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.

A126851 SPM4Sigma(n) = (-1)^(1/2*((Sum_i p_i)-Omega(m'))*Sum_{d|n} (-1)^(1/2*((Sum_j p_j)-Omega(d'))*d =(2^(r+1)-1)*Product_i [Sum_{1<=s_i<=r_i} p_i^s_i +(-1)^((p_i-1)/2)] where n=2^r*m', gcd(2,m')=1, m'=Product_i p_i^r_i, d=2^k*d', gcd(2,d')=1, d'=Product_j p_j^r_j SPM4 for Signed by Prime factors Mod 4.

Original entry on oeis.org

1, 3, 2, 7, 6, 6, 6, 15, 11, 18, 10, 14, 14, 18, 12, 31, 18, 33, 18, 42, 12, 30, 22, 30, 31, 42, 38, 42, 30, 36, 30, 63, 20, 54, 36, 77, 38, 54, 28, 90, 42, 36, 42, 70, 66, 66, 46, 62, 55, 93, 36, 98, 54, 114, 60, 90, 36, 90, 58, 84, 62, 90, 66, 127, 84, 60, 66, 126, 44, 108, 70, 165, 74, 114, 62, 126, 60, 84, 78, 186
Offset: 1

Views

Author

Yasutoshi Kohmoto, Feb 24 2007

Keywords

Comments

The name contains an unmatched parenthesis. - Editors, Mar 12 2024

Examples

			SPM4Sigma(240) = (1+2+4+8+16)*(-1+3)*(1+5).

Crossrefs

Cf. A126852.

Programs

  • Maple
    A126851 := proc(n)
    local r,mprime,piri,iprod,pi,ri,si;
    r := A007814(n) ;
    mprime := n/2^r ;
    iprod := 1 ;
    if mprime > 1 then
        for piri in ifactors(mprime)[2] do
            pi := op(1,piri) ;
            ri := op(2,piri) ;
            add(pi^si,si=1..ri) + (-1)^( (pi-1)/2) ;;
            iprod := iprod*% ;
        end do:
    end if;
    %*A038712(n) ;
end proc:
seq(A126851(n),n=1..40) ; # R. J. Mathar, Mar 13 2024

Formula

SPM4Sigma(n) = (2^r-1)*Product_i (p_i^(r_i+1)-p_i)/(p_i-1)+(-1)^(1/2*(p_i-1)) = (2^r-1)*Product_{i=1 mod 4} ((p_i^(r_i+1)-p_i)/(p_i-1)+1)*Product_{i=3 mod 4} ((p_i^(r_i+1)-p_i)/(p_i-1)-1)
a(2^n) = A000225(n+1). - R. J. Mathar, Mar 13 2024
A038712(n) | a(n). - R. J. Mathar, Mar 13 2024

Extensions

a(2) and a(7) corrected, sequence extended beyond a(20). - R. J. Mathar, Mar 13 2024

A125141 a(1) = 2; for n>1, a(n)=SENSigma(a(n-1)), where SENSigma(m) = (-1)^((Sum_i r_i)+Omega(m))*Sum_{d|m} (-1)^((Sum_j Max(r_j))+Omega(d))*d = Product_i (Sum_{1<=s_i<=r_i} p_i^s_i)+(-1)^(r_i+1) if m=Product_i p_i^r_i, d=Product_j p_j^r_j, p_j^max(r_j) is the largest power of p_j dividing m.

Original entry on oeis.org

2, 3, 4, 5, 6, 12, 20, 30, 72, 165, 288, 693, 1056, 3024, 9280, 22500, 42845, 60480, 240000, 794580, 1814400, 7040040, 26352000, 98654400, 321552000, 1260230400, 5311834416, 17570520000, 75087810000, 325180275840, 1526817600000
Offset: 1

Views

Author

Yasutoshi Kohmoto, Jan 12 2007

Keywords

Comments

By "Max(r_j)" is meant the following: if d|m, d=p^e*q^f, m=p^x*q^y*r^z then Max(e)=x, Max(f)=y.

Crossrefs

Cf. A126851, A126852, A125142.

