A049060 -id:A049060 - cp's OEIS frontend

A049059 Third element t 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))).

52, 156, 780, 1248, 13260, 19968, 468480

Offset: 0

nonn

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

			Factorizations 2^2*13, 2^2*3*13, 2^2*3*5*13, 2^5*3*13, 2^2*3*5*13*17, 2^9*3*13, 2^9*3*5*61

Cf. A049057, A049058, A049060.

A034094 (-1)sigma perfect numbers: (-1)sigma(a) = m*a for some integer m, where if a = Product p(i)^r(i) then (-1)sigma(a) = Product_{i} (-1 + Sum_{s=1..r(i)} p(i)^s).

Original entry on oeis.org

1, 20, 312, 9744, 29280, 53352, 1666224, 5006880, 106798080, 133301760, 980733600, 9099742080, 18262471680, 22794600960, 1556055895680, 3577201689600, 4464942451200, 380428773854896765462278360268800000

Offset: 1

Yasutoshi Kohmoto

The indices of some terms are 1, so these numbers are fixed points of (-1)sigma where (-1)sigma is A049060.

			Factorizations 2^2*5, 2^3*3*13, 2^4*3*7*29, 2^5*3*5*61, 2^3*3^3*13*19, 2^4*3^3*7*19*29, 2^5*3^3*5*19*61, 2^10*3*5*17*409, 2^9*3*5*17*1021, 2^5*3^2*5^2*7*11*29*61, 2^7*3*5*11^2*13*23*131, 2^10*3^3*5*17*19*409, 2^9*3^3*5*17*19*1021, 2^7*3^3*5*11^2*13*19*23*131, 2^10*3^2*5^2*7*11*17*29*409, 2^9*3^2*5^2*7*11*17*29*1021, 2^24*3^3*5^5*7^2*11*17*19*29*61*233*239*467*479*70051.

Cf. A034095, A049060.

f[p_, e_] := (p^(e+1)-2*p+1)/(p-1); s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; Select[Range[10^5], Divisible[s[#], #] &] (* Amiram Eldar, Jul 07 2022 *)

msig(n) = {f = factor(n); for (i=1, #f~, f[i, 1] = (f[i,1]^(f[i,2]+1)-2*f[i,1]+1)/(f[i,1]-1); f[i, 2] = 1;); factorback(f);}
isok(n) = denominator(msig(n)/n) == 1; \\ Michel Marcus, Jun 02 2016

a(1)=1 prepended by Michel Marcus, Jun 02 2016

a(10) and a(11) switched and missing term a(13) inserted by Amiram Eldar, Jul 07 2022

A123732 Decimal expansion of Sum_{m>=1} (-1)Sigma(m)/m^3.

Original entry on oeis.org

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

Offset: 1

Yasutoshi Kohmoto, Nov 18 2006

The sum over (-1)Sigma(m)/m^2 is divergent.

			1.46400354360318202593842489434542708983961995794945...

Cf. A049060, A123733, A123734, A183699.

prodeulerrat(1 + (p^3-p^2+1)/((p-1)^2*(p+1)*(p^2+p+1))) \\ Amiram Eldar, Aug 26 2022

Equals Sum_{m>=1} A049060(m)/m^3.

Equals Product_{p prime} (1 + (p^3-p^2+1)/((p-1)^2*(p+1)*(p^2+p+1))). - Amiram Eldar, Aug 26 2022

40 more digits from R. J. Mathar, Dec 19 2010

More terms from Amiram Eldar, Aug 26 2022

A123733 Decimal expansion of Sum_{m>=1} (-1)Sigma(m)/m^4.

Original entry on oeis.org

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

Offset: 1

Yasutoshi Kohmoto, Nov 18 2006

			1.12634286431455070572497848729992683258464206350262...

Cf. A049060, A123732, A123734, A183700.

prodeulerrat(1 + (p^4-p^3+1)/((p-1)^2*(p^2+p+1)*(p^3+p^2+p+1))) \\ Amiram Eldar, Aug 26 2022

Equals Sum_{m>=1} A049060(m)/m^4.

Equals Product_{p prime} (1 + (p^4- p^3+1)/((p-1)^2*(p^2+p+1)*(p^3+p^2+p+1))). - Amiram Eldar, Aug 26 2022

46 more digits from R. J. Mathar, Dec 19 2010

More terms from Amiram Eldar, Aug 26 2022

A123734 Decimal expansion of Sum_{m>=1} (-1)Sigma(m)/m^5.

