A068349 Numbers k having a partition k=Sum x_i for which Product k/x_i is also a way of factoring k.

1, 4, 27, 32, 36, 256, 288, 324, 432, 500, 512, 576, 800, 864, 900, 972, 1152, 1764, 3125, 3456, 3888, 4000, 4096, 4500, 4608, 5184, 5625, 6400, 6561, 6912, 7200, 7776, 8100, 8192, 8748, 9216, 9261, 9604, 10000, 10125, 10368, 10800, 11664, 12348, 12800, 13500

Offset

1,2

Comments

From Charles L. Hohn, Apr 02 2025: (Start)

Equivalent to: numbers with at least 1 factorization for which the sum of the reciprocals of the factors equals 1, e.g. a(5) = 36: 2 * 3 * 6 = 36 and 1/2 + 1/3 + 1/6 = 1. The first few terms with record counts of such factorizations are: 432 (2: {2, 6, 6, 6} and {3, 3, 4, 12}), 43200 (3), 110592 (4), 115200 (6), 518400 (7).

A000312(k) = k^k for k >= 1 is a subsequence, as all terms factor to k k's, and k*(1/k) = 1. (End)

Links

Table of n, a(n) for n=1..46.

Example

36 is in the sequence because 36 has a partition 18+12+6 for which (36/18)*(36/12)*(36/6)=2*3*6 is also a way of factoring 36.
324 is in the sequence since 54+54+108+108=324 and (324/54)*(324/54)*(324/108)*(324/108)=3*3*6*6=324.

Prog

(PARI) a068349_count(x, f=List())={my(r=x/if(#f, vecprod(Vec(f)), 1)); if(#f && r==1, return(if(sum(i=1, #f, 1/f[i])==1, 1, 0))); my(d, c=0); fordiv(r, d, if(#f, if(d<f[#f], next), d==1 && x>1, next); listput(f, d); c+=a068349_count(x, f); listpop(f)); return(c)} \\ Charles L. Hohn, Apr 02 2025

Crossrefs

Cf. A001055, A066925, A066926.

Cf. A000312.

Sequence in context: A383768 A250095 A085702 * A129204 A239283 A082872

Adjacent sequences: A068346 A068347 A068348 * A068350 A068351 A068352

Keyword

nonn

Author

Naohiro Nomoto, Feb 28 2002

Extensions

More terms from John W. Layman, Apr 16 2002

a(9)-a(44) from Donovan Johnson, May 23 2010

Corrected example and more terms from Charles L. Hohn, Apr 02 2025

Status

approved