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A325991 Heinz numbers of integer partitions such that not every orderless pair of distinct parts has a different sum.

%I A325991 #8 Jun 02 2019 23:41:00
%S A325991 210,420,462,630,840,858,910,924,1050,1155,1260,1326,1386,1470,1680,
%T A325991 1716,1820,1848,1870,1890,1938,2100,2145,2310,2470,2520,2574,2622,
%U A325991 2652,2730,2772,2926,2940,3150,3234,3315,3360,3432,3465,3570,3640,3696,3740,3780,3876
%N A325991 Heinz numbers of integer partitions such that not every orderless pair of distinct parts has a different sum.
%C A325991 The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
%e A325991 The sequence of terms together with their prime indices begins:
%e A325991    210: {1,2,3,4}
%e A325991    420: {1,1,2,3,4}
%e A325991    462: {1,2,4,5}
%e A325991    630: {1,2,2,3,4}
%e A325991    840: {1,1,1,2,3,4}
%e A325991    858: {1,2,5,6}
%e A325991    910: {1,3,4,6}
%e A325991    924: {1,1,2,4,5}
%e A325991   1050: {1,2,3,3,4}
%e A325991   1155: {2,3,4,5}
%e A325991   1260: {1,1,2,2,3,4}
%e A325991   1326: {1,2,6,7}
%e A325991   1386: {1,2,2,4,5}
%e A325991   1470: {1,2,3,4,4}
%e A325991   1680: {1,1,1,1,2,3,4}
%e A325991   1716: {1,1,2,5,6}
%e A325991   1820: {1,1,3,4,6}
%e A325991   1848: {1,1,1,2,4,5}
%e A325991   1870: {1,3,5,7}
%e A325991   1890: {1,2,2,2,3,4}
%t A325991 Select[Range[1000],!UnsameQ@@Plus@@@Subsets[PrimePi/@First/@FactorInteger[#],{2}]&]
%Y A325991 The subset case is A196723.
%Y A325991 The maximal case is A325878.
%Y A325991 The integer partition case is A325857.
%Y A325991 The strict integer partition case is A325877.
%Y A325991 Heinz numbers of the counterexamples are given by A325991.
%Y A325991 Cf. A002033, A056239, A103300, A108917, A112798, A143823, A196724, A325853, A325855, A325858, A325859, A325862, A325992, A325993, A325994.
%K A325991 nonn
%O A325991 1,1
%A A325991 _Gus Wiseman_, Jun 02 2019