A100952 -id:A100952 - cp's OEIS frontend

A100951 Number of ways to write n = p*q + r with three distinct primes p, q and r.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 0, 4, 0, 2, 0, 1, 1, 3, 0, 3, 2, 3, 1, 4, 0, 2, 2, 3, 2, 3, 0, 5, 4, 3, 2, 5, 0, 4, 3, 5, 4, 3, 1, 4, 3, 5, 4, 6, 1, 3, 4, 6, 6, 4, 0, 3, 7, 6, 5, 6, 1, 6, 6, 6, 4, 4, 3, 4, 7, 7, 6, 6, 2, 6, 7, 8, 8, 3, 2, 7, 6, 7, 8, 8, 1, 4, 7, 10, 10, 6, 6, 7, 10, 9, 8, 5, 5, 5, 8

Offset: 1

Reinhard Zumkeller, Nov 23 2004

nonn

a(n)<=A100950(n)<=A100949(n)<=Min{A000720(n), A072000(n)}.
a(n) = A000040(i) + A006881(j) for some i and j, such that A000040(i) is not a factor of A006881(j);
a(A100952(n)) = 0.
a(A160373(n)) = n and a(m) <> n for m < A160373(n). [From Reinhard Zumkeller, May 11 2009]

			A100949(21) = #{7+2*7, 11+2*5, 17+2*2} = 3,
a(21) = #{11+2*5} = 1.

R. Zumkeller, Table of n, a(n) for n = 1..10000

A347933 Positive integers that can't be written in the form a+b+c for some positive integers a, b, and c satisfying gcd(a,b)=1, gcd(a,c)>1, and gcd(b,c)>1.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 16, 18, 20, 22, 24, 30, 36, 42, 48, 60, 84, 90, 210

Offset: 1

José Hernández, Sep 20 2021

			1 and 2 are the first two terms of the sequence because they can't even be written as sums of three positive integers.
3 is the third term of the sequence because there is only one way to express it as a sum of three positive integers (1+1+1).
11 does not belong to the sequence because 11=2+3+6 (and gcd(2,3)=1, gcd(2,6)=2, and gcd(3,6)=3).

Freddy Barrera, Bernardo Recamán, and Stan Wagon, Sums of triples with one pair relatively prime. American Mathematical Monthly, 127 (2020), no. 1, pp. 89-90.

Cf. A224535, A100951, A100952.

Select[Range@210,Select[Flatten[Permutations/@IntegerPartitions[#,{3}],1],GCD[#[[1]],#[[2]]]==1&&GCD[#[[1]],#[[3]]]>1&&GCD[#[[2]],#[[3]]]>1&]=={}&] (* Giorgos Kalogeropoulos, Sep 20 2021 *)

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A100951 Number of ways to write n = p*q + r with three distinct primes p, q and r.

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A347933 Positive integers that can't be written in the form a+b+c for some positive integers a, b, and c satisfying gcd(a,b)=1, gcd(a,c)>1, and gcd(b,c)>1.

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