cp's OEIS Frontend

A356845 Odd numbers with gapless prime indices.

%I A356845 #7 Sep 03 2022 12:19:58
%S A356845 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 A356845 61,67,71,73,75,77,79,81,83,89,97,101,103,105,107,109,113,121,125,127,
%U A356845 131,135,137,139,143,149,151,157,163,167,169,173,175,179,181,191
%N A356845 Odd numbers with gapless prime indices.
%C A356845 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 A356845 A sequence is gapless if it covers an interval of positive integers.
%e A356845 The terms together with their prime indices begin:
%e A356845     1: {}
%e A356845     3: {2}
%e A356845     5: {3}
%e A356845     7: {4}
%e A356845     9: {2,2}
%e A356845    11: {5}
%e A356845    13: {6}
%e A356845    15: {2,3}
%e A356845    17: {7}
%e A356845    19: {8}
%e A356845    23: {9}
%e A356845    25: {3,3}
%e A356845    27: {2,2,2}
%e A356845    29: {10}
%e A356845    31: {11}
%e A356845    35: {3,4}
%e A356845    37: {12}
%e A356845    41: {13}
%e A356845    43: {14}
%t A356845 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A356845 nogapQ[m_]:=Or[m=={},Union[m]==Range[Min[m],Max[m]]];
%t A356845 Select[Range[1,100,2],nogapQ[primeMS[#]]&]
%Y A356845 Consists of the odd terms of A073491.
%Y A356845 These partitions are counted by A264396.
%Y A356845 The strict case is A294674, counted by A136107.
%Y A356845 The version for compositions is A356843, counted by A251729.
%Y A356845 A001221 counts distinct prime factors, sum A001414.
%Y A356845 A056239 adds up prime indices, row sums of A112798, lengths A001222.
%Y A356845 A356069 counts gapless divisors, initial A356224 (complement A356225).
%Y A356845 A356230 ranks gapless factorization lengths, firsts A356603.
%Y A356845 A356233 counts factorizations into gapless numbers.
%Y A356845 Cf. A003963, A034296, A055932, A073493, A107428, A287170, A289508, A325160, A356231, A356234, A356841.
%K A356845 nonn
%O A356845 1,2
%A A356845 _Gus Wiseman_, Sep 03 2022