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%I #12 Jan 27 2022 20:46:12
%S 1,2,3,4,5,7,8,9,10,11,13,14,15,16,17,19,20,22,23,25,26,27,28,29,31,
%T 32,33,34,35,37,38,39,40,41,43,44,45,46,47,49,50,51,52,53,55,56,57,58,
%U 59,61,62,64,67,68,69,70,71,73,74,75,76,77,79,80,81,82,83
%N Heinz numbers of partitions with no adjacent parts of quotient 2.
%C Differs from A320340 in having 105: (4,3,2), 315: (4,3,2,2), 455: (6,4,3), etc.
%C The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are numbers with no adjacent prime indices of quotient 1/2.
%e The terms and their prime indices begin:
%e 1: {} 19: {8} 38: {1,8}
%e 2: {1} 20: {1,1,3} 39: {2,6}
%e 3: {2} 22: {1,5} 40: {1,1,1,3}
%e 4: {1,1} 23: {9} 41: {13}
%e 5: {3} 25: {3,3} 43: {14}
%e 7: {4} 26: {1,6} 44: {1,1,5}
%e 8: {1,1,1} 27: {2,2,2} 45: {2,2,3}
%e 9: {2,2} 28: {1,1,4} 46: {1,9}
%e 10: {1,3} 29: {10} 47: {15}
%e 11: {5} 31: {11} 49: {4,4}
%e 13: {6} 32: {1,1,1,1,1} 50: {1,3,3}
%e 14: {1,4} 33: {2,5} 51: {2,7}
%e 15: {2,3} 34: {1,7} 52: {1,1,6}
%e 16: {1,1,1,1} 35: {3,4} 53: {16}
%e 17: {7} 37: {12} 55: {3,5}
%t primeptn[n_]:=If[n==1,{},Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]];
%t Select[Range[100],And@@Table[FreeQ[Divide@@@Partition[primeptn[#],2,1],2],{i,2,PrimeOmega[#]}]&]
%Y The version with quotients >= 2 is counted by A000929, sets A018819.
%Y <= 2 is A342191, counted by A342094.
%Y < 2 is counted by A342096, sets A045690.
%Y > 2 is counted by A342098, sets A040039.
%Y The sets version (subsets of prescribed maximum) is counted by A045691.
%Y These partitions are counted by A350837.
%Y The strict case is counted by A350840.
%Y For differences instead of quotients we have A350842, strict A350844.
%Y The complement is A350845, counted by A350846.
%Y A000041 = integer partitions.
%Y A000045 = sets containing n with all differences > 2.
%Y A003114 = strict partitions with no successions, ranked by A325160.
%Y A056239 adds up prime indices, row sums of A112798, counted by A001222.
%Y A116931 = partitions with no successions, ranked by A319630.
%Y A116932 = partitions with differences != 1 or 2, strict A025157.
%Y A323092 = double-free integer partitions, ranked by A320340.
%Y A350839 = partitions with gaps and conjugate gaps, ranked by A350841.
%Y Cf. A000302, A001105, A003000, A018819, A094537, A120641, A154402, A319613, A323093, A337135, A342097, A342095.
%K nonn
%O 1,2
%A _Gus Wiseman_, Jan 18 2022
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