A049060 -id:A049060 - cp's OEIS frontend

A258077 Numbers x such that (-1)sigma(x) | sigma(x), where (-1)sigma(x) is defined in A049060 and sigma(x) is the sum of the divisors of x (A000203).

1, 2, 3, 6, 14, 15, 30, 35, 42, 70, 78, 105, 190, 210, 348, 357, 418, 570, 714, 812, 910, 1045, 1254, 2090, 2436, 2730, 3135, 4060, 4522, 4674, 5278, 6270, 9990, 10659, 12180, 12441, 13566, 14630, 15834, 16770, 20026, 21318, 21978, 23374, 24244, 24871, 24882

Offset: 1

Paolo P. Lava, May 19 2015

nonn

			(-1)sigma(1) = 1, sigma(1) = 1 and 1 / 1 = 1;
(-1)sigma(2) = 1, sigma(2) = 3 and 3 / 1 = 3;
(-1)sigma(35) = 24, sigma(35) = 48 and 48 / 24 = 2; etc.

Amiram Eldar, Table of n, a(n) for n = 1..5000 (terms 1..100 from Paolo P. Lava)

Cf. A000203, A049060, A258079.

with(numtheory): P:=proc(q) local a,b,i,j,n; for n from 1 to q do a:=ifactors(n)[2]:
b:=1; for i from 1 to nops(a) do b:=b*(-1+sum(a[i][1]^j,j=1..a[i][2])): od:
if type(sigma(n)/b,integer) then print(n); fi; od; end: P(10^6);

A258101 Number x such that usigma(x) = (-1)sigma(x), where usigma(x) is the sum of unitary divisors of x (A034448) and (-1)sigma(x) is defined in A049060 .

Original entry on oeis.org

1, 4, 15867, 21357, 49887, 63468, 69875, 85428, 86387, 149875, 199548, 247475, 271607, 279500, 293944, 318681, 345548, 599500, 637659, 989900, 1086428, 1169091, 1274724, 1897875, 1913571, 2550636, 2665269, 2801880, 2855691

Offset: 1

Paolo P. Lava, May 20 2015

nonn

			usigma(1) = (-1)sigma(1) = 1;
usigma(4) =  (-1)sigma(4) = 5;
usigma(15867) = (-1)sigma(15867) = 18480; etc.

Amiram Eldar, Table of n, a(n) for n = 1..1000

Cf. A034448, A049060, A258106.

with(numtheory): P:=proc(q) local a,b,d,k,n;
a:=0; for n from 1 to q do a:=divisors(n); d:=0; for k from 1 to nops(a)
do if gcd(a[k],n/a[k])=1 then d:=d+a[k]; fi; od; a:=ifactors(n)[2]; b:=1;
for k from 1 to nops(a) do b:=b*(-1+sum(a[k][1]^j,j=1..a[k][2])); od;
if b=d then print(n); fi; od; end: P(10^9);

aQ[n_] := Module[{f = FactorInteger[n]}, p = f[[;;,1]]; e = f[[;;,2]]; Times@@(p^e+1) == Times@@((p^(e+1)-2*p+1)/(p-1))]; Join[{1}, Select[Range[2, 200000 ], aQ]] (* Amiram Eldar, Jun 23 2019 *)

A258106 Number x such that sigma(x) = usigma(x) + (-1)sigma(x), where sigma(x) is the sum of divisors of x (A000203), usigma(x) is the sum of unitary divisors of x (A034448) and (-1)sigma(x) is defined in A049060.

Original entry on oeis.org

1998, 3876, 4524, 10062, 21582, 45220, 52780, 85428, 125976, 226100, 263900, 271092, 511428, 597012, 602946, 839106, 1033974, 1130500, 1274724, 1280532, 1319500, 1435764, 1469720, 1575860, 1810926, 1895706, 2171364, 2550636, 3162740, 4083366, 4766034, 5652500

Offset: 1

Paolo P. Lava, May 20 2015

nonn

The definition implies that the terms of the sequence could be defined as the numbers x such that (-1)sigma(x) is equal to the sum of the non-unitary divisors of x.

			usigma(1998) = 3192, (-1)sigma(1998) = 1368 and 3191 + 1368 = 4560 = sigma(1998);
usigma(3876) = 7200, (-1)sigma(3876) = 2880 and 7200 + 2880 = 10080 = sigma(3876);
usigma(4524) = 8400, (-1)sigma(4524) = 3360 and 8400 + 3360 = 11760 = sigma(4524); etc.

