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%I #5 Jun 02 2019 23:41:20
%S 42,84,126,168,210,230,252,294,336,378,390,399,420,460,462,504,546,
%T 588,630,672,690,714,742,756,780,798,840,882,920,924,966,1008,1050,
%U 1092,1134,1150,1170,1176,1197,1218,1260,1302,1344,1365,1380,1386,1428,1470,1484
%N Heinz numbers of integer partitions such that not every ordered pair of distinct parts has a different quotient.
%C The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
%e The sequence of terms together with their prime indices begins:
%e 42: {1,2,4}
%e 84: {1,1,2,4}
%e 126: {1,2,2,4}
%e 168: {1,1,1,2,4}
%e 210: {1,2,3,4}
%e 230: {1,3,9}
%e 252: {1,1,2,2,4}
%e 294: {1,2,4,4}
%e 336: {1,1,1,1,2,4}
%e 378: {1,2,2,2,4}
%e 390: {1,2,3,6}
%e 399: {2,4,8}
%e 420: {1,1,2,3,4}
%e 460: {1,1,3,9}
%e 462: {1,2,4,5}
%e 504: {1,1,1,2,2,4}
%e 546: {1,2,4,6}
%e 588: {1,1,2,4,4}
%e 630: {1,2,2,3,4}
%e 672: {1,1,1,1,1,2,4}
%t Select[Range[1000],!UnsameQ@@Divide@@@Subsets[PrimePi/@First/@FactorInteger[#],{2}]&]
%Y The subset case is A325860.
%Y The maximal case is A325861.
%Y The integer partition case is A325853.
%Y The strict integer partition case is A325854.
%Y Heinz numbers of the counterexamples are given by A325994.
%Y Cf. A002033, A056239, A103300, A108917, A112798, A143823, A196724, A325768, A325856, A325868, A325869, A325876.
%K nonn
%O 1,1
%A _Gus Wiseman_, Jun 02 2019
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