A082024 Number of partitions of n into 3 parts which have common divisors.

0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 2, 0, 4, 0, 4, 3, 5, 0, 9, 0, 9, 5, 10, 0, 16, 2, 14, 7, 17, 0, 27, 0, 21, 11, 24, 6, 36, 0, 30, 15, 37, 0, 51, 0, 41, 25, 44, 0, 64, 4, 58, 25, 57, 0, 81, 12, 69, 31, 70, 0, 108, 0, 80, 43, 85, 16, 123, 0, 97, 45, 120, 0, 144, 0, 114, 69, 121, 14, 171, 0

Offset

0,11

Comments

a(p) = 0 if p is a prime. Can anyone suggest a formula?

See example for a method to find a(n). - David A. Corneth, Aug 24 2020

Links

David A. Corneth, Table of n, a(n) for n = 0..9999 (first 5001 terms from Alois P. Heinz)

Example

a(14) = 4 and the partitions are (10,2,2), (8,4,2),(6,6,2) and (6,4,4).
a(13) = 0 as for all r + s + t = 13,r > 0, s > 0,t> 0 gcd(r,s,t) = 1.
From David A. Corneth, Aug 24 2020: (Start)
a(100) = 233. The squarefree part of 100 is 10. The divisors of 10 are 1, 2, 5 and 10.
These are the possible squarefree divisors of parts. As parts must not be coprime, we exclude 1, leaving 2, 5 and 10. We then compute 100/k for each of these numbers.
This gives 50, 20 and 10 respectively. Now a(100) is found by adding -(round(50^2/12)*(-1)^omega(2) + round(20^2/12)*(-1)^omega(5) + round(10^2/12)*(-1)^omega(10)) = -(-208 - 33 + 8) = 233 where omega(m) is the number of distinct divisors of m (Cf. A001221) and round(m^2/12) is the number of partitions of m into 3 parts (Cf. A069905) (End)

Mathematica

a[n_] := Length[Select[Flatten[Table[{a, b, n-a-b}, {a, 1, Floor[n/3]}, {b, a, Floor[(n-a)/2]}], 1], GCD@@#1>1&]]

Prog

(PARI) a(n) = if(n==0, return(0)); cn = factorback(factor(n)[, 1]); d = divisors(cn); -sum(i = 2, #d, round((n/d[i])^2/12) * (-1)^omega(d[i])) \\ David A. Corneth, Aug 24 2020

Crossrefs

Cf. A001221, A069905, A082023, A284825.

Sequence in context: A145893 A137561 A337605 * A337599 A322337 A211008

Adjacent sequences: A082021 A082022 A082023 * A082025 A082026 A082027

Keyword

nonn,easy

Author

Amarnath Murthy, Apr 07 2003

Extensions

More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 20 2003 and Dean Hickerson, Apr 22 2003

Status

approved