A241663 Number of positive integers k less than or equal to n such that gcd(k,n) = gcd(k+1,n) = gcd(k+2,n) = gcd(k+3,n) = 1.
1, 0, 0, 0, 1, 0, 3, 0, 0, 0, 7, 0, 9, 0, 0, 0, 13, 0, 15, 0, 0, 0, 19, 0, 5, 0, 0, 0, 25, 0, 27, 0, 0, 0, 3, 0, 33, 0, 0, 0, 37, 0, 39, 0, 0, 0, 43, 0, 21, 0, 0, 0, 49, 0, 7, 0, 0, 0, 55, 0, 57, 0, 0, 0, 9, 0, 63, 0, 0, 0, 67, 0, 69, 0, 0, 0, 21, 0, 75, 0, 0
Offset: 1
Examples
a(35) = a(5)*a(7) = 1*3 = 3.
Links
- Antti Karttunen, Table of n, a(n) for n = 1..16384
- Colin Defant, On Arithmetic Functions Related to Iterates of the Schemmel Totient Functions, J. Int. Seq. 18 (2015), Article 15.2.1
- Nittiya Pabhapote and Vichian Laohakosol, Combinatorial Aspects of the Generalized Euler's Totient, International Journal of Mathematics and Mathematical Sciences, Volume 2010 (2010), Article ID 648165, 15 p.
Programs
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Mathematica
Table[Boole[n == 1] + Count[Partition[Range@ n, 4, 1], _?(AllTrue[#, CoprimeQ[n, #] &] &)], {n, 81}] (* or *) Array[If[# == 1, 1, Apply[Times, FactorInteger[#] /. {p_, e_} /; p > 1 :> If[p > 3, (p - 4) p^(e - 1), 0]]] &, 81] (* Michael De Vlieger, Nov 05 2017 *) -
PARI
a(n) = {my(f = factor(n)); prod(i=1, #f~, if ((f[i, 1] == 2) || (f[i, 1] == 3), 0, f[i, 1]^(f[i, 2]-1)*(f[i, 1]-4)));} \\ Michel Marcus, May 01 2014 -
Scheme
;; After the given multiplicative formula. Uses memoization-macro definec: (definec (A241663 n) (if (= 1 n) n (let ((p (A020639 n))) (if (<= p 3) 0 (* (- p 4) (expt p (- (A067029 n) 1)) (A241663 (A028234 n))))))) ;; Antti Karttunen, Nov 05 2017
Formula
Multiplicative with a(p^e) = p^(e-1)*(p-4) for p > 3. a(2^e) = a(3^e) = 0 for e > 0.
Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/6) * Product_{p prime >= 5} (1 - 4/p^2) = 0.11357982182683545733... . - Amiram Eldar, Oct 13 2022
Comments