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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.
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%I #16 Mar 13 2024 19:28:01

%S 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,

%T 42,38,42,30,36,30,63,20,54,36,77,38,54,28,90,42,36,42,70,66,66,46,62,

%U 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

%N 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.

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

%F 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)

%F a(2^n) = A000225(n+1). - _R. J. Mathar_, Mar 13 2024

%F A038712(n) | a(n). - _R. J. Mathar_, Mar 13 2024

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

%p A126851 := proc(n)

%p local r,mprime,piri,iprod,pi,ri,si;

%p r := A007814(n) ;

%p mprime := n/2^r ;

%p iprod := 1 ;

%p if mprime > 1 then

%p for piri in ifactors(mprime)[2] do

%p pi := op(1,piri) ;

%p ri := op(2,piri) ;

%p add(pi^si,si=1..ri) + (-1)^( (pi-1)/2) ;;

%p iprod := iprod*% ;

%p end do:

%p end if;

%p %*A038712(n) ;

%p end proc:

%p seq(A126851(n),n=1..40) ; # _R. J. Mathar_, Mar 13 2024

%Y Cf. A126852.

%K nonn,uned

%O 1,2

%A _Yasutoshi Kohmoto_, Feb 24 2007

%E a(2) and a(7) corrected, sequence extended beyond a(20). - _R. J. Mathar_, Mar 13 2024