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.

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

Author

Yasutoshi Kohmoto, Feb 24 2007

Keywords

• nonn
• uned

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