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A367590 Numbers with exactly two distinct prime factors, both appearing with the same exponent.

%I A367590 #12 Aug 04 2025 15:43:44
%S A367590 6,10,14,15,21,22,26,33,34,35,36,38,39,46,51,55,57,58,62,65,69,74,77,
%T A367590 82,85,86,87,91,93,94,95,100,106,111,115,118,119,122,123,129,133,134,
%U A367590 141,142,143,145,146,155,158,159,161,166,177,178,183,185,187,194
%N A367590 Numbers with exactly two distinct prime factors, both appearing with the same exponent.
%C A367590 First differs from A268390 in lacking 210.
%C A367590 First differs from A238748 in lacking 210.
%C A367590 These are the Heinz numbers of the partitions counted by A367588.
%H A367590 Michael De Vlieger, <a href="/A367590/b367590.txt">Table of n, a(n) for n = 1..10000</a>
%F A367590 Union of A006881 and A303661. - _Michael De Vlieger_, Dec 01 2023
%e A367590 The terms together with their prime indices begin:
%e A367590      6: {1,2}         57: {2,8}        106: {1,16}
%e A367590     10: {1,3}         58: {1,10}       111: {2,12}
%e A367590     14: {1,4}         62: {1,11}       115: {3,9}
%e A367590     15: {2,3}         65: {3,6}        118: {1,17}
%e A367590     21: {2,4}         69: {2,9}        119: {4,7}
%e A367590     22: {1,5}         74: {1,12}       122: {1,18}
%e A367590     26: {1,6}         77: {4,5}        123: {2,13}
%e A367590     33: {2,5}         82: {1,13}       129: {2,14}
%e A367590     34: {1,7}         85: {3,7}        133: {4,8}
%e A367590     35: {3,4}         86: {1,14}       134: {1,19}
%e A367590     36: {1,1,2,2}     87: {2,10}       141: {2,15}
%e A367590     38: {1,8}         91: {4,6}        142: {1,20}
%e A367590     39: {2,6}         93: {2,11}       143: {5,6}
%e A367590     46: {1,9}         94: {1,15}       145: {3,10}
%e A367590     51: {2,7}         95: {3,8}        146: {1,21}
%e A367590     55: {3,5}        100: {1,1,3,3}    155: {3,11}
%t A367590 Select[Range[100], SameQ@@Last/@If[#==1, {}, FactorInteger[#]]&&PrimeNu[#]==2&]
%t A367590 Select[Range[200],PrimeNu[#]==2&&Length[Union[FactorInteger[#][[;;,2]]]]==1&] (* _Harvey P. Dale_, Aug 04 2025 *)
%Y A367590 The case of any multiplicities is A007774, counts A002133.
%Y A367590 Partitions of this type are counted by A367588.
%Y A367590 The case of distinct exponents is A367589, counts A182473.
%Y A367590 A000041 counts integer partitions, strict A000009.
%Y A367590 A091602 counts partitions by greatest multiplicity, least A243978.
%Y A367590 A116608 counts partitions by number of distinct parts.
%Y A367590 Cf. A006881, A039956, A071625, A072233, A072774, A109297, A303661, A367580.
%K A367590 nonn
%O A367590 1,1
%A A367590 _Gus Wiseman_, Dec 01 2023