Amiram Eldar, Table of n, a(n) for n = 1..1000

Cf. A000203, A034448, A049060, A258101.

with(numtheory): P:=proc(q) local a,b,c,d,i,k,n; a:=0; b:=0;
for n from 1 to q do a:=divisors(n); d:=0; for k from 1 to nops(a) do
if gcd(a[k],n/a[k])>1 then d:=d+a[k]; fi; od; a:=ifactors(n)[2]; b:=1;
for i from 1 to nops(a) do b:=b*(-1+sum(a[i][1]^j,j=1..a[i][2])); od;
if b=d then print(n); fi; od; end: P(10^9);

aQ[n_] := Module[{f = FactorInteger[n]}, p = f[[;; , 1]]; e = f[[;; , 2]]; Times@@((p^(e+1)-1)/(p-1)) == Times@@(p^e+1) + Times@@((p^(e+1)-2*p+1)/(p-1))]; Select[Range[2, 100000], aQ] (* Amiram Eldar, Jun 25 2019 *)

More terms from Amiram Eldar, Jun 25 2019

A122483 Numbers m such that A049060(m)sigma(m) = kuphi(m)*m for some integer k.

Original entry on oeis.org

6, 140, 312, 1560, 14384, 18018, 40992, 2337400, 7012200, 11027016, 231402600, 534775296, 9866296440, 11453072202

Offset: 1

Yasutoshi Kohmoto, Sep 30 2006

If both 2^n-3 and 2^n-1 are prime then numbers of the form 2^(n-1)*(M_n-2)*M_n appear in the sequence, where M_n means Mersenne prime.

			2^8*7*19*37*73*509, 2^8*5*7*19*37*509, 2^8*5^2*7*19*29*31*37*509, 2^9*3*11*31*1021, 2^9*3*7*11^2*19*31*131*1021, 2^11*3^6*5*7*13*23*137*467*1093*4093, 2^13*3*11*43*127*16381, 2^13*3*7*11^2*19*43*127*131*16381 are terms, but there may be many other terms between 3*10^7 and them.

Cf. A123124, A000203 (sigma), A047994 (uphi), A049060.

f[p_, e_] := (p^(e+1)-2*p+1) * (p^(e+1)-1)/((p-1)^2 * (p^e - 1)); q[n_] := IntegerQ[(Times @@ f @@@ FactorInteger[n])/n]; Select[Range[2, 10^5], q] (* Amiram Eldar, Sep 19 2022 *)

More terms from R. J. Mathar, Oct 01 2006

a(12)-a(14) from Amiram Eldar, Sep 19 2022

A258079 Numbers x such that (-1)sigma(x) | sigma(x) - x, where (-1)sigma(x) is defined in A049060 and sigma(x) - x is the sum of the aliquot parts of x (A001065).

Original entry on oeis.org

1, 2, 6, 10, 36, 168, 1098, 1204, 2840, 19488, 1881124, 3105700, 445577184, 105314212224, 134633963520, 2167649138304

Offset: 1

Paolo P. Lava, May 19 2015

Ratio is equal to 1 for 2, 36, 1204, 2840, 1881124, 3105700, ...

a(17) > 10^13. - Hiroaki Yamanouchi, Sep 11 2015

			(-1)sigma(1) = 1, sigma(1) - 1 = 1 - 1 = 0 and 0 / 1 = 0;
(-1)sigma(2) = 1, sigma(2) - 2 = 3 - 2 = 1 and 1 / 1 = 1;
(-1)sigma(36) = 55, sigma(36) - 36 = 91 - 36 = 55 and 55 / 55 = 1; etc.

Cf. A001065, A049060, A258077.

with(numtheory): P:=proc(q) local a,b,i,j,n; for n from 1 to q do a:=ifactors(n)[2]:
b:=1; for i from 1 to nops(a) do b:=b*(-1+sum(a[i][1]^j,j=1..a[i][2])): od:
if type((sigma(n)-n)/b,integer) then print(n); fi; od; end: P(10^6);

a(13)-a(16) from Hiroaki Yamanouchi, Sep 11 2015

A035479 (-1)sigma sequence: a(n) = (-1)sigma(a(n-1)), where if (-1)sigma(k) = A049060(k) and a(1) = 624.

