A057723 -id:A057723 - cp's OEIS frontend

A340110 Coreful 4-abundant numbers: numbers k such that csigma(k) > 4*k, where csigma(k) is the sum of the coreful divisors of k (A057723).

10584000, 12700800, 15876000, 19051200, 21168000, 22226400, 25401600, 29635200, 31752000, 37044000, 38102400, 42336000, 44452800, 47628000, 50803200, 52920000, 55566000, 57153600, 59270400, 63504000, 64033200, 66679200, 74088000, 76204800, 79380000, 84672000

Offset: 1

Amiram Eldar, Dec 28 2020

nonn

A coreful divisor d of a number k is a divisor with the same set of distinct prime factors as k, or rad(d) = rad(k), where rad(k) is the largest squarefree divisor of k (A007947).

Analogous to A068404 as A308053 is analogous to A005101.

			10584000 is a term since csigma(10584000) = 42653520 > 4 * 10584000.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Subsequence of A308053 and A340109.
Cf. A007947, A057723.
Similar sequences: A068404, A307114.

f[p_, e_] := (p^(e + 1) - 1)/(p - 1) - 1; s[1] = 1; s[n_] := Times @@ (f @@@ FactorInteger[n]); Select[Range[10^8], s[#] > 4*# &]

A364990 Coreful triperfect numbers: numbers k such that csigma(k) = 3*k, where csigma(k) is the sum of the coreful divisors of k (A057723).

Original entry on oeis.org

3600, 25200, 28224, 39600, 46800, 61200, 68400, 82800, 104400, 111600, 133200, 141120, 147600, 154800, 169200, 190800, 212400, 219600, 241200, 255600, 262800, 277200, 284400, 298800, 310464, 320400, 327600, 349200, 363600, 366912, 370800, 385200, 392400, 406800

Offset: 1

Amiram Eldar, Aug 15 2023

nonn

A coreful divisor d of a number k is a divisor with the same set of distinct prime factors as k, or rad(d) = rad(k), where rad(k) is the largest squarefree divisor of k (A007947) (the term "coreful divisor" was used by Hardy and Subbarao, 1983).
If k is a term, then also m*k is, for any squarefree m coprime to k. Thus there are infinitely many coreful triperfect numbers, and all of them can be generated from the sequence of primitive terms, which is the subsequence of powerful terms of this sequence. This sequence is c(n) = A064549(A005820(n)), i.e., the triperfect numbers (A005820), multiplied by their squarefree kernel (A007947): 3600, 28224, 1071645696, 1651818858099200, 532098668445696, 317519577357516800, ...
The asymptotic density of this sequence is Sum_{i>=1} beta(c(i))/c(i), where beta(n) = (6/Pi^2) * Product_{p|n} (p/(p+1)) = 0.0000797856... . If there are only 6 triperfect numbers, then the exact value of this density is 18575679807276818039685539/(23589576231586703755916083200 * Pi^2).

			3600 is in the sequence since its coreful divisors are {30, 60, 90, 120, 150, 180, 240, 300, 360, 450, 600, 720, 900, 1200, 1800, 3600}, whose sum is 10800 = 3 * 3600.

Amiram Eldar, Table of n, a(n) for n = 1..10000
G. E. Hardy and M. V. Subbarao, Highly powerful numbers, Congress. Numer., Vol. 37 (1983), pp. 277-307. (Annotated scanned copy)

Cf. A005820, A007947, A057723, A064549, A307958.

f[p_, e_] := (p^(e+1)-1)/(p-1)-1; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; Select[Range[500000], s[#] == 3*# &]

s(n) = {my(f = factor(n)); prod(i = 1, #f~, sigma(f[i, 1]^f[i, 2]) - 1);}
is(n) = s(n) == 3*n;

A340111 Coreful highly abundant numbers: numbers m such that csigma(m) > csigma(k) for all k < m, where csigma is the sum of the coreful divisors function (A057723).

