A127724 -id:A127724 - 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

A127725 Numbers that are 2-imperfect.

Original entry on oeis.org

2, 12, 40, 252, 880, 10880, 75852, 715816960, 62549517598720

Offset: 1

T. D. Noe, Jan 25 2007

This sequence also contains n = 3074457344902430720 = 2^31*5*17*257*65537, which has the product of four Fermat primes (A019434). For this n, 3*n is a 3-imperfect number (A127726). - T. D. Noe, Apr 03 2009
a(9) > 2*10^11. - Donovan Johnson, Feb 07 2013
62549517598720 is also a term (see the "43 terms > 2*10^11" link by Donovan Johnson in A127724). - Michel Marcus, Nov 05 2017

			40 = 2^3 * 5, (8 - 4 + 2 - 1)(5 - 1) = 20 = 40 / 2, so 40 is in the sequence. - _Jud McCranie_, Aug 17 2019

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

Cf. A127726 (3-imperfect numbers), A127724 (k-imperfect numbers).

okQ[n_] := 2 Sum[d*(-1)^PrimeOmega[n/d], {d, Divisors[n]}] == n;
For[k = 2, k <= 10^9, k = k+2, If[okQ[k], Print[k]]] (* Jean-François Alcover, Jan 27 2019 *)

isok(n) = 2*sumdiv(n, d, d*(-1)^bigomega(n/d)) == n; \\ Michel Marcus, Oct 28 2017

a(9) by Jud McCranie, Aug 17 2019

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

A127727 Primes of the form p^e - p^(e-1) + p^(e-2) - ... + (-1)^e, where p is prime.

Original entry on oeis.org

2, 3, 5, 7, 11, 43, 61, 157, 521, 547, 683, 2731, 4423, 6163, 13421, 19183, 22651, 26407, 37057, 43691, 113233, 121453, 143263, 174763, 208393, 292141, 375157, 398581, 412807, 527803, 590593, 843643, 981091, 1041421, 1193557, 1246573

Offset: 1

T. D. Noe, Jan 25 2007

These primes are important in studying k-imperfect numbers (A127724), see Iannucci-link. Except for the cases p^e = 3 and 8, which yield primes 2 and 5, e is an even number such that e+1 is prime. In fact, except for those two cases, all the primes are of the form (1+p^q)/(1+p), where q is an odd prime; that is, repunit primes with negative prime base.

			From _David A. Corneth_, Oct 28 2017: (Start)
For (p, e) = (3, 1) we have the prime 3^1 - 3^0 = 2.
For (p, e) = (2, 3) we have the prime 2^3 - 2^2 + 2^1 - 2^0 = 5.
The examples above are the cases mentioned in the comments not of the form (1+p^q)/(1+p). A prime of that form is below;
For (p, e) = (2, 4) we have the prime 2^4 - 2^3 + 2^2 - 2^1 + 2^0 = 11 = (1+p^(e+1)) / (1+p) = 33/3. (End)

David A. Corneth, Table of n, a(n) for n = 1..33914 (terms < 10^14) (the first 4799 terms < 10^12 from T. D. Noe)
H. Dubner and T. Granlund, Primes of the Form (b^n+1)/(b+1), J. Integer Sequences, 3 (2000), Article 00.2.7.
Douglas E. Iannucci, On a variation of perfect numbers, INTEGERS: Electronic Journal of Combinatorial Number Theory, 6 (2006), #A41.

Cf. A023195, A065507, A084741, A206369.

upto(n) = {my(res = List([2,5])); forprime(p = 2, sqrtnint(n, 2), forprime(q = 3, logint(n * (1+p), p), r = (1+p^q)/(1+p); if(isprime(r), listput(res, r)))); listsort(res, 1); res} \\ David A. Corneth, Oct 28 2017

A294649 a(n) = numerator(A206369(n))/n.

