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A061026 Smallest number m such that phi(m) is divisible by n, where phi = Euler totient function A000010. 11
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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

REFERENCES

M. J. Knight, Comment with Solution to 10837, American Mathematical Monthly, 2001.

LINKS

T. D. Noe, Table of n, a(n) for n=1..1000

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.

P. Moree, On an arithmetical function related to Euler's totient and the discriminator Fib. Quart. (1995).

József Sándor, On the Euler minimum and maximum functions, Notes on Number Theory and Discrete Mathematics, Volume 15, 2009, Number 3, Pages 1—8.

FORMULA

Sequence is unbounded; a(n) <= n^2 since phi(n^2) is always divisible by n.

If n+1 is prime 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

EXAMPLE

a(48) = 65 because phi(65) = phi(5)phi(13) = (4)(12) = 48 and no smaller integer has phi(n) = 48.

MATHEMATICA

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 *)

PROG

(PARI) for(n=1, 100, s=1; while(eulerphi(s)%n>0, s++); print1(s, ", "))

(Python)

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

CROSSREFS

Cf. A000010, A066674, A066675, A066676, A066678, A067005.

Cf. A233516, A233517 (records).

Cf. A005179 (analog for number of divisors), A070982 (analog for sum of divisors).

Sequence in context: A088514 A254929 A066677 * A064632 A216487 A328984

Adjacent sequences:  A061023 A061024 A061025 * A061027 A061028 A061029

KEYWORD

nonn,changed

AUTHOR

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

STATUS

approved

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Last modified March 5 04:24 EST 2021. Contains 341816 sequences. (Running on oeis4.)