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 1-3 of 3 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

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

A178785 a(n) is the smallest n-perfect number of the form 2^(n+1)*L, where L is an odd number with exponents <= n in its prime power factorization, and a(n)=0 if no such n-perfect number exists.

Original entry on oeis.org

60, 6552, 222768, 288288, 87360, 49585536, 25486965504, 203558400, 683289600, 556121548800
Offset: 1

Views

Author

Vladimir Shevelev, Jun 14 2010, Jun 18 2010

Keywords

Comments

Let k >= 1. In the multiplicative basis Q^(k) = {p^(k+1)^j, p runs through A000040, j=0,1,...} every positive integer m has a unique factorization of the form m = Product_{q is in Q^(k)} q^(m_q), where m_q is in {0,1,...,k}. In particular, in the case of k=1, we have the unique factorization over distinct terms of A050376. Notice that the standard prime basis is the limiting value for k tending to infinity, and, by the definition, Q^(infinity)=A000040. The number d is called a k-divisor of m if the exponents d_q in its factorization in the basis Q^(k) do not exceed m_q. A number m is called k-perfect if it equals to the sum of its proper positive k-divisors. Conjecture: a(11)=0. Note that we also know of n-perfect numbers for n = 12, 14, 15, 16, and 18.

Examples

			In case of n=2, we have the basis ("2-primes"): 2,3,5,7,8,11,13,... By the formula, we construct from the left m and from the right 2*m. By the condition, m begins from "2-prime" 8. From the right we have 8+1=3^2, therefore from the left we have 8*3^2 and from the right 3^2*(3^3-1)/(3-1)=3^2*13. Thus from the left it should be 8*3^2*13 and from the right 3^2*13*14. Finally, from the left we obtain m=8*3^2*13*7=6552 and from the right we have 2*m=3^2*13*14*8. By the construction, it is the smallest 2-perfect number of the required form. Thus a(2)=6552.

Links

Crossrefs

Cf. A000396, A050376, A007357, A092356.

Formula

m = Product_{q is in Q^(k)} q^(m_q) is a k-perfect number iff Product_{q is in Q^(k)} (q^((m_q)+1)-1)/(q-1) = 2*m.
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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