Original entry on oeis.org

1, 2, 3, 4, 7, 8, 12, 16, 24, 32, 36, 48, 56, 64, 72, 96, 108, 128, 144, 192, 200, 216, 288, 360, 400, 432, 504, 576, 648, 720, 792, 800, 864, 1008, 1080, 1152, 1296, 1440, 1512, 1584, 1728, 1800, 1944, 2016, 2160, 2304, 2592, 2880, 3024, 3240, 3456, 3600

Offset: 1

Amiram Eldar, Dec 28 2020

nonn

A coreful divisor d of a number k is a divisor with the same set of distinct prime factors as k, or rad(d) = rad(k), where rad(k) is the largest squarefree divisor of k (A007947).

Analogous to highly abundant numbers (A002093) with the sum of the coreful divisors function (A057723) instead of the sum of divisors function (A000203).

			The first 10 values of A057723(n) for n=1..10 are: 1, 2, 3, 6, 5, 6, 7, 14, 12, 10. The record values, 1, 2, 3, 6, 7 and 14 occur at 1, 2, 3, 4, 7 and 8, the first 6 terms of this sequence.

Amiram Eldar, Table of n, a(n) for n = 1..533 (terms below 10^10)

Cf. A007947, A057723, A283052, A308053.

Similar sequences: A002093, A285614, A292983, A327634, A328134, A329883.

f[p_, e_] := (p^(e + 1) - 1)/(p - 1) - 1; s[1] = 1; s[n_] := Times @@ (f @@@ FactorInteger[n]); seq = {}; sm = 0; Do[s1 = s[n]; If[s1 > sm, sm = s1; AppendTo[seq, n]], {n, 1, 3600}]; seq

A378997 Dirichlet inverse of A057723.

Original entry on oeis.org

1, -2, -3, -2, -5, 6, -7, 2, -3, 10, -11, 6, -13, 14, 15, 6, -17, 6, -19, 10, 21, 22, -23, -6, -5, 26, 6, 14, -29, -30, -31, 2, 33, 34, 35, 6, -37, 38, 39, -10, -41, -42, -43, 22, 15, 46, -47, -18, -7, 10, 51, 26, -53, -12, 55, -14, 57, 58, -59, -30, -61, 62, 21, -10, 65, -66, -67, 34, 69, -70, -71, -6, -73, 74, 15

Offset: 1

Antti Karttunen, Dec 16 2024

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A057723.

A057723(n) = { my(f=factor(n)~); prod(i=1, #f, sigma(f[1, i]^f[2,i])-1); };
memoA378997 = Map();
A378997(n) = if(1==n,1,my(v); if(mapisdefined(memoA378997,n,&v), v, v = -sumdiv(n,d,if(dA057723(n/d)*A378997(d),0)); mapput(memoA378997,n,v); (v)));

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, dA057723(n/d) * a(d).

A339983 Coreful abundant numbers (A308053) with an even sum of coreful divisors (A057723) that are not coreful Zumkeller numbers (A339979).

Original entry on oeis.org

108, 216, 432, 540, 756, 864, 972, 1000, 1080, 1188, 1404, 1512, 1728, 1836, 1944, 2000, 2052, 2160, 2376, 2484, 2744, 2808, 3000, 3024, 3132, 3348, 3456, 3672, 3780, 3888, 3996, 4000, 4104, 4320, 4428, 4644, 4752, 4860, 4968, 5076, 5488, 5616, 5724, 5940, 6000

Offset: 1

Amiram Eldar, Dec 25 2020

nonn

			108 is a term since its set of coreful divisors, {6, 12, 18, 36, 54, 108}, has an even sum, 234 > 2*108, and it cannot be partitioned into two disjoint sets of equal sum.

