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A350838 Heinz numbers of partitions with no adjacent parts of quotient 2.

%I A350838 #12 Jan 27 2022 20:46:12
%S A350838 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 A350838 32,33,34,35,37,38,39,40,41,43,44,45,46,47,49,50,51,52,53,55,56,57,58,
%U A350838 59,61,62,64,67,68,69,70,71,73,74,75,76,77,79,80,81,82,83
%N A350838 Heinz numbers of partitions with no adjacent parts of quotient 2.
%C A350838 Differs from A320340 in having 105: (4,3,2), 315: (4,3,2,2), 455: (6,4,3), etc.
%C A350838 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 A350838 The terms and their prime indices begin:
%e A350838       1: {}            19: {8}             38: {1,8}
%e A350838       2: {1}           20: {1,1,3}         39: {2,6}
%e A350838       3: {2}           22: {1,5}           40: {1,1,1,3}
%e A350838       4: {1,1}         23: {9}             41: {13}
%e A350838       5: {3}           25: {3,3}           43: {14}
%e A350838       7: {4}           26: {1,6}           44: {1,1,5}
%e A350838       8: {1,1,1}       27: {2,2,2}         45: {2,2,3}
%e A350838       9: {2,2}         28: {1,1,4}         46: {1,9}
%e A350838      10: {1,3}         29: {10}            47: {15}
%e A350838      11: {5}           31: {11}            49: {4,4}
%e A350838      13: {6}           32: {1,1,1,1,1}     50: {1,3,3}
%e A350838      14: {1,4}         33: {2,5}           51: {2,7}
%e A350838      15: {2,3}         34: {1,7}           52: {1,1,6}
%e A350838      16: {1,1,1,1}     35: {3,4}           53: {16}
%e A350838      17: {7}           37: {12}            55: {3,5}
%t A350838 primeptn[n_]:=If[n==1,{},Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]];
%t A350838 Select[Range[100],And@@Table[FreeQ[Divide@@@Partition[primeptn[#],2,1],2],{i,2,PrimeOmega[#]}]&]
%Y A350838 The version with quotients >= 2 is counted by A000929, sets A018819.
%Y A350838                            <= 2 is A342191, counted by A342094.
%Y A350838                            < 2 is counted by A342096, sets A045690.
%Y A350838                            > 2 is counted by A342098, sets A040039.
%Y A350838 The sets version (subsets of prescribed maximum) is counted by A045691.
%Y A350838 These partitions are counted by A350837.
%Y A350838 The strict case is counted by A350840.
%Y A350838 For differences instead of quotients we have A350842, strict A350844.
%Y A350838 The complement is A350845, counted by A350846.
%Y A350838 A000041 = integer partitions.
%Y A350838 A000045 = sets containing n with all differences > 2.
%Y A350838 A003114 = strict partitions with no successions, ranked by A325160.
%Y A350838 A056239 adds up prime indices, row sums of A112798, counted by A001222.
%Y A350838 A116931 = partitions with no successions, ranked by A319630.
%Y A350838 A116932 = partitions with differences != 1 or 2, strict A025157.
%Y A350838 A323092 = double-free integer partitions, ranked by A320340.
%Y A350838 A350839 = partitions with gaps and conjugate gaps, ranked by A350841.
%Y A350838 Cf. A000302, A001105, A003000, A018819, A094537, A120641, A154402, A319613, A323093, A337135, A342097, A342095.
%K A350838 nonn
%O A350838 1,2
%A A350838 _Gus Wiseman_, Jan 18 2022