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%I #7 Sep 03 2022 12:19:58
%S 1,3,5,7,9,11,13,15,17,19,23,25,27,29,31,35,37,41,43,45,47,49,53,59,
%T 61,67,71,73,75,77,79,81,83,89,97,101,103,105,107,109,113,121,125,127,
%U 131,135,137,139,143,149,151,157,163,167,169,173,175,179,181,191
%N Odd numbers with gapless prime indices.
%C A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
%C A sequence is gapless if it covers an interval of positive integers.
%e The terms together with their prime indices begin:
%e 1: {}
%e 3: {2}
%e 5: {3}
%e 7: {4}
%e 9: {2,2}
%e 11: {5}
%e 13: {6}
%e 15: {2,3}
%e 17: {7}
%e 19: {8}
%e 23: {9}
%e 25: {3,3}
%e 27: {2,2,2}
%e 29: {10}
%e 31: {11}
%e 35: {3,4}
%e 37: {12}
%e 41: {13}
%e 43: {14}
%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t nogapQ[m_]:=Or[m=={},Union[m]==Range[Min[m],Max[m]]];
%t Select[Range[1,100,2],nogapQ[primeMS[#]]&]
%Y Consists of the odd terms of A073491.
%Y These partitions are counted by A264396.
%Y The strict case is A294674, counted by A136107.
%Y The version for compositions is A356843, counted by A251729.
%Y A001221 counts distinct prime factors, sum A001414.
%Y A056239 adds up prime indices, row sums of A112798, lengths A001222.
%Y A356069 counts gapless divisors, initial A356224 (complement A356225).
%Y A356230 ranks gapless factorization lengths, firsts A356603.
%Y A356233 counts factorizations into gapless numbers.
%Y Cf. A003963, A034296, A055932, A073493, A107428, A287170, A289508, A325160, A356231, A356234, A356841.
%K nonn
%O 1,2
%A _Gus Wiseman_, Sep 03 2022
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