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.

A065218 Consider the subsets of proper divisors of a number that sum to the number. These are numbers that set a record number of such subsets.

Original entry on oeis.org

1, 6, 12, 24, 36, 48, 60, 120, 180, 240, 360, 720, 840, 1260, 1680, 2520, 5040, 7560, 10080, 15120, 20160, 25200, 27720, 45360, 50400, 55440, 83160, 110880, 166320, 221760, 277200, 332640, 498960, 554400, 665280, 720720, 831600, 1081080, 1441440
Offset: 1

Views

Author

Jud McCranie, Oct 21 2001

Keywords

Comments

Indices of records in A065205 and A033630. The corresponding records (number of subsets) are in A065219.
This sequence is not a subset of A002182: 831600 belongs to this sequence but not A002182.

Examples

			Proper divisors of 12 are {1, 2, 3, 4, 6}. Two subsets of this sum to 12: {2, 4, 6} and {1, 2, 3, 6} - more than any smaller number, so 12 is in the sequence.

Links

Crossrefs

Cf. A065219, A064771, A063205, A005835, A002182, A004394.

Programs

  • Mathematica
    With[{s = Table[-1 + SeriesCoefficient[Series[Times @@ ((1 + z^#) & /@ Divisors[n]), {z, 0, n}], n], {n, 2520}]}, FirstPosition[s, #][[1]] & /@ Union@ FoldList[Max, s]] (* Michael De Vlieger, Oct 10 2017 *)

Extensions

More terms from Franklin T. Adams-Watters, Nov 27 2006
Edited and extended by Max Alekseyev, May 29 2009
Offset changed by Andrey Zabolotskiy, Oct 10 2017

A063198 Dimension of the space of weight 2n cuspidal newforms for Gamma_0( 10 ).

Original entry on oeis.org

0, 1, 3, 1, 3, 5, 3, 5, 7, 5, 7, 9, 7, 9, 11, 9, 11, 13, 11, 13, 15, 13, 15, 17, 15, 17, 19, 17, 19, 21, 19, 21, 23, 21, 23, 25, 23, 25, 27, 25, 27, 29, 27, 29, 31, 29, 31, 33, 31, 33
Offset: 1

Views

Author

N. J. A. Sloane, Jul 10 2001

Keywords

Comments

The dimension of weight n is apparently given by 0, 0, 2, 1, 0, 3, 2, 1, 4,... etc as in A063942. - R. J. Mathar, Jul 14 2015

Links

Crossrefs

Cf. A063942.

