A127725 -id:A127725 - cp's OEIS frontend

A206369 a(p^k) = p^k - p^(k-1) + p^(k-2) - ... +- 1, and then extend by multiplicativity.

1, 1, 2, 3, 4, 2, 6, 5, 7, 4, 10, 6, 12, 6, 8, 11, 16, 7, 18, 12, 12, 10, 22, 10, 21, 12, 20, 18, 28, 8, 30, 21, 20, 16, 24, 21, 36, 18, 24, 20, 40, 12, 42, 30, 28, 22, 46, 22, 43, 21, 32, 36, 52, 20, 40, 30, 36, 28, 58, 24, 60, 30, 42, 43, 48, 20, 66, 48, 44, 24, 70, 35

Offset: 1

N. J. A. Sloane, Feb 06 2012

For more information see the Comments in A061020.
a(n) is the number of integers j such that 1 <= j <= n and gcd(n,j) is a perfect square. For example, a(12) = 6 because |{1,4,5,7,8,11}|=6 and the respective GCDs with 12 are 1,4,1,1,4,1, which are squares. - Geoffrey Critzer, Feb 16 2015
If m is squarefree (A005117), then a(m) = A000010(m) where A000010 is the Euler totient function. - Michel Marcus, Nov 08 2017
Also it appears that the primorials (A002110) is the sequence of indices of minimum records for a(n)/n, and these records are A038110(n)/A060753(n). - Michel Marcus, Nov 09 2017
Also called rho(n). When rho(n) | n, then n is called k-imperfect, with k = n/rho(n), cf. A127724. - M. F. Hasler, Feb 13 2020

P. J. McCarthy, Introduction to Arithmetical Functions, Springer Verlag, 1986, page 25.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
László Tóth, A survey of the alternating sum-of-divisors function, arXiv:1111.4842 [math.NT], 2011-2014.

Cf. A061020, A206368.
Cf. A027748 row, A124010, A206475 (first differences).
Cf. A078429.
Cf. A127724 (k-imperfect), A127725 (2-imperfect), A127726 (3-imperfect).

a206369 n = product $
   zipWith h (a027748_row n) (map toInteger $ a124010_row n) where
           h p e = sum $ take (fromInteger e + 1) $
                         iterate ((* p) . negate) (1 - 2 * (e `mod` 2))
-- Reinhard Zumkeller, Feb 08 2012

a:= n-> mul(add(i[1]^(i[2]-j)*(-1)^j, j=0..i[2]), i=ifactors(n)[2]):
seq(a(n), n=1..100);  # Alois P. Heinz, Nov 03 2017

Table[Length[Select[Range[n], IntegerQ[GCD[n, #]^(1/2)] &]], {n, 72}] (* Geoffrey Critzer, Feb 16 2015 *)
a[n_] := n*DivisorSum[n, LiouvilleLambda[#]/#&]; Array[a, 72] (* Jean-François Alcover, Dec 04 2017, after Enrique Pérez Herrero *)
f[p_,e_] := Sum[(-1)^(e-k)*p^k, {k,0,e}]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Jan 01 2020 *)

a(n) = sum(k=1, n, issquare(gcd(n, k)));

ak(p,e)=my(s=1); for(i=1,e, s=s*p + (-1)^i); s
a(n)=my(f=factor(n)); prod(i=1,#f~, ak(f[i,1],f[i,2])) \\ Charles R Greathouse IV, Dec 27 2016

a(n) = sumdiv(n, d, eulerphi(n/d) * issquare(d)); \\ Daniel Suteu, Jun 27 2018

apply( {A206369(n)=vecprod([f[1]^(f[2]+1)\/(f[1]+1)|f<-factor(n)~])}, [1..99]) \\ M. F. Hasler, Feb 13 2020

from math import prod
from sympy import factorint
def A206369(n): return prod((lambda x:x[0]+int((x[1]<<1)>=p+1))(divmod(p**(e+1),p+1)) for p, e in factorint(n).items()) # Chai Wah Wu, Mar 05 2024