Programs

  • Maple
    SENSigma := proc(n) local ifs,i,a,r,p ; ifs := ifactors(n)[2] ; a := 1 ; for i from 1 to nops(ifs) do r := op(2,op(i,ifs)) ; p := op(1,op(i,ifs)) ; a := a*(p*(1-p^r)/(1-p)-(-1)^r) ; od ; RETURN(a) ; end: A125141 := proc(nmax) local a ; a := [2] ; while nops(a)< nmax do a := [op(a),SENSigma(op(-1,a))] ; od ; RETURN(a) ; end: A125141(40) ; # R. J. Mathar, May 18 2007
  • Mathematica
    SENSigma[n_] := Module[{Ifs, i, a, r, p }, Ifs = FactorInteger[n]; a = 1; For[i = 1, i <= Length[Ifs], i++, r = Ifs[[i, 2]]; p = Ifs[[i, 1]]; a = a(p(1 - p^r)/(1 - p) - (-1)^r)]; Return[a]];
A125141[nmax_] := Module[{a}, a = {2}; While[Length[a] < nmax, a = Append[a, SENSigma[a[[-1]]]]]; Return[a]];
A125141[40] (* Jean-François Alcover, Oct 26 2023, after R. J. Mathar *)

Extensions

Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, May 14 2007
More terms from R. J. Mathar, May 18 2007

A125142 a(n) = smallest k such that SEPSigma^{k}(n)=1, or -1 if no such k exists. Here SEPSigma(m) = (-1)^(Sum_i r_i)*Sum_{d|m} (-1)^(Sum_j Max(r_j))*d =Product_i (Sum_{1<=s_i<=r_i} p_i^s_i)+(-1)^r_i where m=Product_i p_i^r_i, d=Product_j p_j^r_j, p_j^max(r_j) is the largest power of p_j dividing m.

Original entry on oeis.org

0, 1, 2, 4, 5, 2, 3, 6, 6, 5, 6, 4, 5, 3, 7, 9, 10, 6, 7, 7, 5, 6, 7, 6, 9, 5, 8, 6, 7, 7, 8, 11, 8, 10, 7, -1, -1, 7, 7, -1, -1, 5, 6, 8, -1, 7, 8, 9, -1, 9, 12, -1, -1, 8, -1, 8, -1, 7, 8, 9, 10, 8, 8, 10, 10, 8, 9, 12, 9, 7, 8, -1, -1, -1, 9, 9, 10, 7, 8, 12, -1, -1, -1, -1, 11, 6, 9, 11, 12, -1
Offset: 1

Views

Author

Yasutoshi Kohmoto, Jan 12 2007, Jan 29 2007

Keywords

Comments

By "Max(r_j)" is meant the following: if d|m, d=p^e*q^f, m=p^x*q^y*r^z then Max(e)=x, Max(f)=y.
For n=36, no k exists which matches the definition since the iteration reaches a cycle that toggles between 168 and 156 ad infinitum: 36->91->72->169->183->120->104->156->168->156-> etc. In the same fashion, no solutions exist for n=37,40,41,45,49,52,53,... - R. J. Mathar, Jun 07 2007

Examples

			SEPSigma^{5}(5)=1, so a(5)=5: 5 -> 4 -> 7 -> 6 -> 2 -> 1

Crossrefs

Cf. A126851-A126852, A125140, A125141.

Programs

  • Maple
    A125140 := proc(n) local ifs,i,a,r,p ; ifs := ifactors(n)[2] ; a := 1 ; for i from 1 to nops(ifs) do r := op(2,op(i,ifs)) ; p := op(1,op(i,ifs)) ; a := a*(p*(1-p^r)/(1-p)+(-1)^r) ; od ; RETURN(a) ; end: A125142 := proc(n) local a,nsep; nsep := n ; a :=0 ; while nsep <> 1 do a := a+1 ; nsep := A125140(nsep) ; od ; RETURN(a) ; end: for n from 1 to 80 do printf("%d, ",A125142(n)) ; od ; # R. J. Mathar, Jun 07 2007

Extensions

Edited by N. J. A. Sloane at the suggestions of Andrew S. Plewe and R. J. Mathar, May 14 2007, Jun 10 2007
More terms from R. J. Mathar, Jun 07 2007
More terms from R. J. Mathar, Oct 20 2009
Showing 1-3 of 3 results.

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