A027598 Numbers k such that the set of prime divisors of k is equal to the set of prime divisors of sigma(k).

1, 6, 28, 120, 270, 496, 672, 1080, 1638, 1782, 3780, 8128, 18600, 20580, 24948, 26208, 30240, 32640, 32760, 35640, 41850, 44226, 55860, 66960, 164640, 167400, 185220, 199584, 273000, 293760, 401310, 441936, 446880, 502740, 523776, 614250, 707616, 802620, 819000

Offset

1,2

Comments

Multiplicities are ignored.

All even perfect numbers are in the sequence. It seems that 1 is the only odd term of the sequence. - Farideh Firoozbakht, Jul 01 2008

sigma() is the multiplicative sum-of-divisors function. - Walter Nissen, Dec 16 2009

Pollack and Pomerance call these "prime-perfect numbers" and show that there are << x^(1/3+e) of these up to x for any e > 0. - Charles R Greathouse IV, May 09 2013

Except for unity for the obvious reason, the primitive terms are the perfect numbers (A000396). - Robert G. Wilson v, Feb 19 2019

If an odd term > 1 exists, it is larger than 5*10^23. - Giovanni Resta, Jun 02 2020

References

R. K. Guy, Unsolved Problems in Number Theory, B19.

Links

Donovan Johnson, Table of n, a(n) for n = 1..500 (first 100 terms from T. D. Noe)

Paul Pollack and Carl Pomerance, Prime-Perfect Numbers, INTEGERS, Electronic J. of Combinatorial Number Theory, Vol. 12a, Paper A14, 2012.

Example

273000 = 2^3*3*5^3*7*13 and sigma(273000) = 1048320 = 2^8*3^2*5*7*13 so 273000 is in the sequence.

Mathematica

Select[Range[1000000], Transpose[FactorInteger[#]][[1]] == Transpose[FactorInteger[DivisorSigma[1, #]]][[1]] &] (* T. D. Noe, Dec 08 2012 *)

Prog

(PARI) a(n) = {for (i=1, n, fn = factor(i); fs = factor(sigma(i)); if (fn[, 1] == fs[, 1], print1(i, ", ")); ); } \\ Michel Marcus, Nov 18 2012
(PARI) is(n)=my(f=factor(n), fs=[], t); for(i=1, #f[, 1], t=factor((f[i, 1]^(f[i, 2]+1)-1)/(f[i, 1]-1))[, 1]; fs=vecsort(concat(fs, t~), , 8); if(#setminus(fs, f[, 1]~), return(0))); fs==f[, 1]~ \\ Charles R Greathouse IV, May 09 2013
(GAP) Filtered([1..1000000], n->Set(Factors(n))=Set(Factors(Sigma(n)))); # Muniru A Asiru, Feb 21 2019

Crossrefs

Intersection of A105402 and A175200. - Amiram Eldar, Jun 02 2020

Cf. A110751, A110819, A055744, A081377, A000203.

Sequence in context: A282775 A325808 A192853 * A183013 A325639 A055717

Adjacent sequences: A027595 A027596 A027597 * A027599 A027600 A027601

Keyword

nonn

Author

Jud McCranie

Extensions

Edited by N. J. A. Sloane, Jul 12 2008 at the suggestion of R. J. Mathar

Status

approved