a(n) = abs(A061020(n)).
a(n) = n*Sum_{d|n} lambda(d)/d, where lambda(n) is A008836(n). - Enrique Pérez Herrero, Sep 23 2012
Dirichlet g.f.: zeta(s - 1)*zeta(2*s)/zeta(s). - Geoffrey Critzer, Feb 25 2015
From Michel Marcus, Nov 05 2017: (Start)
a(2^n) = A001045(n+1);
a(3^n) = A015518(n+1);
a(5^n) = A015531(n+1);
a(7^n) = A015552(n+1);
a(11^n) = A015592(n+1). (End)
a(p^k) = p^k - a(p^(k - 1)) for k > 0 and prime p. - David A. Corneth, Nov 09 2017
a(n) = Sum_{d|n, d is a perfect square} phi(n/d), where phi(k) is the Euler totient function. - Daniel Suteu, Jun 27 2018
a(p^k) = A071324(p^k), for k >= 0 and prime p. - Michel Marcus, Aug 11 2018
Sum_{k=1..n} a(k) ~ Pi^2 * n^2 / 30. - Vaclav Kotesovec, Feb 07 2019
G.f.: Sum_{k>=1} lambda(k)*x^k/(1 - x^k)^2. - Ilya Gutkovskiy, May 23 2019
a(n) = Sum_{i=1..n} A010052(gcd(n,i)). - Ridouane Oudra, Nov 24 2019
a(p^k) = round(p^(k+1)/(p+1)). - M. F. Hasler, Feb 13 2020

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

Original entry on oeis.org

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

T. D. Noe, Jan 25 2007

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

			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

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

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.

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.

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&]

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

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

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

Small correction in name from Michel Marcus, Feb 13 2018

A127726 Numbers that are 3-imperfect.

Original entry on oeis.org

6, 120, 126, 2520, 2640, 30240, 32640, 37800, 37926, 55440, 685440, 758520, 831600, 2600640, 5533920, 6917400, 9102240, 10281600, 11377800, 16687440, 152182800, 206317440, 250311600, 475917120, 866829600, 1665709920, 1881532800, 2082137400, 2147450880

Offset: 1

T. D. Noe, Jan 25 2007

nonn

The new terms come from the paper by Zhou and Zhu. This sequence also contains n = 9223372034707292160 = 2^31*3*5*17*257*65537, which has the product of five Fermat primes (A019434). For this n, n/3 is a 2-imperfect number (A127725). - T. D. Noe, Apr 03 2009
From  M. F. Hasler, Feb 13 2020: (Start)
By definition, n is k-imperfect iff n = k*A206369(n).
So a k-imperfect number is always a multiple of k, and up to the first odd 3-imperfect number (larger than 10^49, if it exists, see Zhou & Zhu (2009)), all terms are a multiple of 6. (End)

			6 = 2*3, so A206369(6) = (2 - 1)(3 - 1) = 2 = 6 / 3, so 6 is a term.
120 = 2^3 * 3 * 5, (8-4+2-1)*(3-1)*(5-1) = 40 = 120 / 3, so 120 is another term.

Michel Marcus, Table of n, a(n) for n = 1..75 (terms < 10^20) (terms 1 to 35 from Donovan Johnson)
Michel Marcus, More 3-imperfect numbers, (includes 15 terms beyond a(75)), Nov 23 2017.
László Tóth, A survey of the alternating sum-of-divisors function, arXiv:1111.4842 [math.NT], 2011-2014.
Weiyi Zhou and Long Zhu, On k-imperfect numbers, INTEGERS: Electronic Journal of Combinatorial Number Theory, 9 (2009), #A01.

Cf. A127724 (k-imperfect), A127725 (2-imperfect), A206369 (the rho function).

okQ[n_] := 3 Sum[d*(-1)^PrimeOmega[n/d], {d, Divisors[n]}] == n;
For[k = 3, k < 10^6, k = k + 3, If[okQ[k], Print[k]]] (* Jean-François Alcover, Feb 01 2019 *)

isok(n) = 3*sumdiv(n, d, d*(-1)^bigomega(n/d)) == n; \\ Michel Marcus, Oct 28 2017

select( {is_A127726(n)=A206369(n)*3==n}, [1..10^5]*6) \\ M. F. Hasler, Feb 13 2020

Extended by T. D. Noe, Apr 03 2009

A309806 Values of k in k-imperfect numbers.

