A143510 -id:A143510 - cp's OEIS frontend

A061026 Smallest number m such that phi(m) is divisible by n, where phi = Euler totient function A000010.

1, 3, 7, 5, 11, 7, 29, 15, 19, 11, 23, 13, 53, 29, 31, 17, 103, 19, 191, 25, 43, 23, 47, 35, 101, 53, 81, 29, 59, 31, 311, 51, 67, 103, 71, 37, 149, 191, 79, 41, 83, 43, 173, 69, 181, 47, 283, 65, 197, 101, 103, 53, 107, 81, 121, 87, 229, 59, 709, 61, 367, 311, 127, 85

Offset: 1

Melvin J. Knight (knightmj(AT)juno.com), May 25 2001

nonn

Conjecture: a(n) is odd for all n. Verified up to n <= 3*10^5. - Jianing Song, Feb 21 2021

The conjecture above is false because a(16842752) = 33817088; see A002181 and A143510. - Flávio V. Fernandes, Oct 08 2023

			a(48) = 65 because phi(65) = phi(5)*phi(13) = 4*12 = 48 and no smaller integer m has phi(m) divisible by 48.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
Ho-joo Lee and Gerald Myerson, Consecutive Integers Whose Totients Are Multiples of n: 10837, The American Mathematical Monthly, Vol. 110, No. 2 (Feb., 2003), pp. 158-159.
Pieter Moree and Hans Roskam, On an arithmetical function related to Euler's totient and the discriminator, Fib. Quart., Vol. 33, No. 4 (1995), pp. 332-340.
József Sándor, On the Euler minimum and maximum functions, Notes on Number Theory and Discrete Mathematics, Volume 15, 2009, Number 3, pp. 1-8.

Cf. A000010, A066674, A066675, A066676, A066678, A067005.
Cf. A233516, A233517 (records).
Cf. A005179 (analog for number of divisors), A070982 (analog for sum of divisors).

a = ConstantArray[1, 64]; k = 1; While[Length[vac = Rest[Flatten[Position[a, 1]]]] > 0, k++; a[[Intersection[Divisors[EulerPhi[k]], vac]]] *= k]; a  (* Ivan Neretin, May 15 2015 *)

a(n) = my(s=1); while(eulerphi(s)%n, s++); s;
vector(100, n, a(n))

from sympy import totient as phi
def a(n):
  k = 1
  while phi(k)%n != 0: k += 1
  return k
print([a(n) for n in range(1, 65)]) # Michael S. Branicky, Feb 21 2021

Sequence is unbounded; a(n) <= n^2 since phi(n^2) is always divisible by n.
If n+1 is prime then a(n) = n+1.
a(n) = min{ k : phi(k) == 0 (mod n) }.
a(n) = a(2n) for odd n > 1. - Jianing Song, Feb 21 2021

A015126 Least k such that phi(k) = phi(n).

Original entry on oeis.org

1, 1, 3, 3, 5, 3, 7, 5, 7, 5, 11, 5, 13, 7, 15, 15, 17, 7, 19, 15, 13, 11, 23, 15, 25, 13, 19, 13, 29, 15, 31, 17, 25, 17, 35, 13, 37, 19, 35, 17, 41, 13, 43, 25, 35, 23, 47, 17, 43, 25, 51, 35, 53, 19, 41, 35, 37, 29, 59, 17, 61, 31, 37, 51, 65, 25, 67, 51, 69, 35, 71, 35, 73

Offset: 1

Vladeta Jovovic, Jan 12 2002

From Jianing Song, Nov 11 2022: (Start)
The first even term is a(33817088) = 16842752 (see A002181 and A143510).
Conjecture: a(n) is always odd for odd n. (End)

Antti Karttunen, Table of n, a(n) for n = 1..10000
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).

Cf. A000010, A028476, A066412, A066659.

Cf. A002181, A143510.

a(n) = {my(en = eulerphi(n)); k = 1; while (eulerphi(k) != en, k++); return (k);} \\ Michel Marcus, Jun 17 2013

a(n) = vecmin(select(x -> x<=n, invphi(eulerphi(n)))); \\ Amiram Eldar, Nov 14 2024, using Max Alekseyev's invphi.gp

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A143511 Least number k such that phi(k) = n, where n runs through the values in A143510.

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A061026 Smallest number m such that phi(m) is divisible by n, where phi = Euler totient function A000010.

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A015126 Least k such that phi(k) = phi(n).

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