cp's OEIS Frontend

A349794 Numbers whose prime signature has an odd term other than the first or last.

%I A349794 #9 Dec 25 2021 02:44:45
%S A349794 30,42,60,66,70,78,84,102,105,110,114,120,130,132,138,140,150,154,156,
%T A349794 165,168,170,174,182,186,190,195,204,210,220,222,228,230,231,238,240,
%U A349794 246,255,258,260,264,266,270,273,276,280,282,285,286,290,294,300,308
%N A349794 Numbers whose prime signature has an odd term other than the first or last.
%C A349794 A number's prime signature (row n of A124010) is the sequence of positive exponents in its prime factorization.
%C A349794 Also numbers whose multiset of prime factors is not weakly alternating, where we define a sequence to be weakly alternating if it is alternately weakly increasing and weakly decreasing, starting with either. This sequence looks at the somewhat degenerate case where no strict decreases are allowed.
%e A349794 The terms and their prime indices begin:
%e A349794    30: {1,2,3}
%e A349794    42: {1,2,4}
%e A349794    60: {1,1,2,3}
%e A349794    66: {1,2,5}
%e A349794    70: {1,3,4}
%e A349794    78: {1,2,6}
%e A349794    84: {1,1,2,4}
%e A349794   102: {1,2,7}
%e A349794   105: {2,3,4}
%e A349794   110: {1,3,5}
%e A349794   114: {1,2,8}
%e A349794   120: {1,1,1,2,3}
%e A349794   130: {1,3,6}
%e A349794   132: {1,1,2,5}
%e A349794   138: {1,2,9}
%t A349794 Select[Range[100],PrimeNu[#]>1&&!And@@EvenQ/@Take[Last/@FactorInteger[#],{2,-2}]&]
%Y A349794 The complement for compositions is A025047, ranked by A345167.
%Y A349794 Signatures of this type are counted by A274230, complement A027383.
%Y A349794 The strong case is A289553, complement A167171.
%Y A349794 The strong case for compositions is A345192, ranked by A345168.
%Y A349794 The version for compositions is A349053, ranked by A349057.
%Y A349794 These partitions are counted by A349061, complement A349060, strong A349801.
%Y A349794 The non-strict case is counted by A349795.
%Y A349794 A001250 counts alternating permutations, complement A348615.
%Y A349794 A096441 counts weakly alternating partitions if 0 is appended.
%Y A349794 A345164 counts alternating permutations of prime indices, weak A349056.
%Y A349794 A345170 counts partitions w/ an alternating permutation, ranked by A345172.
%Y A349794 A349052 counts weakly alternating compositions.
%Y A349794 A349059 counts weakly alternating ordered factorizations, strong A348610.
%Y A349794 Cf. A003242, A112798, A114901, A117298, A128761, A129852, A129853, A344614, A344615, A345165, A349796, A349798.
%K A349794 nonn
%O A349794 1,1
%A A349794 _Gus Wiseman_, Dec 06 2021