Original entry on oeis.org

624, 696, 728, 936, 1716, 1200, 1682, 869, 780, 480, 488, 780, 480, 488, 780, 480, 488, 780, 480, 488, 780, 480, 488, 780, 480, 488, 780, 480, 488, 780, 480, 488, 780, 480, 488, 780, 480, 488, 780, 480, 488, 780, 480, 488, 780, 480, 488, 780, 480, 488, 780, 480

Offset: 1

Yasutoshi Kohmoto

nonn

This sequence becomes a cycle of period 3: 780, 480, 488. It is called a (-1)sigma sociable group of order 3.

			Factorizations: 2^4*3*13, 2^3*3*29, 2^3*7*13, 2^3*3^2*13, 2^2*3*11*13,  2^4*3*5^2, 2*29^2, 11*79, 2^2*3*5*13, 2^5*3*5, 2^3*61.

Cf. A049060.

f[p_, e_] := (p^(e + 1) - 1)/(p - 1) - 2; s[n_] := Times @@ f @@@ FactorInteger[n]; s[1] = 1; NestList[s, 624, 50] (* Amiram Eldar, Aug 26 2022 *)

Offset corrected by Amiram Eldar, Aug 26 2022

A057723 Sum of positive divisors of n that are divisible by every prime that divides n.

Original entry on oeis.org

1, 2, 3, 6, 5, 6, 7, 14, 12, 10, 11, 18, 13, 14, 15, 30, 17, 24, 19, 30, 21, 22, 23, 42, 30, 26, 39, 42, 29, 30, 31, 62, 33, 34, 35, 72, 37, 38, 39, 70, 41, 42, 43, 66, 60, 46, 47, 90, 56, 60, 51, 78, 53, 78, 55, 98, 57, 58, 59, 90, 61, 62, 84, 126, 65, 66, 67, 102, 69, 70

Offset: 1

Leroy Quet, Oct 27 2000

			The divisors of 12 that are divisible by both 2 and 3 are 6 and 12. So a(12) = 6 + 12 = 18.

Ivan Neretin, Table of n, a(n) for n = 1..10000
József Sándor and Krassimir Atanassov, Some new arithmetic functions, Notes on Number Theory and Discrete Mathematics, Volume 30, 2024, Number 4, Pages 851-856. See sigma- function.

Row sums of triangle A284318.
Cf. A000203 (sigma), A007947 (rad), A005361 (number of these divisors).
Cf. A049060 and A060640 (other sigma-like functions).
Cf. A065487, A330596.

[&*PrimeDivisors(n)*SumOfDivisors(n div &*PrimeDivisors(n)): n in [1..70]]; // Vincenzo Librandi, May 14 2015

seq(mul(f[1]*(f[1]^f[2]-1)/(f[1]-1), f = ifactors(n)[2]), n = 1 .. 100); # Robert Israel, May 13 2015

Table[(b = Times @@ FactorInteger[n][[All, 1]])*DivisorSigma[1, n/b], {n, 70}] (* Ivan Neretin, May 13 2015 *)
f[p_, e_] := (p^(e+1)-1)/(p-1) - 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 15 2023 *)

a(n) = {my(f = factor(n)); for (i=1, #f~, f[i,2]=1); my(pp = factorback(f)); sumdiv(n, d, if (! (d % pp), d, 0));} \\ Michel Marcus, May 14 2015

If n = Product p_i^e_i then a(n) = Product (p_i + p_i^2 + ... + p_i^e_i).
a(n) = rad(n)*sigma(n/rad(n)) = A007947(n)*A000203(A003557(n)). - Ivan Neretin, May 13 2015
Dirichlet g.f.: zeta(s) * zeta(s-1) * Product(p prime, 1 - p^(-s) + p^(1-2*s)). - Robert Israel, May 13 2015
Sum_{k=1..n} a(k) ~ c * Pi^2 * n^2 / 12, where c = A330596 = Product_{primes p} (1 - 1/p^2 + 1/p^3) = 0.7485352596823635646442150486379106016416403430053244045... - Vaclav Kotesovec, Dec 18 2019
a(n) = Sum_{d|n, rad(d)=rad(n)} d. - R. J. Mathar, Jun 02 2020
Lim_{n->oo} (1/n) * Sum_{k=1..n} a(k)/k = Product_{p prime}(1 + 1/(p*(p^2-1))) = 1.231291... (A065487). - Amiram Eldar, Jun 10 2020
a(n) = Sum_{d|n, gcd(d, n/d) = 1} (-1)^omega(n/d) * sigma(d). - Ilya Gutkovskiy, Apr 15 2021

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

A049057 First element r of (-1)sigma sociable triple (r,s,t): s=(-1)sigma(r), t=(-1)sigma(s), r=(-1)sigma(t), where if x=Product p(i)^r(i), then (-1)sigma(x)=Product(-1+(Sum p(i)^s(i), s(i)=1 to r(i))).

