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

1,7

Comments

a(n) is the 4th Schemmel totient function.

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.

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

Example

a(35) = a(5)*a(7) = 1*3 = 3.

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 *)

Prog

(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

Crossrefs

Cf. A058026, A241666.

Sequence in context: A160086 A115869 A115859 * A344976 A344876 A129170

Adjacent sequences: A241660 A241661 A241662 * A241664 A241665 A241666

Keyword

nonn,mult

Author

Colin Defant, Apr 26 2014

Status

approved