

A127724


kimperfect numbers for some k >= 1.


12



1, 2, 6, 12, 40, 120, 126, 252, 880, 2520, 2640, 10880, 30240, 32640, 37800, 37926, 55440, 75852, 685440, 758520, 831600, 2600640, 5533920, 6917400, 9102240, 10281600, 11377800, 16687440, 152182800, 206317440, 250311600, 475917120, 715816960, 866829600
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OFFSET

1,2


COMMENTS

For prime powers p^e, define a multiplicative function rho(p^e) = p^e  p^(e1) + p^(e2)  ... + (1)^e. A number n is called kimperfect 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), 2imperfect numbers are also called imperfect numbers. No kimperfect numbers are known for k > 4. As shown by Iannucci, when rho(n) is prime, there is sometimes a technique for generating larger imperfect numbers.
Zhou and Zhu find 5 more terms, which are in the bfile.  T. D. Noe, Mar 31 2009
Does this sequence follow Benford's law?  David A. Corneth, Oct 30 2017
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
For n >= 1, the least nimperfect numbers are 1, 2, 6, 993803899780063855042560 (probably).  Michel Marcus, Feb 13 2018


REFERENCES

R. K. Guy, Unsolved Problems in Theory of Numbers, Springer, 1994, B1.


LINKS

Giovanni Resta, Table of n, a(n) for n = 1..50 (terms < 5*10^12, a(1)a(39) from T. D. Noe (from Iannucci, Zhou, and Zhu), a(40)a(44) from Donovan Johnson)
David A. Corneth, Conjectured to be the terms up to 10^28
Douglas E. Iannucci, On a variation of perfect numbers, INTEGERS: Electronic Journal of Combinatorial Number Theory, 6 (2006), #A41.
Donovan Johnson, 43 terms > 2*10^11
Andrew Lelechenko, 4imperfect numbers
Michel Marcus, More 4imperfect numbers
Weiyi Zhou and Long Zhu, On kimperfect numbers, INTEGERS: Electronic Journal of Combinatorial Number Theory, 9 (2009), #A01.
Michel Marcus, Going toward a 5imperfect number with small denominators


EXAMPLE

126 = 2*3^2*7, rho(126) = (21)*(93+1)*(71) = 42. 3*42 = 126, so 126 is 3imperfect.  Jud McCranie Sep 07 2019.


MATHEMATICA

f[p_, e_]:=Sum[(1)^(ek) p^k, {k, 0, e}]; rho[n_]:=Times@@(f@@@FactorInteger[n]); Select[Range[10^6], Mod[ #, rho[ # ]]==0&]


PROG

(PARI) isok(n) = denominator(n/sumdiv(n, d, d*(1)^bigomega(n/d))) == 1; \\ Michel Marcus, Oct 28 2017
(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}
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))))}
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


CROSSREFS

Cf. A127725 (2imperfect numbers), A127726 (3imperfect numbers), A127727 (related primes), A309806 (the k values).
Cf. A061020 (signed version of rho function), A206369 (the rho function).
Cf. A065508, A074268.
Sequence in context: A327879 A094261 A080497 * A178008 A266005 A056744
Adjacent sequences: A127721 A127722 A127723 * A127725 A127726 A127727


KEYWORD

nice,nonn


AUTHOR

T. D. Noe, Jan 25 2007


EXTENSIONS

Small correction in name from Michel Marcus, Feb 13 2018


STATUS

approved



