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%I #127 Jan 02 2023 12:30:46
%S 1,2,6,12,40,120,126,252,880,2520,2640,10880,30240,32640,37800,37926,
%T 55440,75852,685440,758520,831600,2600640,5533920,6917400,9102240,
%U 10281600,11377800,16687440,152182800,206317440,250311600,475917120,715816960,866829600
%N k-imperfect numbers for some k >= 1.
%C For prime powers p^e, define a multiplicative function rho(p^e) = p^e - p^(e-1) + p^(e-2) - ... + (-1)^e. A number n is called k-imperfect if there is an integer k such that n = k*rho(n). Sequence A061020 gives a signed version of the rho function. As with multiperfect numbers (A007691), 2-imperfect numbers are also called imperfect numbers. As shown by Iannucci, when rho(n) is prime, there is sometimes a technique for generating larger imperfect numbers.
%C Zhou and Zhu find 5 more terms, which are in the b-file. - _T. D. Noe_, Mar 31 2009
%C Does this sequence follow Benford's law? - _David A. Corneth_, Oct 30 2017
%C If a term t has a prime factor p from A065508 with exponent 1 and does not have the corresponding prime factor q from A074268, then t*p*q is also a term. - _Michel Marcus_, Nov 22 2017
%C For n >= 1, the least n-imperfect numbers are 1, 2, 6, 993803899780063855042560. - _Michel Marcus_, Feb 13 2018
%C For any m > 0, if n*p^(2m-1) is k-imperfect, q = rho(p^(2m)) is prime and gcd(pq,n) = 1, then n*p^(2m)*q is also k-imperfect. - _M. F. Hasler_, Feb 13 2020
%D R. K. Guy, Unsolved Problems in Theory of Numbers, Springer, 1994, B1.
%H Giovanni Resta, <a href="/A127724/b127724.txt">Table of n, a(n) for n = 1..50</a> (terms < 10^13, a(1)-a(39) from T. D. Noe (from Iannucci, Zhou, and Zhu), a(40)-a(44) from Donovan Johnson)
%H David A. Corneth, <a href="/A127724/a127724_2.gp.txt">Conjectured to be the terms up to 10^28</a>
%H Douglas E. Iannucci, <a href="http://www.integers-ejcnt.org/g41/g41.Abstract.html">On a variation of perfect numbers</a>, INTEGERS: Electronic Journal of Combinatorial Number Theory, 6 (2006), #A41.
%H Donovan Johnson, <a href="/A127724/a127724.txt">43 terms > 2*10^11</a>
%H Andrew Lelechenko, <a href="/A127724/a127724_1.txt">4-imperfect numbers</a>, Apr 19 2014.
%H Michel Marcus, <a href="/A127724/a127724_2.txt">More 4-imperfect numbers</a>, Nov 07 2017.
%H Michel Marcus, <a href="/A127724/a127724.log.txt">Least known integers with small denominator-fractional k's</a>, Feb 13 2018.
%H Allan Wechsler, <a href="http://list.seqfan.eu/oldermail/seqfan/2020-February/020471.html">Some progress in k-imperfect numbers (A127724)</a>, Seqfan, Feb 13 2020. Gives a first instance of a 5-imperfect number.
%H Weiyi Zhou and Long Zhu, <a href="http://www.emis.de/journals/INTEGERS/papers/j1/j1.Abstract.html">On k-imperfect numbers</a>, INTEGERS: Electronic Journal of Combinatorial Number Theory, 9 (2009), #A01.
%e 126 = 2*3^2*7, rho(126) = (2-1)*(9-3+1)*(7-1) = 42. 3*42 = 126, so 126 is 3-imperfect. - _Jud McCranie_ Sep 07 2019
%t f[p_,e_]:=Sum[(-1)^(e-k) p^k, {k,0,e}]; rho[n_]:=Times@@(f@@@FactorInteger[n]); Select[Range[10^6], Mod[ #,rho[ # ]]==0&]
%o (PARI) isok(n) = denominator(n/sumdiv(n, d, d*(-1)^bigomega(n/d))) == 1; \\ _Michel Marcus_, Oct 28 2017
%o (PARI) upto(ulim) = {res = List([1]); rhomap = Map(); forprime(p = 2, 3, for(i = 1, logint(ulim, p), mapput(rhomap, p^i, rho(p^i)); iterate(p^i, mapget(rhomap, p^i), ulim))); listsort(res, 1); res}
%o iterate(m, rhoo, ulim) = {my(c); if(m / rhoo == m \ rhoo, listput(res, m); my(frho = factor(rhoo)); for(i = 1, #frho~, if(m%frho[i, 1] != 0, for(e = 1, logint(ulim \ m, frho[i, 1]), if(mapisdefined(rhomap, frho[i, 1]^e) == 0, mapput(rhomap, frho[i, 1]^e, rho(frho[i, 1]^e))); iterate(m * frho[i, 1]^e, rhoo * mapget(rhomap, frho[i, 1]^e), ulim)); next(2))))}
%o rho(n) = {my(f = factor(n), res = q = 1); for(i=1, #f~, q = 1; for(j = 1, f[i, 2], q = -q + f[i, 1]^j); res * =q); res} \\ _David A. Corneth_, Nov 02 2017
%o (PARI) A127724_vec=concat(1, select( {is_A127724(n)=!(n%A206369(n))}, [1..10^5]*2))
%o /* It is known that the least odd term > 1 is > 10^49. This code defines an efficient function is_A127724, but A127724_vec is better computed with upto(.) */
%o A127724(n)=A127724_vec[n] \\ Used in other sequences. - _M. F. Hasler_, Feb 13 2020
%Y Cf. A127725 (2-imperfect numbers), A127726 (3-imperfect numbers), A127727 (related primes), A309806 (the k values).
%Y Cf. A061020 (signed version of rho function), A206369 (the rho function).
%Y Cf. A065508, A074268.
%K nice,nonn
%O 1,2
%A _T. D. Noe_, Jan 25 2007
%E Small correction in name from _Michel Marcus_, Feb 13 2018
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