cp's OEIS Frontend

A356069 Number of divisors of n whose prime indices cover an interval of positive integers (A073491).

%I A356069 #8 Aug 30 2022 09:41:22
%S A356069 1,2,2,3,2,4,2,4,3,3,2,6,2,3,4,5,2,6,2,4,3,3,2,8,3,3,4,4,2,7,2,6,3,3,
%T A356069 4,9,2,3,3,5,2,5,2,4,6,3,2,10,3,4,3,4,2,8,3,5,3,3,2,10,2,3,4,7,3,5,2,
%U A356069 4,3,5,2,12,2,3,6,4,4,5,2,6,5,3,2,7,3,3
%N A356069 Number of divisors of n whose prime indices cover an interval of positive integers (A073491).
%C A356069 First differs from A000005 at 10, 14, 20, 21, 22, ... = A307516.
%C A356069 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.
%e A356069 The a(n) counted divisors of n = 1, 2, 4, 6, 12, 16, 24, 30, 36, 48, 72, 90:
%e A356069   1   2   4   6  12  16  24  30  36  48  72  90
%e A356069       1   2   3   6   8  12  15  18  24  36  45
%e A356069           1   2   4   4   8   6  12  16  24  30
%e A356069               1   3   2   6   5   9  12  18  18
%e A356069                   2   1   4   3   6   8  12  15
%e A356069                   1       3   2   4   6   9   9
%e A356069                           2   1   3   4   8   6
%e A356069                           1       2   3   6   5
%e A356069                                   1   2   4   3
%e A356069                                       1   3   2
%e A356069                                           2   1
%e A356069                                           1
%t A356069 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A356069 nogapQ[m_]:=m=={}||Union[m]==Range[Min[m],Max[m]];
%t A356069 Table[Length[Select[Divisors[n],nogapQ[primeMS[#]]&]],{n,100}]
%Y A356069 These divisors belong to A073491, a superset of A055932, complement A073492.
%Y A356069 The initial case is A356224.
%Y A356069 The complement in the initial case is counted by A356225.
%Y A356069 A000005 counts divisors.
%Y A356069 A001223 lists the prime gaps.
%Y A356069 A056239 adds up prime indices, row sums of A112798, lengths A001222.
%Y A356069 A328338 has third-largest divisor prime.
%Y A356069 A356226 gives the lengths of maximal gapless intervals of prime indices.
%Y A356069 Cf. A028334, A029709, A055874, A070824, A119313, A137921, A287170, A307516, A356223, A356233-A356237.
%K A356069 nonn
%O A356069 1,2
%A A356069 _Gus Wiseman_, Aug 28 2022