Programs

  • Maple
    s0star := proc(n)
    local pf,a,p,e ;
    if n = 1 then
        1;
    else
        a :=1 ;
        for pf in ifactors(n)[2] do
            p := op(1,pf) ;
            e := op(2,pf) ;
            if e =1 then
                a := a*(1-1/p) ;
            elif e = 2 then
                a := a*(1-1/p-1/p^2) ;
            else
                a := a*(1-1/p)*(1-1/p^2) ;
            end if;
        end do:
        a ;
    end if;
end proc:
nuInfstar := proc(n)
    local pf,a,p,e ;
    if n = 1 then
        1;
    else
        a :=1 ;
        for pf in ifactors(n)[2] do
            p := op(1,pf) ;
            e := op(2,pf) ;
            if type(e,'odd') then
                return 0;
            elif e = 2 then
                a := a*(p-2) ;
            else
                a := a*(p-1)^2*p^(e/2-2) ;
            end if;
        end do:
        a ;
    end if;
end proc:
nu2star := proc(n)
    local pf,a,p,e ;
    if n = 1 then
        1;
    else
        a :=1 ;
        for pf in ifactors(n)[2] do
            p := op(1,pf) ;
            e := op(2,pf) ;
            if p = 2 then
                if e =1 or e =2  then
                    a := -a ;
                elif e =3 then
                    ;
                else
                    return 0 ;
                end if;
            elif modp(p,4) = 1 then
                if e = 2 then
                    a := -a ;
                else
                    return 0;
                end if;
            else
                if e = 1 then
                    a := -2*a ;
                elif e = 2 then
                    ;
                else
                    return 0;
                end if;
            end if;
        end do:
        a ;
    end if;
end proc:
nu3star := proc(n)
    local pf,a ;
    if n = 1 then
        1;
    else
        a :=1 ;
        for pf in ifactors(n)[2] do
            p := op(1,pf) ;
            e := op(2,pf) ;
            if p = 3 then
                if e =1 or e =2  then
                    a := -a ;
                elif e =3 then
                    ;
                else
                    return 0 ;
                end if;
            elif modp(p,3) = 1 then
                if e = 2 then
                    a := -a ;
                else
                    return 0;
                end if;
            else
                if e = 1 then
                    a := -2*a ;
                elif e = 2 then
                    ;
                else
                    return 0;
                end if;
            end if;
        end do:
        a ;
    end if;
end proc:
c2 := proc(k)
    1/4+floor(k/4)-k/4 ;
end proc:
c3 := proc(k)
    1/3+floor(k/3)-k/3 ;
end proc:
g0star := proc(k,N)
    local a;
    a := (k-1)/12*N*s0star(N) -nuInfstar(N)/2 +c2(k)*nu2star(N)+c3(k)*nu3star(N) ;
    if k/2 = 1 then
        a := a+numtheory[mobius](N) ;
    end if;
    a;
end proc:
A063198 := proc(n)
    g0star(2*n,10) ;
end proc:
A063199 := proc(n)
    g0star(2*n,11) ;
end proc:
A063200 := proc(n)
    g0star(2*n,15) ;
end proc:
A063201 := proc(n)
    g0star(2*n,18) ;
end proc:
A063205 := proc(n)
    g0star(2*n,29) ;
end proc: # R. J. Mathar, Jul 19 2024

Formula

G.f.: x^2*(1+2*x-2*x^2+x^3) / ( (1+x+x^2)*(x-1)^2 ). - R. J. Mathar, Jul 15 2015
For n>1, a(n) = (6*n-3+12*cos(2*n*Pi/3)-4*sqrt(3)*sin(2*n*Pi/3))/9. - Wesley Ivan Hurt, Sep 30 2017

A065219 Number of subsets of proper divisors of numbers in A065218 summing to the number.

Original entry on oeis.org

0, 1, 2, 5, 7, 10, 34, 278, 751, 2157, 22208, 676327, 2225346, 23259536, 265050967, 39161483067, 70455119174849, 776384598617893, 133991542908557129, 21819590324155207874, 263763825614848727692, 12883245190231409112736, 661394651111310011564685
Offset: 1

Views

Author

Jud McCranie, Oct 21 2001

Keywords

Comments

The numbers themselves are in A065218.

Examples

			Proper divisors of 12 are {1, 2, 3, 4, 6}. Two subsets of this sum to 12: {2, 4, 6} and {1, 2, 3, 6} - more than any smaller number, so 2 is in the sequence (and 12 is in A065218).

Crossrefs

Cf. A065218, A064771, A063205, A005835, A002182, A004394.

Programs

  • Mathematica
    Union@ FoldList[Max, Array[Block[{dd = Most@ Divisors@ #, c, cc}, cc = Array[c, Length@ dd]; Length@{ ToRules[Reduce[And @@ (0 <= # <= 1 &) /@ cc && dd.cc == #, cc, Integers]]}] &, 360]] (* Michael De Vlieger, Oct 01 2017, after Jean-François Alcover at A065205 *)

Formula

a(n) = A065205(A065218(n)).

Extensions

Extended by Max Alekseyev, May 29 2009
Initial 0 prepended and offset corrected by Amiram Eldar, Oct 01 2017
Showing 1-3 of 3 results.

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

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