A163272 Numbers k such that k = A074206(k), the number of ordered factorizations of k.

0, 1, 48, 1280, 2496, 28672, 29808, 454656, 2342912, 11534336, 57409536, 218103808, 34753216512, 73014444032, 583041810432, 1305670057984, 2624225017856, 404620279021568, 467515780104192, 1014849232437248, 4446425022726144, 5806013294837760, 46545625738641408

Offset

1,3

Comments

From Mauro Fiorentini, Jul 15 2018: (Start)

If p is an odd prime, 2^(2*p - 2)*p belongs to the sequence, so the sequence is infinite.

If n^2 + 6*n + 6 = 2*p*q is twice the product of two distinct odd primes, 2^n*p*q belongs to the sequence.

No number of the form 2^n*p^2, with p odd prime, belongs to the sequence. (End)

For every possible prime signature (see A025487) there can be at most one number having it in this sequence. - David A. Corneth, Jul 15 2018

2*10^14 < a(18) <= 404620279021568. Also terms: 467515780104192, 1014849232437248, 4446425022726144, 5806013294837760, and 46545625738641408. - Giovanni Resta, Jul 16 2018

These numbers are named "super-perfect numbers" (Miller), "gamma-perfect numbers" (Sandor & Crstici), "factor-perfect numbers" (Knopfmacher & Mays) and "balanced numbers" (Brown). - Amiram Eldar, Aug 22 2018

From David A. Corneth, Aug 23 2018: (Start)

Suppose one searches terms below u. We have A074206(m * t) > A074206(m) for m, t > 1 so if A074206(m) > u we needn't check any value A074206(m * t) where m * t < u.

All terms < 10^25 except 29809 are of the form 4^e * s where s is a squarefree odd number. (End)

References

J. Sandor and B. Crstici, Handbook of Number Theory II, Springer, 2004, pp. 54-55.

Links

David A. Corneth, Table of n, a(n) for n = 1..49 (terms < 10^30)

Peter Brown, Number of Ordered Factorizations, 2004.

Martin Klazar and Florian Luca, On the maximal order of numbers in the "factorisatio numerorum" problem, Journal of Number Theory, Vol. 124, No. 2 (2007), pp. 470-490.

Arnold Knopfmacher and M. E. Mays, A survey of factorization counting functions, International Journal of Number Theory, Vol. 1, No. 4 (2005), pp. 563-581, DOI: 10.1142/S1793042105000315.

Michael D. Miller, A recursively defined divisor function, The Fibonacci Quarterly, Vol. 13 (1975), pp. 199-204.

Project Euler, Problem 548: Gozinta Chains.

Maple

A074206 := proc(n) option remember; if n <= 1 then n; else add(procname(d), d=numtheory[divisors](n) minus {n}) ; end if; end proc: for n from 1 do if n = A074206(n) then printf("%d, \n", n) ; end if; end do: \\ R. J. Mathar, Aug 01 2009

Prog

(PARI) term(n) = {my(f = A074206(n)); if(factor(n)[, 2] == factor(f)[, 2], f, 0) \\ returns 0 if there is no term in the sequence with prime signature of n, or if there is, returns that term. - David A. Corneth, Jul 15 2018

Crossrefs

Cf. A025487, A074206.

Sequence in context: A004425 A082558 A285169 * A165283 A141407 A004341

Adjacent sequences: A163269 A163270 A163271 * A163273 A163274 A163275

Keyword

nonn

Author

Mats Granvik, Jul 24 2009

Extensions

a(6)-a(7) from R. J. Mathar, Aug 01 2009

a(8)-a(9) from Nathaniel Johnston, Dec 04 2010

a(10)-a(12) from Mauro Fiorentini, Dec 07 2015

a(13)-a(17) from Giovanni Resta, Jul 16 2018, following a suggestion from David A. Corneth

a(18)-a(23) from Amiram Eldar, Aug 22 2018, following the same suggestion with an extended list of terms of A025487.

Status

approved