Original entry on oeis.org

1, 1, 2, 3, 4, 1, 6, 5, 7, 2, 10, 1, 12, 3, 8, 11, 16, 7, 18, 3, 4, 5, 22, 5, 21, 6, 20, 9, 28, 4, 30, 21, 20, 8, 24, 7, 36, 9, 8, 1, 40, 2, 42, 15, 28, 11, 46, 11, 43, 21, 32, 9, 52, 10, 8, 15, 12, 14, 58, 2, 60, 15, 2, 43, 48, 10, 66, 12, 44, 12, 70, 35, 72, 18, 14

Offset: 1

Michel Marcus, Nov 06 2017

a(n) = 1 for n in A127724.

Michel Marcus, 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. A127724 (k-imperfect), A206369, A294650 (denominator).

A225679 is a subsequence (on squarefree indices). - Michel Marcus, Dec 22 2017

(* b = A209369 *) b[n_] := n*DivisorSum[n, LiouvilleLambda[#]/# &];
a[n_] := Numerator[b[n]/n];
Array[a, 100] (* Jean-François Alcover, Dec 04 2017 *)

rhope(p, e) = my(s=1); for(i=1, e, s=s*p + (-1)^i); s;
rho(n) = my(f=factor(n)); prod(i=1, #f~, rhope(f[i, 1], f[i, 2]));
a(n) = numerator(rho(n)/n);

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

A344826 Integers k such that k/A097621(k) is an integer.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60, 891, 1584, 1782, 3564, 4032, 4455, 4752, 7920, 8910, 17820, 20160, 22275, 23760, 44550, 49896, 86400, 89100, 100800, 118800, 249480, 349272, 399168, 694008, 1097712, 1746360, 1778400, 1995840, 2181168, 2774304, 2794176, 3470040

Offset: 1

Michel Marcus, May 29 2021, after a suggestion from Allan C. Wechsler

nonn

Allan C. Wechsler remarks that one can derive larger terms from existing terms. For instance, k = 5552064 has q = k/A097621(k) = 18. So multiplying 5552064 by 31 = A000961(18) will give a new term with q = 31.

More precisely, if k = a(n) has q = A343886(k) and m = A000961(q) such that gcd(k, m) = 1, then k*m is also a term. We could call "primitive" those terms not derived from a smaller term in this way. All the listed terms are primitive, but a({35, 36, 38, 42, 43}) allow the sequence to be extended to five larger non-primitive terms. The second and fourth one, having q = 17 resp. q = 23, both lead to a whole chain of many new terms. - M. F. Hasler, Jun 15 2021

Ray Chandler, Table of n, a(n) for n = 1..187 (terms 1..90 from Michel Marcus, terms 91..133 from Lars Blomberg)

Cf. A000961, A095874, A097621, A127724, A343886 (the ratios k/A097621(k)).

f(n) = if(isprimepower(n), sum(i=1, logint(n, 2), primepi(sqrtnint(n, i)))+1, n==1); \\ A095874
ff(n) = my(fr=factor(n)); for (k=1, #fr~, fr[k,1] = f(fr[k,1]^fr[k,2]); fr[k,2] = 1); factorback(fr); \\ A097621
isok(k) = denominator(k/ff(k)) == 1;

mappp(nn) = {my(map = Map()); mapput(map, 1, 1); my(nb=1); for (n=2, nn, if (isprimepower(n), nb++; mapput(map, n, nb));); map;}
ff(n, map) = my(fr=factor(n)); for (k=1, #fr~, fr[k, 1] = mapget(map, fr[k, 1]^fr[k, 2]); fr[k, 2] = 1); factorback(fr); \\ A097621
wa(na, nb) = {my(map = mappp(nb)); for (k=na, nb, if (denominator(k/ff(k, map)) == 1, print1(k, ", ")););}
wa(1, 10^8)

is_A344826(n)=!(n%A097621(n))
extend(n)=n*if(gcd(n, n=A000961(n/A097621(n)))==1,n) \\ Return the larger non-primitive term "derived" from a term n = a(k) with gcd(n,q) = 1, cf. COMMENTS, or zero if gcd(n,q) > 1, i.e., it cannot be "extended" that way. This allows the production of (infinitely?) many new terms from the existing ones. - M. F. Hasler, Jun 15 2021

A294650 a(n) = denominator(A206369(n))/n.

