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

Author

Michel Lagneau, Feb 16 2010

Keywords

• nonn

Comments

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

Examples

			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.
		

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.

Crossrefs

Cf. A000005, A008472, A010553.

Programs

Magma

f:=func; g:=func; [k:k in [2..5000]|f(f(k)) eq g(g(k)) ]; // Marius A. Burtea, Nov 14 2019

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 *)

Formula

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

Extensions

Corrected and edited by Michel Lagneau, Apr 25 2010