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.

A109712 UnitarySigmaUnitaryPhi(n) or USUP(n).

Original entry on oeis.org

1, 3, 2, 5, 4, 6, 6, 9, 8, 12, 10, 10, 12, 18, 8, 17, 16, 24, 18, 20, 12, 30, 22, 18, 24, 36, 26, 30, 28, 24, 30, 33, 20, 48, 24, 40, 36, 54, 24, 36, 40, 36, 42, 50, 32, 66, 46, 34, 48, 72, 32, 60, 52, 78, 40, 54, 36, 84, 58, 40, 60, 90, 48, 65, 48, 60, 66, 80, 44, 72, 70, 72, 72, 108, 48, 90, 60, 72, 78, 68
Offset: 1

Views

Author

Yasutoshi Kohmoto, Aug 08 2005

Keywords

Comments

a(n) is defined as follows. If n = Product p_i^r_i then a(n) = UnitarySigma(2^r_1) *UnitaryPhi(n/2^r_1) = (2^r_1+1)*Product(p_i^r_i-1), 2

Examples

			a(2^4*7^2) = UnitarySigma(2^4) * UnitaryPhi(7^2) = 17*48 = 816.

Links

Crossrefs

Cf. A006519, A047994, A092760, A034448, A065463.

Programs

  • Maple
    A109712 := proc(n)
    local a ;
    a := 1;
    if n > 1 then
        for pe in ifactors(n)[2] do
            if op(1,pe) = 2 then
                a := a*(1+op(1,pe)^op(2,pe)) ;
            else
                a := a*(op(1,pe)^op(2,pe)-1) ;
            end if;
        end do:
    end if;
    a ;
end proc:
seq(A109712(n),n=1..100) ; # R. J. Mathar, Sep 04 2018
  • Mathematica
    A034448[n_] := Sum[If[GCD[d, n/d] == 1, d, 0], {d, Divisors[n]}]; A047994[n_] := Times @@ (Power @@@ FactorInteger[n] - 1); A006519[n_] := 2^IntegerExponent[n, 2]; a[1] = 1; a[n_ /; IntegerQ[Log[2, n]]] := n+1; a[n_] := A034448[ A006519[n] ]*A047994[ n/A006519[n] ]; Table[a[n], {n, 1, 80}] (* Jean-François Alcover, Oct 03 2013 *)
f[p_, e_] := p^e - 1; f[2, e_] := 2^e + 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 17 2022 *)

Formula

a(n) = A034448(t)*A047994(n/t) where t = A006519(n).
Multiplicative with a(2^e) = 1+2^e, a(p^e) = p^e-1 for primes p>2, e>0. - R. J. Mathar, Jun 02 2011
Sum_{k=1..n} a(k) ~ c * n^2, where c = (7/10) * Product_{p prime} (1 - 1/(p*(p+1))) = (7/10) * A065463 = 0.493109... . - Amiram Eldar, Nov 17 2022

A092788 USUP perfect numbers.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 1, 2, 1, 2, 2, 2, 2, 2, 2
Offset: 1

Views

Author

Yasutoshi Kohmoto, Apr 14 2004

Keywords

Comments

USUP stands for UnitarySigmaUnitaryPhi(n) = A109712(n).

Crossrefs

Cf. A092760.

Formula

a(n) = m/A109712(m) where m = A092760(n). - R. J. Mathar, Sep 04 2018

Extensions

Adapted to match A092760. - R. J. Mathar, Sep 04 2018

A093863 Unitary sigma-unitary phi super perfect numbers: USUP(USUP(n))= n/k for some integer k.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 16, 17, 18, 20, 24, 34, 36, 40, 48, 68, 72, 80, 136, 144, 256, 257, 272, 514, 768, 1028, 1280, 2056, 2304, 2808, 4112, 4320, 4352, 20280, 65536, 65537, 65792, 88704, 131074, 196416, 196608, 262148, 327680, 524296, 589824, 998400
Offset: 1

Views

Author

Yasutoshi Kohmoto, May 11 2004

Keywords

Comments

USUP(.)= A109712(.). Where k values are 1, they define fixed points of the function USUP(USUP(n)). k values larger than 1 exist, for example USUP(USUP(4320))= 4320/2.
k = 2 for 4320, 20280, 88704, 196416, 998400, ... - Amiram Eldar, Mar 01 2019

Crossrefs

Cf. A092760, A109712.

Programs

  • Maple
    for n from 1 to 20000 do if n mod A109712(A109712(n)) = 0 then printf("%d,",n); end if; end do:
  • Mathematica
    usigma[1]=1; usigma[n_] := Times @@ (1 + Power @@@ FactorInteger[n]); A047994[n_] := Times @@ (Power @@@ FactorInteger[n] - 1); A006519[n_] := 2^IntegerExponent[ n, 2]; usup[1] = 1; usup[n_ /; IntegerQ[Log[2, n]]] := n+1; usup[n_] := usigma[ A006519[n] ]*A047994[ n/A006519[n] ];  aQ[n_]:=Divisible[n,usup[usup[n]]]; Select[Range[10000], aQ] (* Amiram Eldar, Mar 01 2019 after Jean-François Alcover at A109712 *)

Extensions

More terms from Amiram Eldar, Mar 01 2019
Showing 1-3 of 3 results.

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