cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A069184 Sum of divisors d of n such that d or n/d is odd.

Original entry on oeis.org

1, 3, 4, 5, 6, 12, 8, 9, 13, 18, 12, 20, 14, 24, 24, 17, 18, 39, 20, 30, 32, 36, 24, 36, 31, 42, 40, 40, 30, 72, 32, 33, 48, 54, 48, 65, 38, 60, 56, 54, 42, 96, 44, 60, 78, 72, 48, 68, 57, 93, 72, 70, 54, 120, 72, 72, 80, 90, 60, 120, 62, 96, 104, 65, 84, 144, 68, 90, 96
Offset: 1

Views

Author

Vladeta Jovovic, Apr 10 2002

Keywords

Comments

Might be called UnitaryOrdinarySigma(n): If n=Product p_i^r_i then UOSigma(n)=UnitarySigma(2^r_1)*Sigma(n/2^r_1)=(2^r_1+1)*Product (p_i^(r_i+1)-1)/(p_i-1), p_i is not 2. - Yasutoshi Kohmoto, Jun 11 2005

Examples

			UOSigma(2^4*7^2) = UnitarySigma(2^4)*sigma(7^2) = 17*57 = 969.

Links

Crossrefs

Cf. A069733, A107749, A092356.

Programs

  • Maple
    A069184 := proc(n) local a,f,p,e; a := 1 ; for f in ifactors(n)[2] do p := op(1,f) ; e := op(2,f) ; if p = 2 then a := a*(2^e+1) ; else a := a*(p^(e+1)-1)/(p-1) ; end if; end do; a ; end proc: # R. J. Mathar, Jun 02 2011
  • Mathematica
    Table[ Sum[ d*Boole[ OddQ[d] || OddQ[n/d] ], {d, Divisors[n]}], {n, 1, 69}] (* Jean-François Alcover, Mar 26 2013 *)
f[2, e_] := 2^e+1; f[p_, e_] := (p^(e+1)-1)/(p-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 15 2020 *)
  • PARI
    a(n) = sumdiv(n, d, d*((d % 2) || ((n/d) % 2))); \\ Michel Marcus, Apr 10 2014
  • PARI
    a(n)=my(e=valuation(n,2)); sigma(n>>e) * if(e,2^e+1,1) \\ Charles R Greathouse IV, Apr 10 2014

Formula

Multiplicative with a(2^e) = 2^e+1 and a(p^e) = (p^(e+1)-1)/(p-1) for an odd prime p.
G.f.: Sum_{m>0} m*x^m*(1+x^m+x^(2*m)-x^(3*m))/(1-x^(4*m)).
Dirichlet g.f.: zeta(s) *zeta(s-1) *(1-2^(1-2s)). - R. J. Mathar, Jun 02 2011 [simplified by Michael Shamos, May 14 2025]
Sum_{k=1..n} a(k) ~ 7*Pi^2*n^2 / 96. - Vaclav Kotesovec, Feb 08 2019

Extensions

Edited by N. J. A. Sloane, Aug 29 2008 at the suggestion of R. J. Mathar

A091321 OU-Sigma multiperfect numbers.

Original entry on oeis.org

1, 6, 28, 90, 120, 496, 672, 8128, 10080, 63700, 220500, 523776, 1323000, 1528800, 2056320, 7856640, 33550336, 44553600, 162729000, 252927360, 459818240, 1379454720, 1476304896, 1980840960, 8589869056
Offset: 1

Views

Author

Yasutoshi Kohmoto, Feb 17 2004

Keywords

Comments

The OU-Sigma function is defined as OU-Sigma(n) = A107749(n).
Then an OU-Sigma perfect number satisfies OU-Sigma(n) = k*n for some k.
Every perfect number is here because OE-Sigma(2^(m-1)*M_m) = Sigma(2^(m-1))*UnitarySigma(M_m) = Sigma(2^(m-1))*Sigma(M_m) = 2^m*M_m.
Also in the sequence are 33550336, 8589869056, 22144573440, 51001180160, 153003540480, 243643438080, 583125903360, 71724486113280, 1555825650042470400, but there may be missing terms in between.

Examples

			Sequence begins 2*3, 2*3^2*5, 2^2*7, 2^2*5^2*7^2*13, 2^3*3*5, 2^4*31, 2^5*3^2*5*7, ...

Crossrefs

Cf. A107749, A091322.

Programs

  • Mathematica
    fun[p_,e_] := If[p==2, 2^(e+1)-1, p^e+1]; f[n_] := If[n==1, 1, Times @@ fun @@@ FactorInteger[n]]; aQ[n_] := Divisible[f[n], n]; Select[Range[65000], aQ] (* Amiram Eldar, Mar 17 2019 *)
  • PARI
    f(n)= my(fm=factor(n)); prod(k=1, matsize(fm)[1], if(fm[k, 1]==2, 2^(fm[k, 2]+1)-1, fm[k, 1]^fm[k, 2]+1)); \\ A107749
isok(n) = (f(n) % n) == 0; \\ Michel Marcus, Jan 24 2019

Extensions

Terms 220500 to 2056320 by R. J. Mathar, Jun 02 2011
Corrected and extended by Michel Marcus, Jan 24 2019
a(19)-a(25) from Amiram Eldar, Mar 17 2019

A091322 k-values associated with A091321.

