 1, 1, 0, 0, 3, -2, 1, 1, 2, 0, 0, -1, 3, 1, 0, 0, 3, -2, 1, 1, 2, -1, 0, 0, 2, 0, 1, 1, 5, -1, 1, -2, 0, 0, 0, -2, 3, 1, 0, 0, 3, 1, 1, 1, 4, -4, 1, -1, 2, 0, 0, -1, 2, -2, 0, 1, 1, 0, 1, 0, 5, -2, 1, 1, 3, -1, 0, 0, 2, 1, 0, 1, 2, -3, 0, 0, 5, -2, 1 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,8 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. LINKS Table of n, a(n) for n=1..82. EXAMPLE First positions n such that a(n) = 6, 5, 4, 3, 2, 1, 0, -1, -2, -3, -4, -5, -6, together with their prime indices, are: 192: (2,1,1,1,1,1,1) 32: (1,1,1,1,1) 48: (2,1,1,1,1) 8: (1,1,1) 12: (2,1,1) 2: (1) 1: () 15: (3,2) 9: (2,2) 77: (5,4) 49: (4,4) 221: (7,6) 169: (6,6) MATHEMATICA primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]]; conj[y_]:=If[Length[y]==0, y, Table[Length[Select[y, #>=k&]], {k, 1, Max[y]}]]; Table[Count[primeMS[n], _?OddQ]-Count[conj[primeMS[n]], _?EvenQ], {n, 100}] CROSSREFS The conjugate version is A350849. This is a hybrid of A195017 and A350941. Positions of 0's are A350943. A000041 = integer partitions, strict A000009. A056239 adds up prime indices, counted by A001222, row sums of A112798. A122111 represents conjugation using Heinz numbers. A257991 = # of odd parts, conjugate A344616. A257992 = # of even parts, conjugate A350847. A316524 = alternating sum of prime indices. The following rank partitions: A325698: # of even parts = # of odd parts. A349157: # of even parts = # of odd conjugate parts, counted by A277579. A350848: # even conj parts = # odd conj parts, counted by A045931. A350943: # of even conjugate parts = # of odd parts, counted by A277579. A350944: # of odd parts = # of odd conjugate parts, counted by A277103. A350945: # of even parts = # of even conjugate parts, counted by A350948. Cf. A026424, A028260, A130780, A171966, A239241, A241638, A325700, A350947, A350949, A350950, A350951. Sequence in context: A140736 A284993 A292068 * A140056 A240239 A247044 Adjacent sequences: A350939 A350940 A350941 * A350943 A350944 A350945 KEYWORD sign AUTHOR Gus Wiseman, Jan 28 2022 STATUS approved

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Last modified March 4 22:06 EST 2024. Contains 370532 sequences. (Running on oeis4.)