Subsequence of A308053.
Cf. A057723, A339979, A339982.
Similar sequences: A171641, A323341, A323342, A323343, A323344.

q[n_] := Module[{r = Times @@ FactorInteger[n][[;; , 1]], d, sum, x}, d = r*Divisors[n/r]; (sum = Plus @@ d) >= 2*n && EvenQ[sum] && CoefficientList[Product[1 + x^i, {i, d}], x][[1 + sum/2]] == 0]; Select[Range[10^4], q]

A060640 If n = Product p_i^e_i then a(n) = Product (1 + 2p_i + 3p_i^2 + ... + (e_i+1)*p_i^e_i).

Original entry on oeis.org

1, 5, 7, 17, 11, 35, 15, 49, 34, 55, 23, 119, 27, 75, 77, 129, 35, 170, 39, 187, 105, 115, 47, 343, 86, 135, 142, 255, 59, 385, 63, 321, 161, 175, 165, 578, 75, 195, 189, 539, 83, 525, 87, 391, 374, 235, 95, 903, 162, 430, 245, 459, 107, 710, 253, 735, 273, 295, 119

Offset: 1

N. J. A. Sloane, Apr 17 2001

Equals row sums of triangle A143313. - Gary W. Adamson, Aug 06 2008
Equals row sums of triangle A127099. - Gary W. Adamson, Jul 27 2008
Sum of the divisors d2 from the ordered pairs of divisors of n, (d1,d2) with d1<=d2, such that d1|d2. - Wesley Ivan Hurt, Mar 22 2022

			a(4) = a(2^2) = 1 + (2)*(2) + (3)*(2^2) = 17;
a(6) = a(2)*a(3) = (1 + (2)*(2))*(1+(2)*(3)) = (5)*(7) = 35.
a(6) = tau(1) + 2*tau(2) + 3*tau(3) + 6*tau(6) = 1 + 2*2 + 3*2 + 6*4 = 35.

D. M. Burton, Elementary Number Theory, Allyn and Bacon Inc., Boston, MA, 1976, p. 120.

Seiichi Manyama, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harry J. Smith)

Cf. A000005, A000203, A001001, A006171, A038040 (Mobius transform), A049060, A057660, A057723, A327960 (Dirichlet inverse).

Cf. also triangles A027750, A127099, A143313.

a060640 n = sum [d * a000005 d | d <- a027750_row n]
-- Reinhard Zumkeller, Feb 29 2012

A060640 := proc(n) local ans, i, j; ans := 1: for i from 1 to nops(ifactors(n)[2]) do ans := ans*(1+sum((j+1)*ifactors(n)[2][i][1]^j,j=1..ifactors(n)[2][i][2])): od: RETURN(ans) end:

a[n_] := Total[#*DivisorSigma[1, n/#] & /@ Divisors[n]];
a /@ Range[59] (* Jean-François Alcover, May 19 2011, after Vladeta Jovovic *)
f[p_, e_] := ((e + 1)*p^(e + 2) - (e + 2)*p^(e + 1) + 1)/(p - 1)^2; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Apr 10 2022 *)

j=[]; for(n=1,200,j=concat(j,sumdiv(n,d,n/d*sigma(d)))); j

a(n)=if(n<1,0,direuler(p=2,n,1/(1-X)/(1-p*X)^2)[n]) /* Ralf Stephan */

N=66; default(seriesprecision,N); x=z+O(z^(N+1))
c=sum(j=1,N,j*x^j); t=1/prod(j=1,N, eta(x^(j)));
t=log(t);t=serconvol(t,c);
Vec(t) /* Joerg Arndt, May 03 2008 */

