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A173327
Numbers k such that tau(phi(k))= sopf(k).
2
4, 45, 48, 75, 160, 180, 252, 294, 300, 315, 336, 351, 378, 396, 475, 507, 560, 605, 616, 650, 833, 936, 1216, 1375, 1452, 1690, 1805, 1920, 2023, 2112, 2200, 2349, 2496, 2736, 3211, 3520, 3648, 4095, 4160, 4256, 4332, 4389, 4464, 4477, 4508, 4620, 4693
OFFSET
1,1
COMMENTS
tau(k) is the number of divisors of k (A000005); phi(k) is the Euler totient function (A000010); 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
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.
FORMULA
Numbers n such that A062821(n)= A008472(n)
EXAMPLE
4 is in the sequence because phi(4) = 2, tau(2)=2 and sopf(4)=2 ;
45 is in the sequence because phi(45) = 24, tau(24)=8 and sopf(45)=8.
MAPLE
for n from 1 to 150000 do : t1:= ifactors(n)[2] : t2 :=sum(t1[i][1], i=1..nops(t1)):if tau(phi(n)) = t2 then print (n): else fi : od :
MATHEMATICA
tpsQ[n_]:=DivisorSigma[0, EulerPhi[n]]==Total[Transpose[FactorInteger[n]][[1]]]; Select[Range[5000], tpsQ] (* Harvey P. Dale, Apr 10 2013 *)
PROG
(PARI) sopf(n)=my(f=factor(n)[1, ]); sum(i=1, #f, f[i])
is(n)=numdiv(eulerphi(n))==sopf(n) \\ Charles R Greathouse IV, May 20 2013
(Magma) [ m:m in [2..5100]|#Divisors(EulerPhi(m)) eq &+PrimeDivisors(m)]; // Marius A. Burtea, Jul 10 2019
CROSSREFS
Sequence in context: A327196 A024254 A167781 * A075029 A369228 A070225
KEYWORD
nonn
AUTHOR
Michel Lagneau, Feb 16 2010
EXTENSIONS
Added punctuation to the examples. Corrected and edited by Michel Lagneau, Apr 25 2010
Edited by D. S. McNeil, Nov 20 2010
STATUS
approved