Original entry on oeis.org

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

Offset: 1

Yasutoshi Kohmoto, Nov 18 2006

			1.04712764696328011713161991597879838422134952634243...

Cf. A049060, A123732, A123733.

prodeulerrat(1 + (p^5-p^4+1)/((p-1)^2*(p^3+p^2+p+1)*(p^4+p^3+p^2+p+1))) \\ Amiram Eldar, Aug 26 2022

Equals Sum_{m>=1} A049060(m)/m^5.

Equals Product_{p prime} (1 + (p^5-p^4+1)/((p-1)^2*(p^3+p^2+p+1)*(p^4+p^3+p^2+p+1))). - Amiram Eldar, Aug 26 2022

More terms from Amiram Eldar, Aug 26 2022

A126690 Multiplicative function defined for prime powers by a(p^k) = p + p^2 + p^3 + ... + p^(k-1) - 1 (k >= 1).

Original entry on oeis.org

1, -1, -1, 1, -1, 1, -1, 5, 2, 1, -1, -1, -1, 1, 1, 13, -1, -2, -1, -1, 1, 1, -1, -5, 4, 1, 11, -1, -1, -1, -1, 29, 1, 1, 1, 2, -1, 1, 1, -5, -1, -1, -1, -1, -2, 1, -1, -13, 6, -4, 1, -1, -1, -11, 1, -5, 1, 1, -1, 1, -1, 1, -2, 61, 1, -1, -1, -1, 1, -1, -1, 10, -1, 1, -4, -1, 1, -1, -1, -13, 38, 1, -1, 1, 1, 1, 1, -5, -1, 2, 1, -1, 1, 1, 1, -29, -1, -6, -2, 4

Offset: 1

N. J. A. Sloane, Feb 14 2008, based on a posting to the Sequence Fans Mailing List by Yasutoshi Kohmoto, Feb 02 2005

If we change the definition to a(p^k) = p + p^2 + p^3 + ... + p^k - 1 (k >= 1) we get (-1)sigma(n), A049060.

			a(5) = -1, a(9) = 3-1 = 2, a(45) = (-1)*2 = -2.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A049060.

pksum := proc(L) local p,k ; p := op(1,L) ; k := op(2,L) ; (p^k-p)/(p-1)-1 ; end: A126690 := proc(n) local pe,a ; if n = 1 then RETURN(1) ; else a := 1 ; pe := ifactors(n)[2] ; for d in pe do a := a*pksum(d) ; od: RETURN(a) ; fi; end: for n from 1 to 120 do printf("%d,",A126690(n)) ; od: # R. J. Mathar, Aug 08 2008

a[1] = 1;
a[n_] := a[n] = Product[{p, k} = pk; Total[p^Range[k - 1]] - 1, {pk, FactorInteger[n]}];
Array[a, 100] (* Jean-François Alcover, Mar 31 2020 *)

a(n) = {my(f = factor(n)); prod(i = 1, #f~, (f[i,1]^f[i,2] - 2*f[i,1] + 1)/(f[i,1]-1));} \\ Amiram Eldar, May 26 2025

Dirichlet g.f.: zeta(s-1) * zeta(s) * Product_{p prime} (1 - 1/p^(s-1) - 2/p^s + 3/p^(2*s-1)). - Amiram Eldar, May 26 2025

Extended beyond a(30) by R. J. Mathar, Aug 08 2008

More terms from Antti Karttunen, Sep 23 2017

A327669 Sum of divisors of n that have an odd number of distinct prime factors.

Original entry on oeis.org

0, 2, 3, 6, 5, 5, 7, 14, 12, 7, 11, 9, 13, 9, 8, 30, 17, 14, 19, 11, 10, 13, 23, 17, 30, 15, 39, 13, 29, 40, 31, 62, 14, 19, 12, 18, 37, 21, 16, 19, 41, 54, 43, 17, 17, 25, 47, 33, 56, 32, 20, 19, 53, 41, 16, 21, 22, 31, 59, 104, 61, 33, 19, 126, 18, 82, 67, 23, 26, 84

Offset: 1

Ilya Gutkovskiy, Sep 21 2019

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..20000

Cf. A000203, A030230, A049060, A092248, A285799, A318677, A327670.

with(numtheory):
a:= n-> add(`if`(nops(factorset(d))::odd, d, 0), d=divisors(n)):
seq(a(n), n=1..80);  # Alois P. Heinz, Sep 21 2019

a[n_] := DivisorSum[n, # &, OddQ[PrimeNu[#]] &]; Table[a[n], {n, 1, 70}]

G.f.: Sum_{k>=1} A030230(k) * x^A030230(k) / (1 - x^A030230(k)).
L.g.f.: log(B(x)) = Sum_{n>=1} a(n) * x^n / n, where B(x) = g.f. of A285799.
a(n) = Sum_{d|n} d * A092248(d).
a(n) = A000203(n) - A327670(n).
a(p) = p, where p is prime.