Original entry on oeis.org

1, 2, 2, 2, 3, 2, 3, 2, 3, 2, 2, 3, 2, 3, 3, 3, 2, 3, 2, 3, 3, 4, 3, 3, 2
Offset: 1

Views

Author

Yasutoshi Kohmoto, Feb 17 2004

Keywords

Crossrefs

Cf. A091321

Formula

a(n) = A107749(A091321(n))/ A091321(n). - R. J. Mathar, Jun 02 2011

Extensions

Updated according to the correction of A091321. - R. J. Mathar, Jun 02 2011
a(1) and a(15)-a(25) from Amiram Eldar, Mar 17 2019

A126850 a(n) = OrdinaryUnitarySigma(a(n-1)).

Original entry on oeis.org

2, 3, 4, 7, 8, 15, 24, 60, 168, 480, 1512, 3360, 12096, 28448, 64512, 163760, 401760, 991872, 2399040, 6858000, 13999104, 32752000, 69400800, 172186560, 517867392, 1666990080, 5662137600, 14475575296, 33946612000, 73359820800, 158022774000
Offset: 2

Views

Author

Yasutoshi Kohmoto, Feb 24 2007

Keywords

Crossrefs

Cf. A107749, A126849.
Cf. A107749, A126849, A007497.

Programs

  • Maple
    A034448 := proc(n) local ifs,d ; if n = 1 then 1; else ifs := ifactors(n)[2] ; mul(1+ op(1,op(d,ifs))^op(2,op(d,ifs)),d=1..nops(ifs)) ; fi ; end: A006519 := proc(n) local i ; for i in ifactors(n)[2] do if op(1,i) = 2 then RETURN( op(1,i)^op(2,i) ) ; fi ; od: RETURN(1) ; end: A107749 := proc(n) local p2 ; p2 := A006519(n) ; numtheory[sigma](p2)*A034448(n/p2) ; end: A126850 := proc(n) option remember ; if n = 1 then 2; else A107749(A126850(n-1)) ; fi ; end: seq(A126850(n),n=1..40) ; # R. J. Mathar, Jun 15 2008
  • Mathematica
    f[2, e_] := 2^(e + 1) - 1;
f[p_, e_] := p^e + 1;
A107749[n_] := If[n == 1, 1, Times @@ f @@@ FactorInteger[n]];
a[n_] := a[n] = If[n == 2, 2, A107749[a[n - 1]]];
Table[a[n], {n, 2, 32}] (* Jean-François Alcover, Jul 22 2024, after Amiram Eldar in A107749 *)

Formula

a(n)= A107749(a(n-1)). - R. J. Mathar, Jun 15 2008

Extensions

Edited and extended by R. J. Mathar, Jun 15 2008

A365207 The number of divisors d of n such that gcd(d, n/d) is a power of 2 (A000079).

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 4, 2, 4, 2, 6, 2, 4, 4, 5, 2, 4, 2, 6, 4, 4, 2, 8, 2, 4, 2, 6, 2, 8, 2, 6, 4, 4, 4, 6, 2, 4, 4, 8, 2, 8, 2, 6, 4, 4, 2, 10, 2, 4, 4, 6, 2, 4, 4, 8, 4, 4, 2, 12, 2, 4, 4, 7, 4, 8, 2, 6, 4, 8, 2, 8, 2, 4, 4, 6, 4, 8, 2, 10, 2, 4, 2, 12, 4, 4
Offset: 1

Views

Author

Amiram Eldar, Aug 26 2023

Keywords

Comments

The sum of these divisors is A107749(n).

Links

Crossrefs

Cf. A000005, A000079, A000265, A007814, A034444, A038838, A042968, A107749, A122132.
Cf. A001620, A013661, A073002.

Programs

  • Mathematica
    f[p_, e_] := If[p == 2, e + 1, 2]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
  • PARI
    a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,1] == 2, f[i,2]+1, 2));}

Formula

Multiplicative with a(2^e) = e+1 and a(p^e) = 2 for an odd prime p.
a(n) <= A000005(n), with equality if and only if n is in A122132 (or equivalently, n is not in A038838).
a(n) >= A034444(n), with equality if and only if n is not divisible by 4 (A042968).
a(n) = A000005(A006519(n)) * A034444(A000265(n)).
a(n) = A034444(n) * (A007814(n)+1)/2^(1 - (n mod 2)).
Dirichlet g.f.: (4^s/(4^s-1)) * zeta(s)^2/zeta(2*s).
Sum_{k==1..n} a(k) ~ (8/Pi^2)*n*(log(n) + 2*gamma - 2*log(2)/3 - 2*zeta'(2)/zeta(2) - 1), where gamma is Euler's constant (A001620).
Showing 1-5 of 5 results.

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