A173336 Numbers k such that tau(phi(k)) = sigma(sopf(k)).

8, 9, 25, 36, 49, 54, 96, 100, 320, 441, 495, 704, 891, 1029, 1080, 1089, 1260, 1331, 1386, 1400, 1617, 1701, 1750, 1815, 1848, 1950, 1960, 2079, 2541, 2574, 2704, 2850, 2880, 3000, 3360, 3430, 3510, 3861, 4125, 4275, 4680, 4704, 4719, 4800, 5070, 5096

Offset

1,1

Comments

tau(k) is the number of divisors of k (A000005); phi(k) is the Euler totient function (A000010); sigma(k) is the sum of divisors of k (A000203); and sopf(k) is the sum of the distinct primes dividing k without repetition (A008472).

References

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 840.

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

P. A. MacMahon, Divisors of numbers and their continuations in the theory of partitions, Proc. London Math. Soc., 19 (1919), 75-113.

Wacław Sierpiński, Number Of Divisors And Their Sum, Elementary theory of numbers, Warszawa, 1964.

Wikipedia, Euler's totient function

Formula

k such that A062821(k) = sigma(A008472(k)).

Example

8 is in the sequence because phi(8) = 4, tau(4)=3, sopf(8)=2 and sigma(2) = 3 ;
9 is in the sequence because phi(9) = 6, tau(6)=4, sopf(9)=3 and sigma(3) = 4.

Maple

with(numtheory): for n from 1 to 18000 do : t1:= ifactors(n)[2] : t2 :=sum(t1[i][1], i=1..nops(t1)):if tau(phi(n)) = sigma(t2) then print (n): else fi : od :

Mathematica

sopf[n_] := Plus @@ (First@# & /@ FactorInteger[n]); Select[Range[2, 5100], DivisorSigma[0, EulerPhi[#]] == DivisorSigma[1, sopf[#]] &] (* Amiram Eldar, Jul 09 2019 *)

Prog

(Magma) [m:m in [2..5100]|#Divisors(EulerPhi(m)) eq &+Divisors(&+PrimeDivisors(m))]; // Marius A. Burtea, Jul 10 2019
(PARI) isok(n) = (n>1) && numdiv(eulerphi(n)) == sigma(vecsum(factor(n)[, 1])); \\ Michel Marcus, Jul 10 2019

Crossrefs

Cf. A000005, A000010, A000203, A001157, A001158, A001160, A001065, A002192, A008472, A062821.

Sequence in context: A130100 A226230 A258400 * A277925 A173745 A305828

Adjacent sequences: A173333 A173334 A173335 * A173337 A173338 A173339

Keyword

nonn

Author

Michel Lagneau, Feb 16 2010

Extensions

Corrected and edited by Michel Lagneau, Apr 25 2010

Edited by D. S. McNeil, Nov 20 2010

Status

approved