A241663 - OEIS

%I #32 May 30 2026 16:40:42

%S 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,

%T 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,

%U 0,0,9,0,63,0,0,0,67,0,69,0,0,0,21,0,75,0,0

%N 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.

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

%H Antti Karttunen, <a href="/A241663/b241663.txt">Table of n, a(n) for n = 1..16384</a>

%H Colin Defant, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Defant/defant5.html">On Arithmetic Functions Related to Iterates of the Schemmel Totient Functions</a>, J. Int. Seq. 18 (2015), Article 15.2.1

%H Nittiya Pabhapote and Vichian Laohakosol, <a href="https://doi.org/10.1155/2010/648165">Combinatorial Aspects of the Generalized Euler's Totient</a>, International Journal of Mathematics and Mathematical Sciences, Volume 2010 (2010), Article ID 648165, 15 p.

%F Multiplicative with a(p^e) = p^(e-1)*(p-4) for p > 3. a(2^e) = a(3^e) = 0 for e > 0.

%F 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

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

%t Table[Boole[n == 1] + Count[Partition[Range@ n, 4, 1], _?(AllTrue[#, CoprimeQ[n, #] &] &)], {n, 81}] (* or *)

%t 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 *)

%o (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

%o (Scheme)

%o ;; After the given multiplicative formula. Uses memoization-macro definec:

%o (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

%Y Cf. A058026, A241666.

%K nonn,mult,changed

%O 1,7

%A _Colin Defant_, Apr 26 2014