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Numbers with exactly two distinct prime factors, both appearing with the same exponent.
3

%I #10 Dec 01 2023 15:56:46

%S 6,10,14,15,21,22,26,33,34,35,36,38,39,46,51,55,57,58,62,65,69,74,77,

%T 82,85,86,87,91,93,94,95,100,106,111,115,118,119,122,123,129,133,134,

%U 141,142,143,145,146,155,158,159,161,166,177,178,183,185,187,194

%N Numbers with exactly two distinct prime factors, both appearing with the same exponent.

%C First differs from A268390 in lacking 210.

%C First differs from A238748 in lacking 210.

%C These are the Heinz numbers of the partitions counted by A367588.

%H Michael De Vlieger, <a href="/A367590/b367590.txt">Table of n, a(n) for n = 1..10000</a>

%F Union of A006881 and A303661. - _Michael De Vlieger_, Dec 01 2023

%e The terms together with their prime indices begin:

%e 6: {1,2} 57: {2,8} 106: {1,16}

%e 10: {1,3} 58: {1,10} 111: {2,12}

%e 14: {1,4} 62: {1,11} 115: {3,9}

%e 15: {2,3} 65: {3,6} 118: {1,17}

%e 21: {2,4} 69: {2,9} 119: {4,7}

%e 22: {1,5} 74: {1,12} 122: {1,18}

%e 26: {1,6} 77: {4,5} 123: {2,13}

%e 33: {2,5} 82: {1,13} 129: {2,14}

%e 34: {1,7} 85: {3,7} 133: {4,8}

%e 35: {3,4} 86: {1,14} 134: {1,19}

%e 36: {1,1,2,2} 87: {2,10} 141: {2,15}

%e 38: {1,8} 91: {4,6} 142: {1,20}

%e 39: {2,6} 93: {2,11} 143: {5,6}

%e 46: {1,9} 94: {1,15} 145: {3,10}

%e 51: {2,7} 95: {3,8} 146: {1,21}

%e 55: {3,5} 100: {1,1,3,3} 155: {3,11}

%t Select[Range[100], SameQ@@Last/@If[#==1, {}, FactorInteger[#]]&&PrimeNu[#]==2&]

%Y The case of any multiplicities is A007774, counts A002133.

%Y Partitions of this type are counted by A367588.

%Y The case of distinct exponents is A367589, counts A182473.

%Y A000041 counts integer partitions, strict A000009.

%Y A091602 counts partitions by greatest multiplicity, least A243978.

%Y A116608 counts partitions by number of distinct parts.

%Y Cf. A006881, A039956, A071625, A072233, A072774, A109297, A303661, A367580.

%K nonn

%O 1,1

%A _Gus Wiseman_, Dec 01 2023