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A340933 Numbers whose least prime index is even. Heinz numbers of integer partitions whose last part is even.

%I A340933 #6 Feb 14 2021 01:40:27
%S A340933 3,7,9,13,15,19,21,27,29,33,37,39,43,45,49,51,53,57,61,63,69,71,75,77,
%T A340933 79,81,87,89,91,93,99,101,105,107,111,113,117,119,123,129,131,133,135,
%U A340933 139,141,147,151,153,159,161,163,165,169,171,173,177,181,183
%N A340933 Numbers whose least prime index is even. Heinz numbers of integer partitions whose last part is even.
%C A340933 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. 1 has no prime indices so is not counted.
%F A340933 A055396(a(n)) belongs to A005843.
%F A340933 Closed under multiplication.
%e A340933 The sequence of terms together with their prime indices begins:
%e A340933       3: {2}         51: {2,7}         99: {2,2,5}
%e A340933       7: {4}         53: {16}         101: {26}
%e A340933       9: {2,2}       57: {2,8}        105: {2,3,4}
%e A340933      13: {6}         61: {18}         107: {28}
%e A340933      15: {2,3}       63: {2,2,4}      111: {2,12}
%e A340933      19: {8}         69: {2,9}        113: {30}
%e A340933      21: {2,4}       71: {20}         117: {2,2,6}
%e A340933      27: {2,2,2}     75: {2,3,3}      119: {4,7}
%e A340933      29: {10}        77: {4,5}        123: {2,13}
%e A340933      33: {2,5}       79: {22}         129: {2,14}
%e A340933      37: {12}        81: {2,2,2,2}    131: {32}
%e A340933      39: {2,6}       87: {2,10}       133: {4,8}
%e A340933      43: {14}        89: {24}         135: {2,2,2,3}
%e A340933      45: {2,2,3}     91: {4,6}        139: {34}
%e A340933      49: {4,4}       93: {2,11}       141: {2,15}
%t A340933 Select[Range[2,100],EvenQ[PrimePi[FactorInteger[#][[1,1]]]]&]
%Y A340933 These partitions are counted by A026805.
%Y A340933 Looking at length or at maximum gives A028260/A244990, counted by A027187.
%Y A340933 If all prime indices are even we get A066207, counted by A035363.
%Y A340933 The complement is {1} \/ A340932, counted by A026804.
%Y A340933 A001222 counts prime factors.
%Y A340933 A005843 lists even numbers.
%Y A340933 A031215 lists even-indexed primes.
%Y A340933 A055396 selects least prime index.
%Y A340933 A056239 adds up prime indices.
%Y A340933 A058695 counts partitions of even numbers, ranked by A300061.
%Y A340933 A061395 selects greatest prime index.
%Y A340933 A112798 lists the prime indices of each positive integer.
%Y A340933 Cf. A006141, A030229, A067661, A101708, A236913,  A339846, A340601, A340602, A340605, A340854/A340855.
%K A340933 nonn
%O A340933 1,1
%A A340933 _Gus Wiseman_, Feb 12 2021