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Odd numbers with gapless prime indices.
7

%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