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Numbers whose multiset of prime factors has an alternating permutation.
45

%I #15 Nov 08 2021 04:24:43

%S 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 34,35,36,37,38,39,41,42,43,44,45,46,47,50,51,52,53,55,57,58,59,60,61,

%U 62,63,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,82,83

%N Numbers whose multiset of prime factors has an alternating permutation.

%C First differs from A212167 in containing 72.

%C First differs from A335433 in lacking 270, corresponding to the partition (3,2,2,2,1).

%C 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 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 Complement of A001248 (squares of primes) in A344742.

%e The sequence of terms together with their prime indices begins:

%e 1: {} 20: {1,1,3} 39: {2,6}

%e 2: {1} 21: {2,4} 41: {13}

%e 3: {2} 22: {1,5} 42: {1,2,4}

%e 5: {3} 23: {9} 43: {14}

%e 6: {1,2} 26: {1,6} 44: {1,1,5}

%e 7: {4} 28: {1,1,4} 45: {2,2,3}

%e 10: {1,3} 29: {10} 46: {1,9}

%e 11: {5} 30: {1,2,3} 47: {15}

%e 12: {1,1,2} 31: {11} 50: {1,3,3}

%e 13: {6} 33: {2,5} 51: {2,7}

%e 14: {1,4} 34: {1,7} 52: {1,1,6}

%e 15: {2,3} 35: {3,4} 53: {16}

%e 17: {7} 36: {1,1,2,2} 55: {3,5}

%e 18: {1,2,2} 37: {12} 57: {2,8}

%e 19: {8} 38: {1,8} 58: {1,10}

%t wigQ[y_]:=Length[Split[y]]== Length[y]&&Length[Split[Sign[Differences[y]]]]==Length[y]-1;

%t Select[Range[100],Select[Permutations[ Flatten[ConstantArray@@@FactorInteger[#]]],wigQ[#]&]!={}&]

%Y Including squares of primes A001248 gives A344742, counted by A344740.

%Y This is a subset of A335433, which is counted by A325534.

%Y Positions of nonzero terms in A345164.

%Y The partitions with these Heinz numbers are counted by A345170.

%Y The complement is A345171, which is counted by A345165.

%Y A345173 = A345171 /\ A335433 is counted by A345166.

%Y A000041 counts partitions of 2n with alternating sum 0, ranked by A000290.

%Y A001250 counts alternating permutations.

%Y A003242 counts anti-run compositions.

%Y A025047 counts alternating or wiggly compositions, also A025048, A025049.

%Y A325535 counts inseparable partitions, ranked by A335448.

%Y A344604 counts alternating compositions with twins.

%Y A344606 counts alternating permutations of prime indices with twins.

%Y A345192 counts non-alternating compositions.

%Y Cf. A001222, A071321, A071322, A316523, A316524, A335126, A344605, A344614, A344616, A344653, A344654, A345163, A345167, A347706, A348379.

%K nonn

%O 1,2

%A _Gus Wiseman_, Jun 13 2021