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for(n=1, 121, print1(a(n), ", "))

(PARI) {a(n) = local(L, X=x+x*O(x^n)); L = sum(m=0, n\8+1, log( (1-x^(3+8*m))*(1-x^(5+8*m))*(1-x^(8+8*m)) / ( (1-x^(1+8*m))*(1-x^(4+8*m))*(1-x^(7+8*m) +x*O(x^n)) ))); n*polcoeff(L, n)}

(PARI) {a(n) = local(L, X=x+x*O(x^n)); L = sum(m=0, n\8, log( (1-x^(3+8*n))*(1-x^(5+8*n))*(1-x^(8+8*n)) / ( (1-x^(1+8*n))*(1-x^(4+8*n))*(1-x^(7+8*n)) ) ).

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From formulas given by Michael Somos in A093709: (Start)

L.g.f.: log( (theta_3(x) + theta_3(x^2)) / 2).

L.g.f.: logLog( psi(q^4) * f( 1 + Sum_{n>=1} x-q^3, -q^A0289825) / f(n-q, -q^7) ), in powers of q where A028982 lists the squares and twice squarespsi(), f() are Ramanujan theta functions.

L.g.f.: Log( f(-q^3, -q^5)^2 / psi(-q) ) in powers of q where psi(), f() are Ramanujan theta functions.

(End)

(PARI) {a(n) = local(L, X=x+x*O(x^n)); L = sum(m=0, n\8, log( (1-x^(3+8*n))*(1-x^(5+8*n))*(1-x^(8+8*n)) / ( (1-x^(1+8*n))*(1-x^(4+8*n))*(1-x^(7+8*n)) ) ).

Cf. A258655, A258328, A028982, A093709.

Cooper, Shaun; Hirschhorn, Michael. <a href="http://projecteuclid.org/download/pdf_1/euclid.rmjm/1181070243">On Some Finite Product Identities.</a> Rocky Mountain J. Math. 31 (2001), no. 1, 131--139.

L.g.f.: Sum_{n>=0} log( (1-x^(3+8*n))*(1-x^(5+8*n))*(1-x^(8+8*n)) / ( (1-x^(1+8*n))*(1-x^(4+8*n))*(1-x^(7+8*n)) ) ). [See Cooper and Hirschhorn reference]

a(n) = -sigma(n) + [Sum_{d|n, d==2 (mod 4)} d] + [Sum_{d|n, d==1,4,7 (mod 8)} 2*d].

for(n=1, 121, print1(a(n), ", "))

(PARI) {a(n) = -sigma(n) + sumdiv(n, d, if(d%4==2, d)) + 2*sumdiv(n, d, if((d%8)%3==1, d))}

Cf. A258655, A258328, A028982.

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a(n) == 1 (mod 2) iff n is a square or twice square (A028982).

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L.g.f.: L(x) = x + x^2/2 - 2*x^3/3 + 5*x^4/4 - 4*x^5/5 - 2*x^6/6 + 8*x^7/7 - 3*x^8/8 + 7*x^9/9 - 4*x^10/10 - 10*x^11/11 + 14*x^12/12 - 12*x^13/13 + 8*x^14/14 + 8*x^15/15 - 19*x^16/16 +...+ a(n)*x^n/n +...

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