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

Author

Mats Granvik, Jul 24 2009

Keywords

• nonn

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.

Crossrefs

Cf. A025487, A074206.

Programs

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

PARI

term(n) = {my(f = A074206(n)); if(factor(n)[, 2] == factor(f)[, 2], f, 0)};
isok(n) = term(n) == n;  \\ David A. Corneth, Jul 15 2018

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.