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 A027598 Numbers k such that the set of prime divisors of k is equal to the set of prime divisors of sigma(k). 15
 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. - 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 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); b:=ifactors(sigma(n));   if nops(a)=nops(b) then     if product(a[k], k=1..nops(a))=product(b[k], k=1..nops(a)) then print(n); fi; fi; od; end: A027598(100000); # Paolo P. Lava, Jan 09 2013 MATHEMATICA Select[Range, Transpose[FactorInteger[#]][] == Transpose[FactorInteger[DivisorSigma[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 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 July 15 02:31 EDT 2020. Contains 335762 sequences. (Running on oeis4.)