Original entry on oeis.org

1, 2, 3, 4, 5, 3, 7, 8, 9, 5, 11, 2, 13, 7, 15, 16, 17, 18, 19, 5, 7, 11, 23, 12, 25, 13, 27, 14, 29, 15, 31, 32, 33, 17, 35, 12, 37, 19, 13, 2, 41, 7, 43, 22, 45, 23, 47, 24, 49, 50, 51, 13, 53, 27, 11, 28, 19, 29, 59, 5, 61, 31, 3, 64, 65, 33, 67, 17, 69, 35

Offset: 1

Michel Marcus, Nov 06 2017

Michel Marcus, 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. A127724 (k-imperfect), A206369, A294649 (numerator).

A225680 is a subsequence (on squarefree indices). - Michel Marcus, Dec 22 2017

(* b = A209369 *) b[n_] := n*DivisorSum[n, LiouvilleLambda[#]/# &];
a[n_] := Denominator[b[n]/n];
Array[a, 100] (* Jean-François Alcover, Dec 04 2017 *)

rhope(p, e) = my(s=1); for(i=1, e, s=s*p + (-1)^i); s;
rho(n) = my(f=factor(n)); prod(i=1, #f~, rhope(f[i, 1], f[i, 2]));
a(n) = denominator(rho(n)/n);

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

A295236 Hemi-imperfect numbers: numbers such that the denominator of k/A206369(k) is equal to 2.

Original entry on oeis.org

3, 10, 42, 60, 63, 840, 1260, 12642, 18480, 18900, 18963, 154350, 228480, 252840, 379260, 3458700, 5562480, 5688900, 68772480, 1041068700, 15032156160, 53621568000, 4524679004160, 9812746944000

Offset: 1

Michel Marcus, Nov 19 2017

This is to rho (A206369) what hemiperfect numbers are to sigma (A000203).
After 3, 10 and 42, whose quotients are resp. 3/2, 5/2 and 7/2, 373316437260251755241798182764378479569038727298776522806597255168000000 is an instance of a term with quotient 9/2. - Michel Marcus, Dec 17 2017
a(25) > 10^13. - Giovanni Resta, Feb 17 2020

			3 is a term since rho(3) = 2, so 3/rho(3) is 3/2.
10 is a term since rho(10) = 4, so 10/rho(10) is 5/2.
42 is a term since rho(42) = 12, so 42/rho(42) is 7/2.

Douglas E. Iannucci, On a variation of perfect numbers, INTEGERS: Electronic Journal of Combinatorial Number Theory, 6 (2006), #A41.
László Tóth, A survey of the alternating sum-of-divisors function, arXiv:1111.4842 [math.NT], 2011-2014.
Wikipedia, Hemiperfect number

Cf. A127724 (k-imperfect), A206369 (rho).

Cf. A159907 (hemiperfect).

rho:= proc(n) local f;
  mul((f[1]^(f[2]+1)+(-1)^f[2])/(f[1]+1), f = ifactors(n)[2]);
end proc:
select(t -> denom(t/rho(t)) = 2, [$1..10^6]); # Robert Israel, Nov 20 2017

(* b = A209369 *) b[n_] := n*DivisorSum[n, LiouvilleLambda[#]/# &];
Select[Range[10^6], If[Denominator[#/b[#]] == 2, Print[#]; True, False]&] (* Jean-François Alcover, Dec 04 2017 *)

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;}
isok(n) = denominator(n/rho(n))==2;

a(20) from Jinyuan Wang, Feb 15 2020

a(21)-a(24) from Giovanni Resta, Feb 17 2020

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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A127725 Numbers that are 2-imperfect.

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A127726 Numbers that are 3-imperfect.

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A127727 Primes of the form p^e - p^(e-1) + p^(e-2) - ... + (-1)^e, where p is prime.

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A294649 a(n) = numerator(A206369(n))/n.

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A309806 Values of k in k-imperfect numbers.

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A344826 Integers k such that k/A097621(k) is an integer.

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