A327670 Sum of divisors of n that have an even number of distinct prime factors.

Original entry on oeis.org

1, 1, 1, 1, 1, 7, 1, 1, 1, 11, 1, 19, 1, 15, 16, 1, 1, 25, 1, 31, 22, 23, 1, 43, 1, 27, 1, 43, 1, 32, 1, 1, 34, 35, 36, 73, 1, 39, 40, 71, 1, 42, 1, 67, 61, 47, 1, 91, 1, 61, 52, 79, 1, 79, 56, 99, 58, 59, 1, 64, 1, 63, 85, 1, 66, 62, 1, 103, 70, 60, 1, 169, 1, 75, 91

Offset: 1

Ilya Gutkovskiy, Sep 21 2019

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..20000

Cf. A000961 (positions of 1's), A000203, A030231, A049060, A285798, A318676, A327669.

with(numtheory):
a:= n-> add(`if`(nops(factorset(d))::even, d, 0), d=divisors(n)):
seq(a(n), n=1..80);  # Alois P. Heinz, Sep 21 2019

a[n_] := DivisorSum[n, # &, EvenQ[PrimeNu[#]] &]; Table[a[n], {n, 1, 75}]

G.f.: Sum_{k>=1} A030231(k) * x^A030231(k) / (1 - x^A030231(k)).
L.g.f.: log(B(x)) = Sum_{n>=1} a(n) * x^n / n, where B(x) = g.f. of A285798.
a(n) = A000203(n) - A327669(n).

A034095 Indices of (-1)sigma perfect numbers.

Original entry on oeis.org

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

Offset: 1

Yasutoshi Kohmoto

Cf. A034094, A049060.

f[p_, e_] := (p^(e+1)-2*p+1)/(p-1); r[1] = 1; r[n_] := (Times @@ f @@@ FactorInteger[n])/n; Select[r /@ Range[10^5], IntegerQ] (* Amiram Eldar, Jul 07 2022 *)

a(n) = (-1)sigma(A034094(n))/A034094(n) where (-1)sigma(n) = A049060(n).

a(1)=1 prepended and offset corrected by Michel Marcus, Jun 02 2016

a(10) and a(11) switched and missing term a(13) inserted by Amiram Eldar, Jul 07 2022

A051152 (-1)sigma sociable number of order 2: (-1)sigma((-1)sigma(x))=x, but (-1)sigma(x)<>x, 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)); (-1)sigma(1)=1.

Original entry on oeis.org

4, 5, 216, 494, 16800, 21228, 246400, 440220

Offset: 0

Yasutoshi Kohmoto

Factorizations: 4, 5, 2^3*3^3, 2*13*19, 2^5*3*5^2*7, 2^2*3*29*61

Cf. A049057, A049058, A049059, A049060, A034094, A051153.

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) ; }
isA051152(r)={ local(s,t) ; s=A049060(r) ; t=A049060(s) ; if( r == t && s !=r, return(1), return(0) ) ; }
{ for(n=1,30000000, if( isA051152(n), print(n," ") ) ; ) ; } \\ R. J. Mathar, Oct 12 2006

More terms from R. J. Mathar, Oct 12 2006

cp's OEIS Frontend

A049059 Third element t 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))).

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A034094 (-1)sigma perfect numbers: (-1)sigma(a) = m*a for some integer m, where if a = Product p(i)^r(i) then (-1)sigma(a) = Product_{i} (-1 + Sum_{s=1..r(i)} p(i)^s).

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A123732 Decimal expansion of Sum_{m>=1} (-1)Sigma(m)/m^3.

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A123733 Decimal expansion of Sum_{m>=1} (-1)Sigma(m)/m^4.

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A123734 Decimal expansion of Sum_{m>=1} (-1)Sigma(m)/m^5.

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A126690 Multiplicative function defined for prime powers by a(p^k) = p + p^2 + p^3 + ... + p^(k-1) - 1 (k >= 1).

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Maple

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A327669 Sum of divisors of n that have an odd number of distinct prime factors.

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Maple

Mathematica

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A327670 Sum of divisors of n that have an even number of distinct prime factors.

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