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.

A328258 a(n) = Sum_{d|n, gcd(d,n/d) = 1} (-1)^(d + 1) * d.

Original entry on oeis.org

1, -1, 4, -3, 6, -4, 8, -7, 10, -6, 12, -12, 14, -8, 24, -15, 18, -10, 20, -18, 32, -12, 24, -28, 26, -14, 28, -24, 30, -24, 32, -31, 48, -18, 48, -30, 38, -20, 56, -42, 42, -32, 44, -36, 60, -24, 48, -60, 50, -26, 72, -42, 54, -28, 72, -56, 80, -30, 60, -72, 62, -32, 80, -63, 84
Offset: 1

Views

Author

Ilya Gutkovskiy, Oct 09 2019

Keywords

Comments

Excess of sum of odd unitary divisors of n over sum of even unitary divisors of n.
a(n) = n+1 iff n is in A061345 \ {1}. - Bernard Schott, Mar 05 2023

Links

Crossrefs

Cf. A002129, A034448, A077610, A109712, A192066, A254503, A306633, A360156.

Programs

  • Magma
    [&+[(-1)^(d+1)*d:d in Divisors(n)|Gcd(d, n div d) eq 1]:n in [1..70]]; // Marius A. Burtea, Oct 10 2019
  • Maple
    f:= proc(n) local t;
  mul(1 - (-1)^t[1] * t[1]^t[2], t=ifactors(n)[2])
end proc:
map(f, [$1..100]); # Robert Israel, Oct 10 2019
  • Mathematica
    a[n_] := Sum[Boole[GCD[d, n/d] == 1] (-1)^(d + 1) d, {d, Divisors[n]}]; Table[a[n], {n, 1, 65}]
a[1] = 1; a[n_] := Times @@ (1 - (-1)^First[#] First[#]^Last[#] & /@ FactorInteger[n]); Table[a[n], {n, 1, 65}]
  • PARI
    a(n) = sumdiv(n, d, if (gcd(d,n/d) == 1, (-1)^(d + 1) * d)); \\ Michel Marcus, Oct 10 2019

Formula

If n = Product (p_j^k_j) then a(n) = Product (1 - (-1)^p_j * p_j^k_j).
If n odd, a(n) = usigma(n), where usigma = A034448.
Sum_{k=1..n} a(k) ~ c * n^2, where c = zeta(2)/(14*zeta(3)) = A306633 / 14 = 0.0977451... . - Amiram Eldar, Nov 17 2022
From Amiram Eldar, Jan 28 2023: (Start)
a(n) = 2 * A192066(n) - A034448(n).
a(n) = A192066(n) - A360156(n/2) if n is even, and A192066(n) otherwise.
Dirichlet g.f.: (zeta(s)*zeta(s-1)/zeta(2*s-1))*(2^(2*s)-2^(s+2)+2)/(2^(2*s)-2). (End)

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

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 1-4 of 4 results.

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