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.

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

Offset

1,2

Comments

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

References

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.

Links

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S. W. Golomb, Equality among number-theoretic functions, Unpublished manuscript.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)

Mathematica

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

Prog

(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

Crossrefs

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

Sequence in context: A186055 A243580 A219723 * A183724 A221312 A098139

Adjacent sequences: A033628 A033629 A033630 * A033632 A033633 A033634

Keyword

nonn

Author

Jud McCranie

Extensions

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

Status

approved