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A033631 Numbers k such that sigma(phi(k)) = sigma(k) where sigma is the sum of divisors function A000203 and phi is the Euler totient function A000010. 14
1, 87, 362, 1257, 1798, 5002, 9374, 21982, 22436, 25978, 35306, 38372, 41559, 50398, 51706, 53098, 53314, 56679, 65307, 68037, 89067, 108946, 116619, 124677, 131882, 136551, 136762, 138975, 144014, 160629, 165554, 170037, 186231, 192394, 197806 (list; graph; refs; listen; history; text; internal format)



For corresponding values of phi(k) and sigma(k), see A115619 and A115620.

This sequence is infinite because for each positive integer k, 3^k*7*1979 and 3^k*7*2699 are in the sequence (the proof is easy). A108510 gives primes p like 1979 and 2699 such that for each positive integer k, 3^k*7*p is in this sequence. - Farideh Firoozbakht, Jun 07 2005

There is another class of [conjecturally] infinite subsets connected to A005385 (safe primes). Examples: Let s,t be safe primes, s<>t, then 3^2*5*251*s, 2^2*61*71*s, 2*61*s*t and 2*19*311*s are in this sequence. So is 3*s*A108510(m). (There are some obvious exceptions for small s, t.) - Vim Wenders, Dec 27 2006


J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 87, p. 29, Ellipses, Paris 2008.

R. K. Guy, Unsolved Problems in Number Theory, B42.

D. Wells, The Penguin Dictionary of Curious and Interesting Numbers, Penguin Books 1997.

David Wells, Curious and Interesting Numbers (Revised), Penguin Books, page 114.


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

J.-M. De Koninck, F. Luca, Positive integers n such that sigma(phi(n))=sigma(n), JIS 11 (2008) 08.1.5.

S. W. Golomb, Equality among number-theoretic functions, Unpublished manuscript. (Annotated scanned copy)


Do[If[DivisorSigma[1, EulerPhi[n]]==DivisorSigma[1, n], Print[n]], {n, 1, 10^5}]


(PARI) is(n)=sigma(eulerphi(n))==sigma(n) \\ Charles R Greathouse IV, Feb 13 2013

(MAGMA) [k:k in [1..200000]| DivisorSigma(1, EulerPhi(k)) eq DivisorSigma(1, k)]; // Marius A. Burtea, Feb 09 2020


Cf. A000203, A000010, A006872, A115619, A115620.

Sequence in context: A186055 A243580 A219723 * A183724 A221312 A098139

Adjacent sequences:  A033628 A033629 A033630 * A033632 A033633 A033634




Jud McCranie


Entry revised by N. J. A. Sloane, Apr 10 2006



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Last modified December 4 12:08 EST 2021. Contains 349489 sequences. (Running on oeis4.)