{ for (n=1, 1000, write("b060640.txt", n, " ", direuler(p=2, n, 1/(1 - X)/(1 - p*X)^2)[n]); ) } /* Harry J. Smith, Jul 08 2009 */

def A060640(n) :
    sigma = sloane.A000203
    return add(sigma(k)*(n/k) for k in divisors(n))
[A060640(i) for i in (1..59)] # Peter Luschny, Sep 15 2012

a(n) = Sum_{d|n} d*tau(d), where tau(d) is the number of divisors of d, cf. A000005. a(n) = Sum_{d|n} d*sigma(n/d), where sigma(n)=sum of divisors of n, cf. A000203. - Vladeta Jovovic, Apr 23 2001
Multiplicative with a(p^e) = ((e+1)*p^{e+2} - (e+2)*p^{e+1} + 1) / (p-1)^2. Dirichlet g.f.: zeta(s)*zeta(s-1)^2. - Franklin T. Adams-Watters, Aug 03 2006
L.g.f.: Sum(A060640(n)*x^n/n) = -log( Product_{j>=1} P(x^j) ) where P(x) = Product_{k>=1} (1-x^k). - Joerg Arndt, May 03 2008
G.f.: Sum_{k>=1} k*tau(k)*x^k/(1 - x^k). - Ilya Gutkovskiy, Sep 06 2018
Sum_{k=1..n} a(k) ~ n^2/24 * ((4*gamma - 1)*Pi^2 + 2*Pi^2 * log(n) + 12*Zeta'(2)), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Feb 01 2019

More terms from James Sellers, Vladeta Jovovic and Matthew Conroy, Apr 17 2001

A049060 a(n) = (-1)^omega(n)Sum_{d|n} d(-1)^omega(d), where omega(n) = A001221(n) is number of distinct primes dividing n.

Original entry on oeis.org

1, 1, 2, 5, 4, 2, 6, 13, 11, 4, 10, 10, 12, 6, 8, 29, 16, 11, 18, 20, 12, 10, 22, 26, 29, 12, 38, 30, 28, 8, 30, 61, 20, 16, 24, 55, 36, 18, 24, 52, 40, 12, 42, 50, 44, 22, 46, 58, 55, 29, 32, 60, 52, 38, 40, 78, 36, 28, 58, 40, 60, 30, 66, 125, 48, 20, 66, 80, 44, 24, 70

Offset: 1

N. J. A. Sloane

Might be called (-1)sigma(n). If x = Product p_i^r_i, then (-1)sigma(x) = Product (-1 + Sum p_i^s_i, s_i = 1 to r_i) = Product ((p_i^(r_i+1)-1)/(p_i-1)-2), with (-1)sigma(1) = 1. - Yasutoshi Kohmoto, May 23 2005

R. J. Mathar, Table of n, a(n) for n = 1..100000

Used in A049057, A049058, A049059.

Cf. A000203, A001221, A057723, A060640, A126602, A126690.

A049060 := proc(n) local it, ans, i, j; it := ifactors(n): ans := 1: for i from 1 to nops(ifactors(n)[2]) do ans := ans*(-1+sum(ifactors(n)[2][i][1]^j, j=1..ifactors(n)[2][i][2])): od: RETURN(ans) end: [seq(A049060(i),i=1..n)];

a[p_?PrimeQ] := p-1; a[1] = 1; a[n_] := Times @@ ((#[[1]]^(#[[2]] + 1) - 2*#[[1]] + 1)/(#[[1]] - 1) & ) /@ FactorInteger[n]; Table[a[n], {n, 1, 71}] (* Jean-François Alcover, May 21 2012 *)

A049060(n)={ local(i,resul,rmax,p) ; if(n==1, return(1) ) ; i=factor(n) ; rmax=matsize(i)[1] ; resul=1 ; for(r=1,rmax, p=0 ; for(j=1,i[r,2], p += i[r,1]^j ; ) ; resul *= p-1 ; ) ; return(resul) ; } { for(n=1,40, print(n," ",A049060(n)) ) ; } \\ R. J. Mathar, Oct 12 2006

apply( A049060(n)=vecprod([(f[1]^(f[2]+1)-1)\(f[1]-1)-2 | f<-factor(n)~]), [1..99]) \\ M. F. Hasler, Sep 21 2022

from math import prod
from sympy import factorint
def A049060(n): return prod((p**(e+1)-2*p+1)//(p-1) for p,e in factorint(n).items()) # Chai Wah Wu, Sep 13 2021

