A173338 Numbers n such that tau(tau(n)) = sopf(sopf(n)), sopf = A008472.

2, 4, 14, 16, 27, 64, 158, 168, 196, 216, 312, 378, 384, 440, 456, 482, 546, 680, 702, 744, 770, 1024, 1026, 1032, 1160, 1454, 1608, 1640, 1674, 1880, 2024, 2058, 2295, 2322, 2472, 2750, 2805, 2944, 3336, 3560, 3608, 3618, 3768, 3828, 3944, 3960, 4040, 4096

Offset

1,1

Comments

sopf(n) is the sum of distinct primes dividing n (without repetition, A008472), tau(n) is the number of divisors of n (A000005).

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 (terms 1..1000 from Alois P. Heinz)

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

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

Formula

{ n : A010553(n) = A008472(A008472(n)) }.

Example

4 is in the sequence: tau(4) = 3, tau(3) = 2; sopf(4) = 2, sopf(2) = 2.
546 is in the sequence: tau(546) = 16, tau(16) = 5; sopf(546) = 25, sopf(25) = 5.

Maple

with(numtheory): for n from 1 to 60000 do : t1:= ifactors(n)[2] : t2 :=sum(t1[i][1], i=1..nops(t1)): tt1:= ifactors(t2)[2] : tt2 :=sum(tt1[i][1], i=1..nops(tt1)):if tau(tau(n))= tt2 then print (n): else fi : od :
# second Maple program:
with(numtheory): sopf:= n-> add(i, i=factorset(n)):
a:= proc(n) option remember; local k;
      for k from 1+ `if`(n=1, 0, a(n-1))
      while tau(tau(k)) <> sopf(sopf(k)) do od; k
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Aug 26 2010

Mathematica

Select[Range[4100], DivisorSigma[0, DivisorSigma[0, #]]==Total[ Transpose[ FactorInteger[ Total[Transpose[FactorInteger[#]][[1]]]]][[1]]]&] (* Harvey P. Dale, Aug 05 2013 *)

Prog

(Magma) f:=func<n|NumberOfDivisors(n)>; g:=func<n|&+PrimeDivisors(n)>; [k:k in [2..5000]|f(f(k)) eq g(g(k)) ]; // Marius A. Burtea, Nov 14 2019

Crossrefs

Cf. A000005, A008472, A010553.

Sequence in context: A087420 A194388 A141654 * A054600 A077570 A360713

Adjacent sequences: A173335 A173336 A173337 * A173339 A173340 A173341

Keyword

nonn

Author

Michel Lagneau, Feb 16 2010

Extensions

Corrected and edited by Michel Lagneau, Apr 25 2010

Status

approved