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A027598 Numbers n such that the set of prime divisors of n is equal to the set of prime divisors of sigma(n). 9
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 (list; graph; refs; listen; history; text; internal format)
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. [From 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

REFERENCES

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

LINKS

T. D. Noe and 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.

MAPLE

with(numtheory);

A027598:=proc(q)

local a, b, k, n;

for n from 1 to q do

  a:=ifactors(n)[2]; b:=ifactors(sigma(n))[2];

  if nops(a)=nops(b) then

    if product(a[k][1], k=1..nops(a))=product(b[k][1], k=1..nops(a)) then print(n);

fi; fi; od; end:

A027598(100000); # Paolo P. Lava, Jan 09 2013

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

CROSSREFS

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

Sequence in context: A183016 A282775 A192853 * A183013 A055717 A090777

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

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Last modified October 21 17:06 EDT 2018. Contains 316427 sequences. (Running on oeis4.)