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%I #77 Feb 16 2024 10:09:47
%S 1,6,28,120,270,496,672,1080,1638,1782,3780,8128,18600,20580,24948,
%T 26208,30240,32640,32760,35640,41850,44226,55860,66960,164640,167400,
%U 185220,199584,273000,293760,401310,441936,446880,502740,523776,614250,707616,802620,819000
%N Numbers k such that the set of prime divisors of k is equal to the set of prime divisors of sigma(k).
%C Multiplicities are ignored.
%C 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
%C sigma() is the multiplicative sum-of-divisors function. - _Walter Nissen_, Dec 16 2009
%C 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
%C Except for unity for the obvious reason, the primitive terms are the perfect numbers (A000396). - _Robert G. Wilson v_, Feb 19 2019
%C If an odd term > 1 exists, it is larger than 5*10^23. - _Giovanni Resta_, Jun 02 2020
%D R. K. Guy, Unsolved Problems in Number Theory, B19.
%H Donovan Johnson, <a href="/A027598/b027598.txt">Table of n, a(n) for n = 1..500</a> (first 100 terms from T. D. Noe)
%H Paul Pollack and Carl Pomerance, <a href="http://www.emis.de/journals/INTEGERS/papers/a14self/a14self.Abstract.html">Prime-Perfect Numbers</a>, INTEGERS, Electronic J. of Combinatorial Number Theory, Vol. 12a, Paper A14, 2012.
%e 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.
%t Select[Range[1000000], Transpose[FactorInteger[#]][[1]] == Transpose[FactorInteger[DivisorSigma[1, #]]][[1]] &] (* _T. D. Noe_, Dec 08 2012 *)
%o (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
%o (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
%o (GAP) Filtered([1..1000000],n->Set(Factors(n))=Set(Factors(Sigma(n)))); # _Muniru A Asiru_, Feb 21 2019
%Y Intersection of A105402 and A175200. - _Amiram Eldar_, Jun 02 2020
%Y Cf. A110751, A110819, A055744, A081377, A000203.
%K nonn
%O 1,2
%A _Jud McCranie_
%E Edited by _N. J. A. Sloane_, Jul 12 2008 at the suggestion of _R. J. Mathar_
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