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 1-4 of 4 results.

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

A107749 OrdinaryUnitarySigma(n): If n = Product p_i^r_i then OUSigma(n) = Sigma(2^r_1)*UnitarySigma(n/2^r_1).

Original entry on oeis.org

1, 3, 4, 7, 6, 12, 8, 15, 10, 18, 12, 28, 14, 24, 24, 31, 18, 30, 20, 42, 32, 36, 24, 60, 26, 42, 28, 56, 30, 72, 32, 63, 48, 54, 48, 70, 38, 60, 56, 90, 42, 96, 44, 84, 60, 72, 48, 124, 50, 78, 72, 98, 54, 84, 72, 120, 80, 90, 60, 168, 62, 96, 80, 127, 84, 144, 68, 126, 96
Offset: 1

Views

Author

Yasutoshi Kohmoto, Jun 11 2005, Feb 24 2007

Keywords

Comments

The sum of divisors d of n such that gcd(d, n/d) is a power of 2 (A000079). - Amiram Eldar, Aug 26 2023

Examples

			OUSigma(2^4*7^2) = Sigma(2^4)*UnitarySigma(7^2) = 31*50 = 1550.

Links

Crossrefs

Cf. A000079, A069184, A091321, A000203, A034448, A006519, A306633.

Programs

  • Maple
    A107749 := 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)-1) ; else a := a*(p^e+1) ; end if; end do; a ; end proc: # R. J. Mathar, Jun 02 2011
  • Mathematica
    f[2, e_] := 2^(e+1)-1; f[p_, e_] := p^e+1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 15 2020 *)
  • PARI
    a(n)=local(fm);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))

Formula

a(n) = A000203(p2) * A034448(n/p2), where p2 = A006519(n). - R. J. Mathar, Jun 15 2008
Multiplicative with a(2^e) = 2^(e+1)-1, a(p^e) = p^e+1 for p>2, e>0.
Sum_{k=1..n} a(k) ~ c * n^2, where c = (4/7) * zeta(2)/zeta(3) = (4/7) * A306633 = 0.781961... . - Amiram Eldar, Nov 01 2022
Dirichlet g.f.: (4^s/(4^s-2)) * zeta(s)*zeta(s-1)/zeta(2*s-1). - Amiram Eldar, Aug 26 2023

Extensions

Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, Jun 08 2007
More terms from R. J. Mathar, Jun 15 2008
Name corrected by Franklin T. Adams-Watters, Aug 24 2013

A092356 UO-sigma multiperfect numbers: n such that A069184(n)/n is an integer.

Original entry on oeis.org

1, 6, 60, 1080, 6552, 36720, 47520, 87360, 222768, 288288, 8173440, 49585536, 203558400, 683289600, 920387520, 4201148160, 25486965504, 556121548800, 1610457666048, 3633511924224, 4399770343643136, 6075071799091200, 9926754576979968, 27220195859304960, 66800080530869760, 629720915643477504
Offset: 1

Views

Author

Yasutoshi Kohmoto, Mar 20 2004

Keywords

Comments

The UO-sigma function is defined by UO-sigma(n) = A069184(n).
E.g., UO-sigma(2^4*7^2) = UnitarySigma(2^4)*sigma(7^2) = 17*57 = 969. So UO-sigma(n) = UnitarySigma(n) if n=2^r, or = sigma(n) if GCD(2,n)=1.
A UO-sigma perfect number satisfies UO-sigma(n) = k*n for some k.
The initial values of k are 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2. However, I conjecture that every positive integer >= 2 must appear.
Some interesting subsequences exist: s(n) := {a(1), a(4), a(9), a(11), ...} has the property that s(n-1)|s(n): 2*3, 2^3*3^2*7*13, 2^5*3^2*7*13*11, 2^7*3^2*7*11*13*43, 2^8*3^2*7*11*13*43*257, ...

Examples

			Sequence begins: 2*3, 2^2*3*5, 2^3*3^3*5, 2^3*3^2*7*13, 2^4*3^3*5*17, 2^5*3^3*5*11, 2^6*3*5*7*13, 2^4*3^2*7*13*17, 2^5*3^2*7*13*11, 2^7*3^3*5*11*43, 2^7*3^2*7*11*13*43, ...

Links

Crossrefs

Cf. A091321.

Programs

  • PARI
    is(n)=my(e=valuation(n, 2)); (sigma(n>>e) * if(e, 2^e+1, 1)) % n == 0 \\ Charles R Greathouse IV, Apr 10 2014

Extensions

Corrected by Andrew Lelechenko, Apr 10 2014

A092760 Unitary-sigma unitary-phi perfect numbers.

Original entry on oeis.org

6, 20, 72, 272, 2808, 5280, 12480, 65792, 251719680, 4295032832, 39462420480, 2151811200000, 375297105592320, 4238621367336960, 20203489717239782783648394117120, 84353101158454670682576150304666023245622804480
Offset: 1

Views

Author

Yasutoshi Kohmoto, Apr 14 2004

Keywords

Comments

USUP(n) = n/k for some integer k where USUP(n) = A109712(n).

Examples

			USUP(2^4*7^2)=UnitarySigma(2^4)*UnitaryPhi(7^2)=17*48= 816
So USUP(n) = UnitarySigma(n) if n=2^r = UnitaryPhi(n) if GCD(2,n)=1
Examples : a(1)=2*F_0, a(5)=2^5*11*F_0*F_1, ...., a(12)=2^40*4278255361*F_0*F_1*F_2*F_3*F_4.
Factorizations : 2*3; 2^2*5; 2^3*3^2; 2^4*17; 2^5*3*11*5; 2^6*5*13*3; 2^8*257; 2^12*3*5*17*241; 2^16*65537; 2^14*3*5*7^2*29*113; 2^10*3*5^5*7*11*41*71; 2^17*3*5*17*257*43691; 2^20*3*5*17*257*61681; 2^40*3*5*17*257*65537*4278255361; 2^48*3^6*5*7*11*13*17*23*47*137*193*65537*115903*22253377; 2^48*3^7*5*7*11*13*17*23*47*137*193*1093*65537*115903*22253377

Crossrefs

Cf. A092788, A091321, A092356

Programs

  • Maple
    A047994 := proc(n) local ifs,d ; if n = 1 then 1; else ifs := ifactors(n)[2] ; mul(op(1,op(d,ifs))^op(2,op(d,ifs))-1,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: Usup := proc(n) local p2 ; p2 := A006519(n) ; (p2+1)*A047994(n/p2) ; end: for n from 1 do if n mod Usup(n) = 0 then print(n) ; fi; od: # R. J. Mathar, Dec 15 2008

Formula

Numbers of form 2^(2^m)*F_m appear in the sequence, where F_m means Fermat prime 2^(2^m)+1. Because USUP(2^(2^m)*F_m)=UnitarySigma(2^(2^m))*UnitaryPhi(F_m)=(2^(2^m)+1)*(F_m-1)= F_m*2^(2^m)).
Numbers of the following form exist in the sequence. For j=0 to 4, k*product F_i, i=0 to j, F_i means Fermat prime 2^(2^n)+1, k is an integer.

Extensions

2808 inserted by R. J. Mathar, Dec 15 2008
39462420480 and 2151811200000 inserted by Andrew Lelechenko, Apr 10 2014
Showing 1-4 of 4 results.

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