cp's OEIS Frontend

A345173 Numbers whose multiset of prime factors is separable but has no alternating permutation.

%I A345173 #12 Nov 05 2021 22:20:16
%S A345173 270,378,594,702,918,1026,1242,1566,1620,1674,1750,1998,2214,2268,
%T A345173 2322,2538,2625,2750,2862,3186,3250,3294,3564,3618,3834,3942,4050,
%U A345173 4125,4212,4250,4266,4482,4750,4806,4875,5238,5454,5508,5562,5670,5750,5778,5886,6102
%N A345173 Numbers whose multiset of prime factors is separable but has no alternating permutation.
%C A345173 A multiset is separable if it has an anti-run permutation (no adjacent parts equal).
%C A345173 A sequence is alternating if it is alternately strictly increasing and strictly decreasing, starting with either. For example, the partition (3,2,2,2,1) has no alternating permutations, even though it has the anti-run permutations (2,3,2,1,2) and (2,1,2,3,2).
%C A345173 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.
%F A345173 Equals A345171 /\ A335433.
%e A345173 The terms together with their prime indices begin:
%e A345173    270: {1,2,2,2,3}
%e A345173    378: {1,2,2,2,4}
%e A345173    594: {1,2,2,2,5}
%e A345173    702: {1,2,2,2,6}
%e A345173    918: {1,2,2,2,7}
%e A345173   1026: {1,2,2,2,8}
%e A345173   1242: {1,2,2,2,9}
%e A345173   1566: {1,2,2,2,10}
%e A345173   1620: {1,1,2,2,2,2,3}
%e A345173   1674: {1,2,2,2,11}
%e A345173   1750: {1,3,3,3,4}
%e A345173   1998: {1,2,2,2,12}
%e A345173   2214: {1,2,2,2,13}
%e A345173   2268: {1,1,2,2,2,2,4}
%e A345173   2322: {1,2,2,2,14}
%t A345173 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A345173 wigQ[y_]:=Or[Length[y]==0,Length[Split[y]]== Length[y]&&Length[Split[Sign[Differences[y]]]]==Length[y]-1];
%t A345173 sepQ[y_]:=!MatchQ[y,{___,x_,x_,___}];
%t A345173 Select[Range[1000],Select[Permutations[primeMS[#]],wigQ]=={}&&!Select[Permutations[primeMS[#]],sepQ]=={}&]
%Y A345173 The partitions with these Heinz numbers are counted by A345166.
%Y A345173 Permutations of this type are ranked by A345169.
%Y A345173 Numbers with a factorization of this type are counted by A348609.
%Y A345173 A000041 counts integer partitions.
%Y A345173 A001250 counts alternating permutations, complement A348615.
%Y A345173 A003242 counts anti-run compositions.
%Y A345173 A025047 counts alternating compositions, ascend A025048, descend A025049.
%Y A345173 A325534 counts separable partitions, ranked by A335433.
%Y A345173 A325535 counts inseparable partitions, ranked by A335448.
%Y A345173 A344606 counts alternating permutations of prime indices with twins.
%Y A345173 A344740 counts twins and partitions with an alternating permutation.
%Y A345173 A345164 counts alternating permutations of prime factors.
%Y A345173 A345165 counts partitions without an alternating permutation.
%Y A345173 A345170 counts partitions with an alternating permutation.
%Y A345173 A345192 counts non-alternating compositions, without twins A348377.
%Y A345173 A348379 counts factorizations with an alternating permutation.
%Y A345173 Cf. A001222, A071321, A316524, A335126, A344614, A344616, A344652, A344653, A345163, A345168, A345193, A347706, A348380, A348613.
%K A345173 nonn
%O A345173 1,1
%A A345173 _Gus Wiseman_, Jun 13 2021