Original entry on oeis.org

1, 2, 3, 2, 2, 3, 3, 2, 2, 3, 3, 2, 3, 3, 3, 3, 3, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3

Offset: 1

Jud McCranie, Aug 17 2019

The value of k in k-imperfect numbers, A127724. Except for the first term, all terms are > 1.
It appears that the first instance of a(n)=4 is for n=98 (993803899780063855042560), while no integer is currently known to be 5-imperfect. - Michel Marcus, Aug 20 2019
A number n is called k-imperfect iff k := n/rho(n) is an integer, where rho = A206369 is a sum-of divisors function with alternating signs. - M. F. Hasler, Feb 14 2020

			The first 3 terms of A127724 are 1, 2, and 6, that are respectively 1-, 2-, and 3-imperfect. So the first 3 terms of this sequence are 1, 2 and 3.

Michel Marcus, PARI script

Cf. A127724, A127725 (2-imperfect), A127726 (3-imperfect), A206369 (rho).

{1}~Join~Map[If[IntegerQ@ #, #, Nothing] &[#/Times @@ (Sum[(-1)^(#2 - k) #1^k, {k, 0, #2}] & @@@ FactorInteger[#])] &, Range[2, 10^6]] (* Michael De Vlieger, Feb 15 2020 *)

lista(lim) = {my(v = []); for (i=1, 4, my(vi = solveIMP(1, i, lim)); v = concat (v, vi);); apply(x->x/rhon(x), vecsort(v));} \\ uses the script in links section
lista(10^24) \\ to get 98 terms; Michel Marcus, Aug 20 2019

A309806(n)=(n=A127724(n))/A206369(n) \\ M. F. Hasler, Feb 14 2020

a(n) = m/A206369(m) with m = A127724(n). - M. F. Hasler, Feb 14 2020

a(45)-a(50) from Giovanni Resta, Aug 19 2019

A307038 Unitary imperfect numbers: numbers n that equal to their unitary analog of the alternating sum of divisors, A307037(n).

Original entry on oeis.org

1, 20, 272, 65792, 2901600, 4596800, 29016000, 4295032832, 5789534400, 49085337600, 585248256000, 960935040000

Offset: 1

Amiram Eldar, Mar 21 2019

Includes all the numbers of the form F(k)*(F(k)-1) with k > 0, where F(k) = 2^(2^k) + 1, the k-th Fermat number, is prime.
a(9) > 5*10^9.
Also in the sequence are 49085337600, 585248256000, 960935040000 and 46127952000000.
a(13) > 2*10^12. - Giovanni Resta, Mar 21 2019

Cf. A000215, A127724 (k-imperfect), A127725 (2-imperfect), A127726 (3-imperfect), A307037.

f[p_,e_] := p^e + (-1)^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Select[Range[10^6], a[#] == # &]

a(9)-a(12) from Giovanni Resta, Mar 21 2019

A206367 (2,3)-imperfect numbers.

Original entry on oeis.org

3, 15, 18, 36, 72, 255, 1152, 2709, 65535, 294912, 4294967295, 4380480000

Offset: 1

N. J. A. Sloane, Feb 06 2012

a(13) > 10^10, if it exists. - Amiram Eldar, Sep 12 2023

László Tóth, A survey of the alternating sum-of-divisors function, arXiv:1111.4842 [math.NT], 2011-2014. See p. 7.

Cf. A127725 (2-imperfect).

Cf. A206369 (beta).

f[p_, e_] := ((-1)^e + p^(e + 1))/(p + 1); s[n_] := Times @@ f @@@ FactorInteger[n]; s[1] = 1; Select[Range[300000], # == 3 * s[s[#]] &] (* Amiram Eldar, Sep 12 2023 *)

beta(n) = sumdiv(n, d, d*(-1)^bigomega(n/d));
isok(k) = 3*beta(beta(k)) == k; \\ Michel Marcus, Sep 11 2023

a(7)-a(10) from Michel Marcus, Sep 11 2023

a(11)-a(12) from Amiram Eldar, Sep 12 2023

A328860 Numbers that are 4-imperfect.

