cp's OEIS Frontend

A345193 Heinz numbers of non-twin (x,x) inseparable partitions.

%I A345193 #7 Jun 18 2021 01:13:53
%S A345193 8,16,24,27,32,40,48,54,56,64,80,81,88,96,104,112,125,128,135,136,144,
%T A345193 152,160,162,176,184,189,192,208,224,232,240,243,248,250,256,272,288,
%U A345193 296,297,304,320,324,328,336,343,344,351,352,368,375,376,384,400,405
%N A345193 Heinz numbers of non-twin (x,x) inseparable partitions.
%C A345193 The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
%C A345193 A multiset is separable if it has an anti-run permutation (no adjacent parts equal). This is equivalent to having maximal multiplicity greater than one plus the sum of the remaining multiplicities. For example, the partition (3,2,2,2,1) has the anti-run permutations (2,3,2,1,2) and (2,1,2,3,2), so is separable.
%F A345193 Complement of A001248 in A335448.
%e A345193 The sequence of terms together with their prime indices begins:
%e A345193       8: {1,1,1}          112: {1,1,1,1,4}        232: {1,1,1,10}
%e A345193      16: {1,1,1,1}        125: {3,3,3}            240: {1,1,1,1,2,3}
%e A345193      24: {1,1,1,2}        128: {1,1,1,1,1,1,1}    243: {2,2,2,2,2}
%e A345193      27: {2,2,2}          135: {2,2,2,3}          248: {1,1,1,11}
%e A345193      32: {1,1,1,1,1}      136: {1,1,1,7}          250: {1,3,3,3}
%e A345193      40: {1,1,1,3}        144: {1,1,1,1,2,2}      256: {1,1,1,1,1,1,1,1}
%e A345193      48: {1,1,1,1,2}      152: {1,1,1,8}          272: {1,1,1,1,7}
%e A345193      54: {1,2,2,2}        160: {1,1,1,1,1,3}      288: {1,1,1,1,1,2,2}
%e A345193      56: {1,1,1,4}        162: {1,2,2,2,2}        296: {1,1,1,12}
%e A345193      64: {1,1,1,1,1,1}    176: {1,1,1,1,5}        297: {2,2,2,5}
%e A345193      80: {1,1,1,1,3}      184: {1,1,1,9}          304: {1,1,1,1,8}
%e A345193      81: {2,2,2,2}        189: {2,2,2,4}          320: {1,1,1,1,1,1,3}
%e A345193      88: {1,1,1,5}        192: {1,1,1,1,1,1,2}    324: {1,1,2,2,2,2}
%e A345193      96: {1,1,1,1,1,2}    208: {1,1,1,1,6}        328: {1,1,1,13}
%e A345193     104: {1,1,1,6}        224: {1,1,1,1,1,4}      336: {1,1,1,1,2,4}
%t A345193 Select[Range[100],!MatchQ[FactorInteger[#],{{_,2}}]&&Select[Permutations[Flatten[ConstantArray@@@FactorInteger[#]]],!MatchQ[#,{___,x_,x_,___}]&]=={}&]
%Y A345193 A000041 counts integer partitions.
%Y A345193 A001248 lists Heinz numbers of twins (x,x).
%Y A345193 A001250 counts wiggly permutations.
%Y A345193 A003242 counts anti-run compositions.
%Y A345193 A025047 counts wiggly compositions (ascend: A025048, descend: A025049).
%Y A345193 A056239 adds up prime indices,  row sums of A112798.
%Y A345193 A325534 counts separable partitions, ranked by A335433.
%Y A345193 A325535 counts inseparable partitions, ranked by A335448.
%Y A345193 A344740 counts twins and partitions w/ wiggly permutation, rank: A344742.
%Y A345193 A345164 counts wiggly permutations of prime indices (with twins: A344606).
%Y A345193 A345165 counts partitions without a wiggly permutation, ranked by A345171.
%Y A345193 A345170 counts partitions with a wiggly permutation, ranked by A345172.
%Y A345193 A345192 counts non-wiggly compositions.
%Y A345193 Cf. A001222, A071321, A316523, A316524, A335126, A344614, A344616, A344652, A344654, A345169, A345173.
%K A345193 nonn
%O A345193 1,1
%A A345193 _Gus Wiseman_, Jun 17 2021