This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A340784 #22 Jul 28 2024 16:38:56
%S A340784 1,4,9,10,16,21,22,25,34,36,39,40,46,49,55,57,62,64,81,82,84,85,87,88,
%T A340784 90,91,94,100,111,115,118,121,129,133,134,136,144,146,155,156,159,160,
%U A340784 166,169,183,184,187,189,194,196,198,203,205,206,210,213,218,220
%N A340784 Heinz numbers of even-length integer partitions of even numbers.
%C A340784 The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are positive integers whose number of prime indices and sum of prime indices are both even, counting multiplicity in both cases.
%C A340784 A multiplicative semigroup: if m and n are in the sequence, then so is m*n. - _Antti Karttunen_, Jul 28 2024
%H A340784 Antti Karttunen, <a href="/A340784/b340784.txt">Table of n, a(n) for n = 1..20000</a>
%H A340784 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%F A340784 Intersection of A028260 and A300061.
%e A340784 The sequence of partitions together with their Heinz numbers begins:
%e A340784 1: () 57: (8,2) 118: (17,1)
%e A340784 4: (1,1) 62: (11,1) 121: (5,5)
%e A340784 9: (2,2) 64: (1,1,1,1,1,1) 129: (14,2)
%e A340784 10: (3,1) 81: (2,2,2,2) 133: (8,4)
%e A340784 16: (1,1,1,1) 82: (13,1) 134: (19,1)
%e A340784 21: (4,2) 84: (4,2,1,1) 136: (7,1,1,1)
%e A340784 22: (5,1) 85: (7,3) 144: (2,2,1,1,1,1)
%e A340784 25: (3,3) 87: (10,2) 146: (21,1)
%e A340784 34: (7,1) 88: (5,1,1,1) 155: (11,3)
%e A340784 36: (2,2,1,1) 90: (3,2,2,1) 156: (6,2,1,1)
%e A340784 39: (6,2) 91: (6,4) 159: (16,2)
%e A340784 40: (3,1,1,1) 94: (15,1) 160: (3,1,1,1,1,1)
%e A340784 46: (9,1) 100: (3,3,1,1) 166: (23,1)
%e A340784 49: (4,4) 111: (12,2) 169: (6,6)
%e A340784 55: (5,3) 115: (9,3) 183: (18,2)
%t A340784 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A340784 Select[Range[100],EvenQ[PrimeOmega[#]]&&EvenQ[Total[primeMS[#]]]&]
%o A340784 (PARI)
%o A340784 A056239(n) = { my(f); if(1==n, 0, f=factor(n); sum(i=1, #f~, f[i, 2] * primepi(f[i, 1]))); }
%o A340784 A353331(n) = ((!(bigomega(n)%2)) && (!(A056239(n)%2)));
%o A340784 isA340784(n) = A353331(n); \\ _Antti Karttunen_, Apr 14 2022
%Y A340784 Note: A-numbers of Heinz-number sequences are in parentheses below.
%Y A340784 The case of prime powers is A056798.
%Y A340784 These partitions are counted by A236913.
%Y A340784 The odd version is A160786 (A340931).
%Y A340784 A000009 counts partitions into odd parts (A066208).
%Y A340784 A001222 counts prime factors.
%Y A340784 A047993 counts balanced partitions (A106529).
%Y A340784 A056239 adds up prime indices.
%Y A340784 A058695 counts partitions of odd numbers (A300063).
%Y A340784 A061395 selects the maximum prime index.
%Y A340784 A072233 counts partitions by sum and length.
%Y A340784 A112798 lists the prime indices of each positive integer.
%Y A340784 - Even -
%Y A340784 A027187 counts partitions of even length/maximum (A028260/A244990).
%Y A340784 A034008 counts compositions of even length.
%Y A340784 A035363 counts partitions into even parts (A066207).
%Y A340784 A058696 counts partitions of even numbers (A300061).
%Y A340784 A067661 counts strict partitions of even length (A030229).
%Y A340784 A339846 counts factorizations of even length.
%Y A340784 A340601 counts partitions of even rank (A340602).
%Y A340784 A340785 counts factorizations into even factors.
%Y A340784 A340786 counts even-length factorizations into even factors.
%Y A340784 Cf. A026424, A257541, A300272, A326837, A326845, A340385 (A340386), A340604, A353331 (characteristic function), A353332, A353333, A353334.
%Y A340784 Squares (A000290) is a subsequence.
%Y A340784 Not a subsequence of A329609 (30 is the first term of A329609 not occurring here, and 210 is the first term here not present in A329609).
%Y A340784 Positions of even terms in A373381.
%K A340784 nonn
%O A340784 1,2
%A A340784 _Gus Wiseman_, Jan 30 2021