cp's OEIS Frontend

A375930 Numbers k such that A005117(k+1) - A005117(k) > 1. In other words, the k-th squarefree number is more than 1 less than the next.

%I A375930 #15 Sep 15 2024 14:40:53
%S A375930 3,6,8,11,12,13,16,17,20,23,26,29,31,32,33,34,37,39,42,45,47,50,52,55,
%T A375930 56,57,60,61,64,67,70,73,75,77,78,81,83,86,89,91,92,93,95,98,99,100,
%U A375930 103,104,106,109,112,115,117,120,121,122,125,127,130,133,136,139
%N A375930 Numbers k such that A005117(k+1) - A005117(k) > 1. In other words, the k-th squarefree number is more than 1 less than the next.
%C A375930 The asymptotic density of this sequence is 1 - Product_{p prime} (1 - 1/(p^2-1)) = 1 - A065469 = 0.46928817... . - _Amiram Eldar_, Sep 15 2024
%H A375930 Amiram Eldar, <a href="/A375930/b375930.txt">Table of n, a(n) for n = 1..10000</a>
%e A375930 The squarefree numbers are 1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 30, ... which first increase by more than one after positions 3, 6, 8, 11, ...
%t A375930 Join@@Position[Differences[Select[Range[100],SquareFreeQ[#]&]],_?(#>1&)]
%o A375930 (PARI) lista(kmax) = {my(is1 = 1, is2, c = 1); for(k = 2, kmax, is2 = issquarefree(k); if(is2, c++); if(is1 && !is2, print1(c, ", ")); is1 = is2);} \\ _Amiram Eldar_, Sep 15 2024
%Y A375930 For nonprime numbers: A014689, complement A375926, differences A373403.
%Y A375930 For composite numbers: A065890 shifted, complement A375929.
%Y A375930 Positions of terms > 1 in A076259.
%Y A375930 First differences are A120992, complement A373127.
%Y A375930 The complement is A375927.
%Y A375930 A005117 lists the squarefree numbers, first differences A076259.
%Y A375930 A013929 lists the nonsquarefree numbers, first differences A078147.
%Y A375930 A053797 gives lengths of runs of nonsquarefree numbers, firsts A373199.
%Y A375930 Cf. A007674, A061399, A065469, A068781, A072284, A110969, A294242, A375709.
%K A375930 nonn
%O A375930 1,1
%A A375930 _Gus Wiseman_, Sep 12 2024