Original entry on oeis.org

993803899780063855042560, 2028353759451110328141864960, 6476620014866676143312363520

Offset: 1

Michel Marcus, Feb 16 2020

Douglas E. Iannucci, On a variation of perfect numbers, INTEGERS: Electronic Journal of Combinatorial Number Theory, 6 (2006), #A41.
Andrew Lelechenko, 4-imperfect numbers, Apr 19 2014.
Michel Marcus, More 4-imperfect numbers, Nov 07 2017.
Michel Marcus, solveIMP PARI script
Michel Marcus, List of 4-imperfect numbers (including the 2 files above), last edit Jun 2019.
Weiyi Zhou and Long Zhu, On k-imperfect numbers, INTEGERS: Electronic Journal of Combinatorial Number Theory, 9 (2009), #A01.

Cf. A127724 (k-imperfect), A127725 (2-imperfect), A127726 (3-imperfect).
Cf. A309806 (the k values).
Cf. A206369 (the rho function).

PARI
```
solveIMP(1, 4, 10^24)
```

A365470 (2,6)-imperfect numbers.

Original entry on oeis.org

6, 12, 24, 30, 60, 120, 252, 384, 510, 864, 1020, 1260, 1920, 2040, 2640, 5418, 12960, 21420, 21672, 32640, 75852, 98304, 108864, 131070, 262140, 346752, 379260, 491520, 524280, 1982988, 2296728, 5504940, 6447420, 8355840, 8388480, 36747648, 39009600, 57024000

Offset: 1

Michel Marcus, Sep 11 2023

nonn

Based on testing the first 38 terms, it seems this is a subsequence of Zumkeller numbers (A083207). - Ivan N. Ianakiev, Sep 12 2023

Amiram Eldar, Table of n, a(n) for n = 1..56
László Tóth, A survey of the alternating sum-of-divisors function, arXiv:1111.4842 [math.NT], 2011-2014. See p. 7.

Cf. A127725 (2-imperfect), A206367 ((2,3)-imperfect).
Cf. A206369 (beta).
Subsequence of A008588.

beta[n_]:=Sum[d*LiouvilleLambda[n/d],{d,Divisors[n]}]; Select[Range[5500],#==6*beta[beta[#]]&] (* Ivan N. Ianakiev, Sep 12 2023 *)

beta(n) = sumdiv(n, d, d*(-1)^bigomega(n/d));
isok(k) = 6*beta(beta(k)) == k;

A200758 Superimperfect numbers.

Original entry on oeis.org

2, 4, 8, 128, 32768, 2147483648

Offset: 1

Laszlo Toth, Nov 22 2011

A number n is said to be superimperfect if 2*beta(beta(n)) = n, where beta is the multiplicative function defined by beta(p^e) = p^e - p^(e-1) + p^(e-2) - ... + (-1)^e for every prime power p^e. The function beta is called the alternating sum-of-divisors function. Here beta(n) is the absolute value of A061020(n). There are no other superimperfect numbers up to 10^7. The number 2^(2^k-1) is superimperfect if and only if k=1,2,3,4,5.

Laszlo Toth, A survey of the alternating sum-of-divisors function, arXiv:1111.4842 [math.NT], 2011-2014.
Laszlo Toth, The alternating sum-of-divisors function, 9th Joint Conf. on Math. and Comp. Sci., February 9-12, 2012, Siofok, Hungary.

Cf. A061020, A127724, A127725, A019279, A058891, A206369.

beta(n)=sumdiv(n,d,(-1)^bigomega(n/d)*d)
for(n=1,1e8,if(2*beta(beta(n))==n,print1(n", "))) \\ Charles R Greathouse IV, Nov 22 2011

ak(p,e)=my(s=1); for(i=1,e, s=s*p + (-1)^i); s
beta(n)=my(f=factor(n)); prod(i=1,#f~, ak(f[i,1],f[i,2]))
is(n)=my(b=beta(n)); 2*b-2 >= n && 2*beta(b)==n \\ Charles R Greathouse IV, Dec 27 2016

cp's OEIS Frontend

A206369 a(p^k) = p^k - p^(k-1) + p^(k-2) - ... +- 1, and then extend by multiplicativity.

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

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A127726 Numbers that are 3-imperfect.

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A309806 Values of k in k-imperfect numbers.

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A307038 Unitary imperfect numbers: numbers n that equal to their unitary analog of the alternating sum of divisors, A307037(n).

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A206367 (2,3)-imperfect numbers.

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