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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.
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
1,2
COMMENTS
The name contains an unmatched parenthesis. - Editors, Mar 12 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
EXAMPLE
SPM4Sigma(240) = (1+2+4+8+16)*(-1+3)*(1+5).
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
CROSSREFS
Cf. A126852.
Sequence in context: A255067 A139285 A080398 * A082321 A057502 A071656
KEYWORD
nonn,uned
AUTHOR
Yasutoshi Kohmoto, Feb 24 2007
EXTENSIONS
a(2) and a(7) corrected, sequence extended beyond a(20). - R. J. Mathar, Mar 13 2024
STATUS
approved