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A186099 Sum of divisors of n congruent to 1 or 5 mod 6. 8
1, 1, 1, 1, 6, 1, 8, 1, 1, 6, 12, 1, 14, 8, 6, 1, 18, 1, 20, 6, 8, 12, 24, 1, 31, 14, 1, 8, 30, 6, 32, 1, 12, 18, 48, 1, 38, 20, 14, 6, 42, 8, 44, 12, 6, 24, 48, 1, 57, 31, 18, 14, 54, 1, 72, 8, 20, 30, 60, 6, 62, 32, 8, 1, 84, 12, 68, 18, 24, 48, 72, 1, 74, 38, 31, 20, 96, 14, 80, 6 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,5

COMMENTS

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

LINKS

Seiichi Manyama, Table of n, a(n) for n = 1..10000

FORMULA

Expansion of (1 + a(x)^2 - 2*a(x^2)^2) / 12 in powers of x where a() is a cubic AGM function.

a(n) is multiplicative with a(2^e) = a(3^e) = 1, a(p^e) = (p^(e+1) - 1) / (p - 1) if p>3.

Equals the logarithmic derivative of A003105, where A003105(n) = number of partitions of n into parts 6*n+1 or 6*n-1. - Paul D. Hanna, Feb 17 2013

L.g.f.: Sum_{n>=1} a(n)*x^n/n  =  Sum_{n>=1} S(n,x)*x^n/n  where S(n,x) = Sum_{d|n} d*(1-x^d)^(n/d). - Paul D. Hanna, Feb 17 2013

a(n) = A284098(n) + A284104(n). - Seiichi Manyama, Mar 24 2017

EXAMPLE

G.f.: x + x^2 + x^3 + x^4 + 6*x^5 + x^6 + 8*x^7 + x^8 + x^9 + 6*x^10 + 12*x^11 +...

L.g.f.: L(x) = x + x^2/2 + x^3/3 + x^4/4 + 6*x^5/5 + x^6/6 + 8*x^7/7 + x^8/8 +...

where exp(L(x)) = 1 + x + x^2 + x^3 + x^4 + 2*x^5 + 2*x^6 + 3*x^7 + 3*x^8 + 3*x^9 +...+ A003105(n)*x^n +...

MATHEMATICA

Table[Total[Select[Divisors[n], MemberQ[{1, 5}, Mod[#, 6]]&]], {n, 0, 100}]  (* From Harvey P. Dale, Feb 24 2011 *)

a[ n_] := If[ n < 1, 0, DivisorSum[n, If[ 1 == GCD[#, 6], #, 0] &]]; (* Michael Somos, Jun 27 2017 *)

a[ n_] := If[n < 1, 0, Times @@ (Which[# < 5, 1, True, (#^(#2 + 1) - 1) / (# - 1)] & @@@ FactorInteger[n])]; (* Michael Somos, Jun 27 2017 *)

PROG

(PARI) {a(n) = sumdiv( n, d, d * (1 == gcd(d, 6) ))};

(PARI) {a(n) = direuler( p=2, n, 1 / (1 - X) / (1 - (p>3) * p * X)) [n]};

(PARI) a(n)=sigma(n/2^valuation(n, 2)/3^valuation(n, 3)) \\ Charles R Greathouse IV, Dec 07 2011

(PARI)

{S(n, x)=sumdiv(n, d, d*(1-x^d)^(n/d))}

{a(n)=n*polcoeff(sum(k=1, n, S(k, x)*x^k/k)+x*O(x^n), n)}

for(n=1, 80, print1(a(n), ", "))

/* From Paul D. Hanna, Feb 17 2013 */

CROSSREFS

Cf. A000593, A046913, A113957, A116073, A003105, A284098, A284104.

Sequence in context: A156921 A094214 A001622 * A021622 A073228 A256853

Adjacent sequences:  A186096 A186097 A186098 * A186100 A186101 A186102

KEYWORD

nonn,mult

AUTHOR

Michael Somos, Feb 12 2011

STATUS

approved

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Last modified December 12 06:05 EST 2017. Contains 295937 sequences.