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 A014197 Number of numbers m with Euler phi(m) = n. 53
 2, 3, 0, 4, 0, 4, 0, 5, 0, 2, 0, 6, 0, 0, 0, 6, 0, 4, 0, 5, 0, 2, 0, 10, 0, 0, 0, 2, 0, 2, 0, 7, 0, 0, 0, 8, 0, 0, 0, 9, 0, 4, 0, 3, 0, 2, 0, 11, 0, 0, 0, 2, 0, 2, 0, 3, 0, 2, 0, 9, 0, 0, 0, 8, 0, 2, 0, 0, 0, 2, 0, 17, 0, 0, 0, 0, 0, 2, 0, 10, 0, 2, 0, 6, 0, 0, 0, 6, 0, 0, 0, 3 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS Carmichael conjectured that there are no 1's in this sequence. - Jud McCranie, Oct 10 2000 Number of cyclotomic polynomials of degree n. - T. D. Noe, Aug 15 2003 Let v == 0 (mod 24), w = v + 24, and v < k < q < w, where k and q are integer. It seems that, for most values of v, there is no b such that b = a(k) + a(q) and b > a(v) + a(w). The first case where b > a(v) + a(w) occurs at v = 888: b = a(896) + a(900) = 15 + 4, b > a(888) + a(912), or 19 > 8 + 7. The first case where v < n < w and a(n) > a(v) + a(w) occurs at v = 2232: a(2240) > a(2232) + a(2256), or 27 > 7 + 8. - Sergey Pavlov, Feb 05 2017 One elementary result relating to phi(m) is that if m is odd, then phi(m)=phi(2m) because 1 and 2 both have phi value 1 and phi is multiplicative. - Roderick MacPhee, Jun 03 2017 REFERENCES R. K. Guy, Unsolved Problems in Number Theory, section B39. J. Roberts, Lure of The Integers, entry 32, page 182. LINKS T. D. Noe, Table of n, a(n) for n = 1..10000 Max A. Alekseyev, Computing the Inverses, their Power Sums, and Extrema for Euler's Totient and Other Multiplicative Functions. Journal of Integer Sequences, Vol. 19 (2016), Article 16.5.2 K. Ford, The number of solutions of phi(x)=m, arXiv:math/9907204 [math.NT], 1999. S. Sivasankaranarayana Pillai, On some functions connected with phi(n), Bull. Amer. Math. Soc. 35 (1929), 832-836. Eric Weisstein's World of Mathematics, Totient Function Eric Weisstein's World of Mathematics, Totient Valence Function FORMULA Dirichlet g.f.: Sum_{n>=1} a(n)*n^-s = zeta(s)*Product_(1+1/(p-1)^s-1/p^s). - Benoit Cloitre, Apr 12 2003 Lim_{n->infinity} (1/n) * Sum_{k=1..n} a(k) = zeta(2)*zeta(3)/zeta(6) = 1.94359643682075920505707036... (see A082695). - Benoit Cloitre, Apr 12 2003 From Christopher J. Smyth, Jan 08 2017: (Start) Euler transform = Product_{n>=1} (1-x^n)^(-a(n)) = g.f. of A120963. Product_{n>=1} (1+x^n)^a(n) = Product_{n>=1} ((1-x^(2n))/(1-x^n))^a(n) = Product_{n>=1} (1-x^n)^(-A280712(n)) = Euler transform of A280712 = g.f. of A280611. (End) a(A000010(n)) = A066412(n). - Antti Karttunen, Jul 18 2017 From Antti Karttunen, Dec 04 2018: (Start) a(A000079(n)) = A058321(n). a(A000142(n)) = A055506(n). a(A017545(n)) = A063667(n). a(n) = Sum_{d|n} A008683(n/d)*A070633(d). a(n) = A056239(A322310(n)). (End) MAPLE with(numtheory): A014197:=n-> nops(invphi(n)): seq(A014197(n), n=1..200); MATHEMATICA a = 2; a[m_?OddQ] = 0; a[m_] := Module[{p, nmax, n, k}, p = Select[ Divisors[m]+1, PrimeQ]; nmax = m*Times @@ (p/(p - 1)); n = m; k = 0; While[n <= nmax, If[EulerPhi[n] == m, k++]; n++]; k]; Array[a, 92] (* Jean-François Alcover, Dec 09 2011, updated Apr 25 2016 *) With[{nn = 116}, Function[s, Function[t, Take[#, nn] &@ ReplacePart[t, Map[# -> Length@ Lookup[s, #] &, Keys@ s]]]@ ConstantArray[0, Max@ Keys@ s]]@ KeySort@ PositionIndex@ Array[EulerPhi, Floor[nn^(3/2)] + 10]] (* Michael De Vlieger, Jul 19 2017 *) PROG (PARI) A014197(n, m=1) = { n==1 && return(1+(m<2)); my(p, q); sumdiv(n, d, if( d>=m && isprime(d+1), sum( i=0, valuation(q=n\d, p=d+1), A014197(q\p^i, p))))} \\ M. F. Hasler, Oct 05 2009 (Python) from sympy import totient, divisors, isprime from operator import mul def a(m):     if m==1: return 2     if m%2==1: return 0     X=[x + 1 for x in divisors(m)]     p=[i for i in X if isprime(i)]     nmax=m*reduce(mul, [i/(i - 1) for i in p])     n=m     k=0     while n<=nmax:         if totient(n)==m:k+=1         n+=1     return k print map(a, xrange(1, 101)) # Indranil Ghosh, Jul 18 2017, after Mathematica code (MAGMA) [#EulerPhiInverse(n): n in [1..100]]; // Marius A. Burtea, Sep 08 2019 CROSSREFS Cf. A000010, A002202, A032446 (bisection), A049283, A051894, A055506, A057635, A057826, A058277 (nonzero terms), A058341, A063439, A066412, A070243 (partial sums), A070633, A071386 (positions of odd terms), A071387, A071388 (positions of primes), A071389 (where prime(n) occurs for the first time), A082695, A097942 (positions of records), A097946, A120963, A134269, A219930, A280611, A280709, A280712, A296655 (positions of positive even terms), A305353, A305656, A319048, A322019. For records see A131934. Column 1 of array A320000. Sequence in context: A164917 A166238 A293275 * A181308 A292246 A277141 Adjacent sequences:  A014194 A014195 A014196 * A014198 A014199 A014200 KEYWORD nonn,nice,easy AUTHOR STATUS approved

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Last modified October 15 20:04 EDT 2019. Contains 328037 sequences. (Running on oeis4.)