A306488 Number of ways of expressing n as a + b + c, with a, b, and c positive integers, gcd(a, b) = 1, but gcd(a, c) and gcd(b, c) both greater than 1.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 4, 0, 4, 0, 1, 0, 9, 0, 7, 1, 4, 1, 15, 0, 13, 1, 4, 2, 16, 0, 24, 4, 10, 1, 29, 0, 32, 4, 5, 3, 41, 0, 38, 2, 17, 6, 54, 1, 43, 6, 26, 10, 70, 0, 65, 9, 20, 11, 68, 1, 86, 14, 35, 2, 99, 1, 99, 15, 18, 16, 104, 1, 125, 10, 53, 19, 134, 0, 114, 21, 58

Offset

0,18

References

F. Barrera, B. Recamán and S. Wagon, Problem 12044, Amer. Math. Monthly 125 (2018), p. 466.

Links

Freddy Barrera, Table of n, a(n) for n = 0..1000

Example

a(11) = 1 because of the ten partitions of 11 into three parts, only 6 + 3 + 2 satisfies the conditions. But a(210) = 0, because 210 does not have any partition that satisfies the conditions.

Mathematica

a[n_] := Length@ Select[ IntegerPartitions[ n, {3}], (t = Sort[GCD @@@ Subsets[#, {2}]]; t[[1]] == 1 && t[[2]] > 1 && t[[3]] > 1) &]; a /@ Range[0, 87] (* Giovanni Resta, Feb 20 2019 *)

Prog

(Sage)
def a(n):
    if n < 3: return 0
    r = 0
    t = [False, True, True]
    for p in Partitions(n, length=3, min_part=2, max_slope=-1):
        s = sorted(gcd(a, b) > 1 for a, b in Subsets(p, 2))
        r += int(s == t)
    return r
[a(n) for n in (0..100)]

Crossrefs

Sequence in context: A028618 A147986 A147988 * A230031 A019920 A246130

Adjacent sequences: A306485 A306486 A306487 * A306489 A306490 A306491

Keyword

nonn

Author

Freddy Barrera, Feb 18 2019

Status

approved