cp's OEIS Frontend

A356229 Number of maximal gapless submultisets of the prime indices of 2n.

%I A356229 #12 Jan 19 2025 09:26:33
%S A356229 1,1,1,1,2,1,2,1,1,2,2,1,2,2,1,1,2,1,2,2,2,2,2,1,2,2,1,2,2,1,2,1,2,2,
%T A356229 2,1,2,2,2,2,2,2,2,2,1,2,2,1,2,2,2,2,2,1,3,2,2,2,2,1,2,2,2,1,3,2,2,2,
%U A356229 2,2,2,1,2,2,1,2,2,2,2,2,1,2,2,2,3,2,2,2,2,1,3,2,2,2,3,1,2,2,2,2,2,2,2,2,1
%N A356229 Number of maximal gapless submultisets of the prime indices of 2n.
%C A356229 A sequence is gapless if it covers an unbroken interval of positive integers. For example, the multiset {2,3,5,5,6,9} has three maximal gapless submultisets: {2,3}, {5,5,6}, {9}.
%C A356229 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 A356229 This is a bisection of A287170, but is important in its own right because the even numbers are exactly those whose prime indices begin with 1.
%H A356229 Antti Karttunen, <a href="/A356229/b356229.txt">Table of n, a(n) for n = 1..100000</a>
%F A356229 a(n) = A287170(2n).
%e A356229 The prime indices of 2*9282 are {1,1,2,4,6,7}, with maximal gapless submultisets {1,1,2}, {4}, {6,7}, so a(9282) = 3.
%t A356229 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A356229 Table[Length[Split[primeMS[2n],#1>=#2-1&]],{n,100}]
%o A356229 (PARI)
%o A356229 A287170(n) = { my(f=factor(n)); if(#f~==0, return (0), return(#f~ - sum(i=1, #f~-1, if (primepi(f[i, 1])+1 == primepi(f[i+1, 1]), 1, 0)))); };
%o A356229 A356229(n) = A287170(2*n); \\ _Antti Karttunen_, Jan 19 2025
%Y A356229 This is the even (bisected) case of A287170, firsts A066205.
%Y A356229 Alternate row-lengths of A356226, minima A356227(2n), maxima A356228(2n).
%Y A356229 A001221 counts distinct prime factors, sum A001414.
%Y A356229 A001222 counts prime indices, listed by A112798, sum A056239.
%Y A356229 A003963 multiplies together the prime indices of n.
%Y A356229 A073093 counts the prime indices of 2n.
%Y A356229 A073491 lists numbers with gapless prime indices, cf. A073492-A073495.
%Y A356229 Cf. A000005, A060680, A060681, A132747, A132881, A286470, A289509, A356230, A356231, A356232.
%K A356229 nonn
%O A356229 1,5
%A A356229 _Gus Wiseman_, Aug 16 2022
%E A356229 Data section extended to a(105) by _Antti Karttunen_, Jan 19 2025