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A120992 Number of integers in n-th run of squarefree positive integers.

%I A120992 #35 Feb 28 2025 12:06:08
%S A120992 3,3,2,3,1,1,3,1,3,3,3,3,2,1,1,1,3,2,3,3,2,3,2,3,1,1,3,1,3,3,3,3,2,2,
%T A120992 1,3,2,3,3,2,1,1,2,3,1,1,3,1,2,3,3,3,2,3,1,1,3,2,3,3,3,3,2,3,1,1,3,1,
%U A120992 2,1,1,3,3,2,3,1,1,2,2,3,3,2,1,1,2,3,1
%N A120992 Number of integers in n-th run of squarefree positive integers.
%C A120992 The values 1, 2 and 3 occur 309008, 251134 and 439858 times, respectively, in the first 1000000 terms. - _Rick L. Shepherd_, Jul 25 2006
%C A120992 From _Reinhard Zumkeller_, Jan 20 2008: (Start)
%C A120992 1 <= a(n) <= 3.
%C A120992 A136742(n) = Product{k=0..a(n)} (A072284(n)+k).
%C A120992 A136743(n) = Sum_{k=0..a(n)} A001221(A072284(n)+k).
%C A120992 (End)
%C A120992 Also the lengths of runs in A243348, differences of the n-th squarefree number and n. - _Antti Karttunen_, Jun 06 2014
%H A120992 A. Karttunen & R. Zumkeller (the first 1000 terms), <a href="/A120992/b120992.txt">Table of n, a(n) for n = 1..10000</a>
%H A120992 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Squarefree.html">Squarefree</a>
%e A120992 The runs of squarefree integers are as follows: (1,2,3), (5,6,7), (10,11), (13,14,15), (17), (19), (21,22,23),...
%p A120992 with(numtheory): a:=proc(n) if mobius(n)=0 then n else fi end: A:=[0,seq(a(n),n=1..500)]: b:=proc(n) if A[n]-A[n-1]>1 then A[n]-A[n-1]-1 else fi end: seq(b(n),n=2..nops(A)); # _Emeric Deutsch_, Jul 24 2006
%t A120992 t = {}; cnt = 0; Do[If[SquareFreeQ[n], cnt++, If[cnt > 0, AppendTo[t, cnt]; cnt = 0]], {n, 500}]; t (* _T. D. Noe_, Mar 19 2013 *)
%o A120992 (PARI) n=1; while(n<1000, c=0; while(issquarefree(n), n++; c++); print1(c,", "); while(!issquarefree(n), n++)) \\ _Rick L. Shepherd_, Jul 25 2006
%o A120992 (Scheme)
%o A120992 ;; With _Antti Karttunen_'s IntSeq-library.
%o A120992 (define (A120992 n) (if (= n 1) (Aincr_points_of_A243348 n) (- (Aincr_points_of_A243348 n) (Aincr_points_of_A243348 (- n 1)))))
%o A120992 ;; Using these two auxiliary functions, not submitted separately:
%o A120992 (define Aincr_points_of_A243348 (COMPOSE -1+ (NONZERO-POS 1 1 Afirst_diffs_of_A243348)))
%o A120992 (define (Afirst_diffs_of_A243348 n) (if (< n 2) (- n 1) (- (A243348 n) (A243348 (- n 1)))))
%Y A120992 Cf. A005117, A076259, A013928, A013929, A053797, A053806, A120993, A243348.
%K A120992 nonn
%O A120992 1,1
%A A120992 _Leroy Quet_, Jul 21 2006
%E A120992 More terms from _Emeric Deutsch_ and _Rick L. Shepherd_, Jul 25 2006