a(n) = Sum_{d|n} d*(-1)^A001221(d).
Multiplicative with a(p^e) = (p^(e+1)-2*p+1)/(p-1).
Simpler: a(p^e) = (p^(e+1)-1)/(p-1)-2. - M. F. Hasler, Sep 21 2022
Sum_{k=1..n} a(k) ~ c * n^2, where c = (Pi^2/12) * Product_{p prime} (1 - 2/p^2 + 2/p^3) = 0.4478559359... . - Amiram Eldar, Oct 25 2022

More terms from James Sellers, May 03 2000

Better description from Vladeta Jovovic, Apr 06 2002

A330596 Decimal expansion of Product_{primes p} (1 - 1/p^2 + 1/p^3).

Original entry on oeis.org

7, 4, 8, 5, 3, 5, 2, 5, 9, 6, 8, 2, 3, 6, 3, 5, 6, 4, 6, 4, 4, 2, 1, 5, 0, 4, 8, 6, 3, 7, 9, 1, 0, 6, 0, 1, 6, 4, 1, 6, 4, 0, 3, 4, 3, 0, 0, 5, 3, 2, 4, 4, 0, 4, 5, 1, 5, 8, 5, 2, 7, 9, 3, 9, 2, 5, 9, 2, 5, 5, 8, 6, 8, 9, 5, 4, 9, 5, 8, 8, 3, 4, 2, 1, 2, 6, 2, 0, 6, 8, 1, 4, 6, 4, 7, 0, 9, 8, 1, 3, 1, 4, 3, 3, 5, 4

Offset: 0

Vaclav Kotesovec, Dec 19 2019

The asymptotic density of A337050. - Amiram Eldar, Aug 13 2020

			0.748535259682363564644215048637910601641640343005324404515852793925925...

Cf. A057723, A065464, A065473, A065476, A065487, A328017, A330594, A330595, A337050, A372636.

Do[Print[N[Exp[-Sum[q = Expand[(p^2 - p^3)^j]; Sum[PrimeZetaP[Exponent[q[[k]], p]] * Coefficient[q[[k]], p^Exponent[q[[k]], p]], {k, 1, Length[q]}]/j, {j, 1, t}]], 110]], {t, 20, 200, 20}]

prodeulerrat(1 - 1/p^2 + 1/p^3) \\ Amiram Eldar, Mar 17 2021

Equals (6/Pi^2) * A065487. - Amiram Eldar, Jun 10 2020

A336563 Sum of proper divisors of n that are divisible by every prime that divides n.

Original entry on oeis.org

0, 0, 0, 2, 0, 0, 0, 6, 3, 0, 0, 6, 0, 0, 0, 14, 0, 6, 0, 10, 0, 0, 0, 18, 5, 0, 12, 14, 0, 0, 0, 30, 0, 0, 0, 36, 0, 0, 0, 30, 0, 0, 0, 22, 15, 0, 0, 42, 7, 10, 0, 26, 0, 24, 0, 42, 0, 0, 0, 30, 0, 0, 21, 62, 0, 0, 0, 34, 0, 0, 0, 96, 0, 0, 15, 38, 0, 0, 0, 70, 39, 0, 0, 42, 0, 0, 0, 66, 0, 30, 0, 46, 0, 0, 0, 90

Offset: 1

Antti Karttunen, Jul 27 2020

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10800
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A001065, A003557, A007947, A057723, A033879, A065487, A335341, A336564, A336565, A336566, A336567.