Original entry on oeis.org

20, 40, 52, 60, 104, 120, 156, 312, 480, 488, 780, 1248, 1464, 1560, 5856, 6240, 7320, 7680, 8168, 9744, 13260, 19968, 24504, 26520, 29280, 93696, 98016, 99840, 106080, 122520, 124440, 468480, 490080, 497760, 1568256, 1697280, 2082840, 7841280

Offset: 0

Yasutoshi Kohmoto

nonn

Since the definition is circular and the definition does not specify that r is the largest number in the triple, for each r in the sequence also the s and t show up. - R. J. Mathar, Oct 12 2006
Otherwise, if the definition is supposed to mean "smallest r of a triple....", the list is 20, 40, 104, 312, 480, 1248, 5856, 7680, 9744, 19968, 29280, ... - R. J. Mathar, Oct 12 2006
If, as a third interpretation, the sequence is "Smallest r of a triple of pairwise different numbers r,s,t with..." then the sequence is 40, 104, 480, 1248, 5856, 7680, ... - R. J. Mathar, Oct 12 2006

			Factorizations 2^3*5, 2^3*13, 2^3*61, 2^3*3*5*13, 2^3*1021, 2^3*3*5*13*17, 2^5*3*5*17*61.
(r,s,t)=(20,20,20), (40,52,60), (52,60,40), (60,40,52), (104,156,120), (120,104,156), ..., (29280,29280,29280).

Cf. A049058, A049059, A049060.

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) ; } isA049057(r)={ local(s,t) ; s=A049060(r) ; t=A049060(s) ; if( r == A049060(t), return(1), return(0) ) ; } { for(n=1,30000000, if( isA049057(n), print(n," ",factor(n)) ) ; ) ; } \\ R. J. Mathar, Oct 12 2006

Corrected and extended by R. J. Mathar, Oct 12 2006

A049058 Second element s of (-1)sigma sociable triple (r,s,t): s=(-1)sigma(r), t=(-1)sigma(s), r=(-1)sigma(t), where if x=Product p(i)^r(i), then (-1)sigma(x)=Product(-1+(Sum p(i)^s(i), s(i)=1 to r(i))).

Original entry on oeis.org

60, 120, 480, 1464, 7680, 24504, 490080

Offset: 0

Yasutoshi Kohmoto

nonn

Definition unclear, see comments in A049057. - Sean A. Irvine, Jul 17 2021

			Factorizations 2^2*3*5, 2^3*3*5, 2^5*3*5, 2^3*3*61, 2^9*3*5, 2^3*3*1021, 2^5*3*5*1021

Cf. A049057, A049059, A049060.

cp's OEIS Frontend

A258077 Numbers x such that (-1)sigma(x) | sigma(x), where (-1)sigma(x) is defined in A049060 and sigma(x) is the sum of the divisors of x (A000203).

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A258101 Number x such that usigma(x) = (-1)sigma(x), where usigma(x) is the sum of unitary divisors of x (A034448) and (-1)sigma(x) is defined in A049060 .

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A258106 Number x such that sigma(x) = usigma(x) + (-1)sigma(x), where sigma(x) is the sum of divisors of x (A000203), usigma(x) is the sum of unitary divisors of x (A034448) and (-1)sigma(x) is defined in A049060.

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A122483 Numbers m such that A049060(m)*sigma(m) = k*uphi(m)*m for some integer k.

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A258079 Numbers x such that (-1)sigma(x) | sigma(x) - x, where (-1)sigma(x) is defined in A049060 and sigma(x) - x is the sum of the aliquot parts of x (A001065).

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A035479 (-1)sigma sequence: a(n) = (-1)sigma(a(n-1)), where if (-1)sigma(k) = A049060(k) and a(1) = 624.

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A057723 Sum of positive divisors of n that are divisible by every prime that divides n.

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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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A122483 Numbers m such that A049060(m)sigma(m) = kuphi(m)*m for some integer k.

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).