cp's OEIS Frontend

A357707 Numbers whose prime indices have equal number of parts congruent to each of 1 and 3 (mod 4).

%I A357707 #7 Oct 12 2022 19:44:51
%S A357707 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 A357707 79,81,87,89,90,91,94,100,101,102,107,111,113,115,117,129,130,131,133,
%U A357707 134,139,147,151,159,163,165,166,169,171,173,181,183,186,187,189
%N A357707 Numbers whose prime indices have equal number of parts congruent to each of 1 and 3 (mod 4).
%C A357707 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 A357707 The terms together with their prime indices begin:
%e A357707      1: {}
%e A357707      3: {2}
%e A357707      7: {4}
%e A357707      9: {2,2}
%e A357707     10: {1,3}
%e A357707     13: {6}
%e A357707     19: {8}
%e A357707     21: {2,4}
%e A357707     27: {2,2,2}
%e A357707     29: {10}
%e A357707     30: {1,2,3}
%t A357707 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A357707 Select[Range[100],Count[primeMS[#],_?(Mod[#,4]==1&)]==Count[primeMS[#],_?(Mod[#,4]==3&)]&]
%Y A357707 These partitions are counted by A035544.
%Y A357707 Includes A066207 = products of primes of even index.
%Y A357707 The conjugate partitions are ranked by A357636, reverse A357632.
%Y A357707 The conjugate reverse version is A357640 (aerated).
%Y A357707 A056239 adds up prime indices, row sums of A112798.
%Y A357707 A316524 gives alternating sum of prime indices, reverse A344616.
%Y A357707 A344651 counts partitions by alternating sum, ordered A097805.
%Y A357707 A357705 counts reversed partitions by skew-alternating sum, half A357704.
%Y A357707 Cf. A035363, A035550, A035594, A053251, A298311, A357486, A357623, A357638.
%K A357707 nonn
%O A357707 1,2
%A A357707 _Gus Wiseman_, Oct 12 2022