A127724 k-imperfect numbers for some k >= 1.

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

Offset: 1

Author

T. D. Noe, Jan 25 2007

Keywords

• nice
• nonn

Comments

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.

Zhou and Zhu find 5 more terms, which are in the b-file. - 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 n-imperfect numbers are 1, 2, 6, 993803899780063855042560. - Michel Marcus, Feb 13 2018

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

Examples

			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
		

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 < 10^13, 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, 4-imperfect numbers, Apr 19 2014.
• Michel Marcus, More 4-imperfect numbers, Nov 07 2017.
• Michel Marcus, Least known integers with small denominator-fractional k's, Feb 13 2018.
• Allan Wechsler, Some progress in k-imperfect numbers (A127724), Seqfan, Feb 13 2020. Gives a first instance of a 5-imperfect number.
• Weiyi Zhou and Long Zhu, On k-imperfect numbers, INTEGERS: Electronic Journal of Combinatorial Number Theory, 9 (2009), #A01.

Crossrefs

Cf. A127725 (2-imperfect numbers), A127726 (3-imperfect numbers), A127727 (related primes), A309806 (the k values).

Cf. A061020 (signed version of rho function), A206369 (the rho function).

Cf. A065508, A074268.

Programs

Mathematica

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&]

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

PARI

A127724_vec=concat(1, select( {is_A127724(n)=!(n%A206369(n))}, [1..10^5]*2))
  /* 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(.) */
  A127724(n)=A127724_vec[n] \\ Used in other sequences. - M. F. Hasler, Feb 13 2020

Extensions

Small correction in name from Michel Marcus, Feb 13 2018