cp's OEIS Frontend

A070243 a(n) = Card{ k, phi(k) <= n }.

%I A070243 #40 Apr 24 2025 01:55:07
%S A070243 2,5,5,9,9,13,13,18,18,20,20,26,26,26,26,32,32,36,36,41,41,43,43,53,
%T A070243 53,53,53,55,55,57,57,64,64,64,64,72,72,72,72,81,81,85,85,88,88,90,90,
%U A070243 101,101,101,101,103,103,105,105,108,108,110,110,119,119,119,119,127,127
%N A070243 a(n) = Card{ k, phi(k) <= n }.
%D A070243 Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 115-118.
%D A070243 Gérald Tenenbaum and Jie Wu, Exercices corrigés de théorie analytique et probabiliste des nombres, Collection SMF, Cours spécialisés, Numero 2, pp. 78-79.
%H A070243 Amiram Eldar, <a href="/A070243/b070243.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from T. D. Noe)
%H A070243 Max Alekseyev, <a href="https://oeis.org/wiki/User:Max_Alekseyev/gpscripts">PARI/GP Scripts for Miscellaneous Math Problems</a> (invphi.gp).
%H A070243 Matteo Caorsi and Sergio Cecotti, <a href="https://doi.org/10.1007/JHEP07(2018)138">Geometric classification of 4d N=2 SCFTs</a>, Journal of High Energy Physics, Vol. 2018 (2018), Article number 138; <a href="https://arxiv.org/abs/1801.04542">arXiv preprint</a>, arXiv:1801.04542 [hep-th], 2018.
%F A070243 Limit_{n->oo} a(n)/n = zeta(2)*zeta(3)/zeta(6) = 1.943596436820759205057... = A082695.
%F A070243 From _Benoit Cloitre_, Apr 12 2003: (Start)
%F A070243 a(n) = Sum_{k=1..n} A014197(k).
%F A070243 a(n) = (zeta(2)*zeta(3)/zeta(6))*n + O(n*exp(-c*sqrt(log(n)))) for a suitable constant c > 0. (End)
%o A070243 (PARI) for(n=1,150,print1(sum(i=1,100*n,if(sign(eulerphi(i)-n)+1,0,1)+if((eulerphi(i)-n),0,1)),","))
%o A070243 (PARI) list(nmax) = my(s = 0); for(n = 1, nmax, s += invphiNum(n); print1(s, ", ")); \\ _Amiram Eldar_, Dec 23 2024, using _Max Alekseyev_'s invphi.gp
%Y A070243 Partial sums of A014197.
%Y A070243 Cf. A000010, A082695.
%K A070243 easy,nonn
%O A070243 1,1
%A A070243 _Benoit Cloitre_, May 11 2002