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Numbers whose prime indices have equal number of parts congruent to each of 1 and 3 (mod 4).
1

%I #7 Oct 12 2022 19:44:51

%S 1,3,7,9,10,13,19,21,27,29,30,34,37,39,43,49,53,55,57,61,62,63,70,71,

%T 79,81,87,89,90,91,94,100,101,102,107,111,113,115,117,129,130,131,133,

%U 134,139,147,151,159,163,165,166,169,171,173,181,183,186,187,189

%N Numbers whose prime indices have equal number of parts congruent to each of 1 and 3 (mod 4).

%C 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.

%e The terms together with their prime indices begin:

%e 1: {}

%e 3: {2}

%e 7: {4}

%e 9: {2,2}

%e 10: {1,3}

%e 13: {6}

%e 19: {8}

%e 21: {2,4}

%e 27: {2,2,2}

%e 29: {10}

%e 30: {1,2,3}

%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];

%t Select[Range[100],Count[primeMS[#],_?(Mod[#,4]==1&)]==Count[primeMS[#],_?(Mod[#,4]==3&)]&]

%Y These partitions are counted by A035544.

%Y Includes A066207 = products of primes of even index.

%Y The conjugate partitions are ranked by A357636, reverse A357632.

%Y The conjugate reverse version is A357640 (aerated).

%Y A056239 adds up prime indices, row sums of A112798.

%Y A316524 gives alternating sum of prime indices, reverse A344616.

%Y A344651 counts partitions by alternating sum, ordered A097805.

%Y A357705 counts reversed partitions by skew-alternating sum, half A357704.

%Y Cf. A035363, A035550, A035594, A053251, A298311, A357486, A357623, A357638.

%K nonn

%O 1,2

%A _Gus Wiseman_, Oct 12 2022