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A163272 Numbers k such that k = A074206(k), the number of ordered factorizations of k. 9
0, 1, 48, 1280, 2496, 28672, 29808, 454656, 2342912, 11534336, 57409536, 218103808, 34753216512, 73014444032, 583041810432, 1305670057984, 2624225017856, 404620279021568, 467515780104192, 1014849232437248, 4446425022726144, 5806013294837760, 46545625738641408 (list; graph; refs; listen; history; text; internal format)
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

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Last modified October 15 01:40 EDT 2019. Contains 328025 sequences. (Running on oeis4.)