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Numbers k such that k = A074206(k), the number of ordered factorizations of k.
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%I #78 Feb 05 2025 09:23:55

%S 0,1,48,1280,2496,28672,29808,454656,2342912,11534336,57409536,

%T 218103808,34753216512,73014444032,583041810432,1305670057984,

%U 2624225017856,404620279021568,467515780104192,1014849232437248,4446425022726144,5806013294837760,46545625738641408

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

%C From _Mauro Fiorentini_, Jul 15 2018: (Start)

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

%C 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.

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

%C 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

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

%C 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

%C From _David A. Corneth_, Aug 23 2018: (Start)

%C 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.

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

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

%H David A. Corneth, <a href="/A163272/b163272.txt">Table of n, a(n) for n = 1..49</a> (terms < 10^30)

%H Peter Brown, <a href="http://www.mountainman.com.au/harmonics_01.htm">Number of Ordered Factorizations</a>, 2004.

%H Martin Klazar and Florian Luca, <a href="https://doi.org/10.1016/j.jnt.2006.10.003">On the maximal order of numbers in the "factorisatio numerorum" problem</a>, Journal of Number Theory, Vol. 124, No. 2 (2007), pp. 470-490.

%H Arnold Knopfmacher and M. E. Mays, <a href="https://pdfs.semanticscholar.org/d7ed/31ad7c11cad37442838d6614f658af539ef5.pdf">A survey of factorization counting functions</a>, International Journal of Number Theory, Vol. 1, No. 4 (2005), pp. 563-581, DOI: 10.1142/S1793042105000315.

%H Michael D. Miller, <a href="https://fq.math.ca/Scanned/13-3/miller.pdf">A recursively defined divisor function</a>, The Fibonacci Quarterly, Vol. 13 (1975), pp. 199-204.

%H Project Euler, <a href="https://projecteuler.net/problem=548">Problem 548: Gozinta Chains</a>.

%p 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

%o (PARI) term(n) = {my(f = A074206(n)); if(factor(n)[, 2] == factor(f)[, 2], f, 0)};

%o isok(n) = term(n) == n; \\ _David A. Corneth_, Jul 15 2018

%Y Cf. A025487, A074206.

%K nonn

%O 1,3

%A _Mats Granvik_, Jul 24 2009

%E a(6)-a(7) from _R. J. Mathar_, Aug 01 2009

%E a(8)-a(9) from _Nathaniel Johnston_, Dec 04 2010

%E a(10)-a(12) from _Mauro Fiorentini_, Dec 07 2015

%E a(13)-a(17) from _Giovanni Resta_, Jul 16 2018, following a suggestion from _David A. Corneth_

%E a(18)-a(23) from _Amiram Eldar_, Aug 22 2018, following the same suggestion with an extended list of terms of A025487.