Cf. A005117 (positions of zeros).

f[p_, e_] := (p^(e + 1) - 1)/(p - 1) - 1; a[1] = 0; a[n_] := Times @@ f @@@ FactorInteger[n] - n; Array[a, 100] (* Amiram Eldar, May 06 2023 *)

A007947(n) = factorback(factorint(n)[, 1]);
A057723(n) = { my(r=A007947(n)); (r*sigma(n/r)); };
A336563(n) = (A057723(n)-n);
\\ Or just as:
A336563(n) = { my(x=A007947(n),y = n/x); (x*(sigma(y)-y)); };

a(n) = A057723(n) - n.
a(n) = A007947(n) * A336567(n) = A007947(n) * A001065(A003557(n)).
a(n) = A336564(n) - A033879(n).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = A065487 - 1 = 0.231291... . - Amiram Eldar, Dec 07 2023

A308135 Sum of non-coreful divisors of n.

Original entry on oeis.org

0, 1, 1, 1, 1, 6, 1, 1, 1, 8, 1, 10, 1, 10, 9, 1, 1, 15, 1, 12, 11, 14, 1, 18, 1, 16, 1, 14, 1, 42, 1, 1, 15, 20, 13, 19, 1, 22, 17, 20, 1, 54, 1, 18, 18, 26, 1, 34, 1, 33, 21, 20, 1, 42, 17, 22, 23, 32, 1, 78, 1, 34, 20, 1, 19, 78, 1, 24, 27, 74, 1, 27, 1, 40

Offset: 1

Amiram Eldar and Paolo P. Lava, May 14 2019

nonn

Non-coreful divisor d of a number k is a divisor such that rad(d) != rad(k), where rad(k) is the largest squarefree divisor of k (A007947).

			a(15) = 9. Prime factors of 15 are 3, 5 and its divisors are 1, 3, 5, 15. The non-coreful divisors are 1, 3, 5 and their sum is 9.

Amiram Eldar, Table of n, a(n) for n = 1..10000
G. E. Hardy and M. V. Subbarao, Highly powerful numbers, Congress. Numer., Vol. 37 (1983), pp. 277-307. (Annotated scanned copy)

Cf. A000203, A005101, A007947, A057723, A307888, A307958, A308029, A308127.

Cf. A013661, A065487.

with(numtheory): P:=proc(k) local a,n; a:=mul(n,n=factorset(k));
sigma(k)-a*sigma(k/a); end: seq(P(i),i=1..74);

f[p_, e_] := (p^(e + 1) - 1)/(p - 1); fc[p_, e_] := f[p, e] - 1; a[1] = 0; a[n_] := Times @@ (f @@@ FactorInteger[n]) - Times @@ (fc @@@ FactorInteger[n]); Array[a, 100]

a(n) = A000203(n) - A057723(n).

Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = A013661 - A065487 = 0.413642... . - Amiram Eldar, Dec 08 2023

cp's OEIS Frontend

A340110 Coreful 4-abundant numbers: numbers k such that csigma(k) > 4*k, where csigma(k) is the sum of the coreful divisors of k (A057723).

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A364990 Coreful triperfect numbers: numbers k such that csigma(k) = 3*k, where csigma(k) is the sum of the coreful divisors of k (A057723).

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A340111 Coreful highly abundant numbers: numbers m such that csigma(m) > csigma(k) for all k < m, where csigma is the sum of the coreful divisors function (A057723).

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A378997 Dirichlet inverse of A057723.

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A339983 Coreful abundant numbers (A308053) with an even sum of coreful divisors (A057723) that are not coreful Zumkeller numbers (A339979).

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A060640 If n = Product p_i^e_i then a(n) = Product (1 + 2*p_i + 3*p_i^2 + ... + (e_i+1)*p_i^e_i).

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A049060 a(n) = (-1)^omega(n)*Sum_{d|n} d*(-1)^omega(d), where omega(n) = A001221(n) is number of distinct primes dividing n.

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A060640 If n = Product p_i^e_i then a(n) = Product (1 + 2p_i + 3p_i^2 + ... + (e_i+1)*p_i^e_i).

A049060 a(n) = (-1)^omega(n)Sum_{d|n} d(-1)^omega(d), where omega(n) = A001221(n) is number of distinct primes dividing n.