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A345172 Numbers whose multiset of prime factors has an alternating permutation.

%I A345172 #15 Nov 08 2021 04:24:43
%S A345172 1,2,3,5,6,7,10,11,12,13,14,15,17,18,19,20,21,22,23,26,28,29,30,31,33,
%T A345172 34,35,36,37,38,39,41,42,43,44,45,46,47,50,51,52,53,55,57,58,59,60,61,
%U A345172 62,63,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,82,83
%N A345172 Numbers whose multiset of prime factors has an alternating permutation.
%C A345172 First differs from A212167 in containing 72.
%C A345172 First differs from A335433 in lacking 270, corresponding to the partition (3,2,2,2,1).
%C A345172 A sequence is alternating if it is alternately strictly increasing and strictly decreasing, starting with either. For example, the partition (3,3,2,2,2,2,1) has no alternating permutations, even though it has the anti-run permutations (2,3,2,3,2,1,2), (2,3,2,1,2,3,2), and (2,1,2,3,2,3,2).
%C A345172 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 A345172 Complement of A001248 (squares of primes) in A344742.
%e A345172 The sequence of terms together with their prime indices begins:
%e A345172       1: {}          20: {1,1,3}       39: {2,6}
%e A345172       2: {1}         21: {2,4}         41: {13}
%e A345172       3: {2}         22: {1,5}         42: {1,2,4}
%e A345172       5: {3}         23: {9}           43: {14}
%e A345172       6: {1,2}       26: {1,6}         44: {1,1,5}
%e A345172       7: {4}         28: {1,1,4}       45: {2,2,3}
%e A345172      10: {1,3}       29: {10}          46: {1,9}
%e A345172      11: {5}         30: {1,2,3}       47: {15}
%e A345172      12: {1,1,2}     31: {11}          50: {1,3,3}
%e A345172      13: {6}         33: {2,5}         51: {2,7}
%e A345172      14: {1,4}       34: {1,7}         52: {1,1,6}
%e A345172      15: {2,3}       35: {3,4}         53: {16}
%e A345172      17: {7}         36: {1,1,2,2}     55: {3,5}
%e A345172      18: {1,2,2}     37: {12}          57: {2,8}
%e A345172      19: {8}         38: {1,8}         58: {1,10}
%t A345172 wigQ[y_]:=Length[Split[y]]== Length[y]&&Length[Split[Sign[Differences[y]]]]==Length[y]-1;
%t A345172 Select[Range[100],Select[Permutations[ Flatten[ConstantArray@@@FactorInteger[#]]],wigQ[#]&]!={}&]
%Y A345172 Including squares of primes A001248 gives A344742, counted by A344740.
%Y A345172 This is a subset of A335433, which is counted by A325534.
%Y A345172 Positions of nonzero terms in A345164.
%Y A345172 The partitions with these Heinz numbers are counted by A345170.
%Y A345172 The complement is A345171, which is counted by A345165.
%Y A345172 A345173 = A345171 /\ A335433 is counted by A345166.
%Y A345172 A000041 counts partitions of 2n with alternating sum 0, ranked by A000290.
%Y A345172 A001250 counts alternating permutations.
%Y A345172 A003242 counts anti-run compositions.
%Y A345172 A025047 counts alternating or wiggly compositions, also A025048, A025049.
%Y A345172 A325535 counts inseparable partitions, ranked by A335448.
%Y A345172 A344604 counts alternating compositions with twins.
%Y A345172 A344606 counts alternating permutations of prime indices with twins.
%Y A345172 A345192 counts non-alternating compositions.
%Y A345172 Cf. A001222, A071321, A071322, A316523, A316524, A335126, A344605, A344614, A344616, A344653, A344654, A345163, A345167, A347706, A348379.
%K A345172 nonn
%O A345172 1,2
%A A345172 _Gus Wiseman_, Jun 13 2021