A141099 -id:A141099 - cp's OEIS frontend

A141095 Number of unordered pairs of coprime nonprime numbers that sum to 2n.

1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 2, 1, 0, 1, 0, 1, 2, 0, 1, 2, 0, 1, 3, 0, 1, 4, 1, 1, 1, 0, 3, 4, 1, 0, 4, 0, 4, 3, 0, 4, 5, 1, 1, 3, 1, 4, 6, 0, 0, 4, 1, 6, 6, 1, 5, 7, 1, 5, 3, 0, 8, 7, 0, 0, 10, 3, 9, 9, 0, 6, 5, 2, 8, 5, 2, 9, 10, 1, 5, 11, 2, 9, 12, 0, 3, 12, 4, 11, 5, 2, 12, 9, 1, 4, 14

Offset: 1

T. D. Noe, Jun 02 2008

Nonprime numbers are 1 and the composite numbers. See A141097 for pairs of coprime composite numbers. It appears that a(n) > 0 except for the 26 values of 2n given in A141096.

			a(17)=2 because 34 = 1+33 = 9+25.

T. D. Noe, Table of n, a(n) for n=1..10000
Carlos Rivera, Conjecture 33: The Goldbach Temptation, The Prime Puzzles & Problems Connection.

Cf. A141099, A141100.

Table[cnt=0; Do[If[GCD[2n-i,i]==1 && !PrimeQ[i] && !PrimeQ[2n-i], cnt++ ], {i,1,n,2}]; cnt, {n,100}]

A141100 Number of unordered pairs of odd composite numbers that sum to 2n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 1, 2, 0, 1, 3, 1, 1, 3, 2, 1, 4, 1, 2, 5, 1, 3, 5, 1, 4, 5, 3, 3, 6, 3, 3, 7, 3, 3, 9, 3, 4, 7, 4, 6, 9, 5, 5, 8, 6, 6, 10, 5, 5, 12, 4, 6, 12, 5, 9, 11, 7, 7, 11, 9, 9, 13, 8, 8, 16, 7, 11, 14, 8, 11, 14, 9, 9, 17, 13, 10, 16, 11, 11, 19, 11, 12, 18, 10

Offset: 1

T. D. Noe, Jun 02 2008, Jun 05 2008

nonn

See A141099 for pairs of odd nonprime numbers. We have a(n) > 0 except for the 14 values of 2n given in A118081.

			a(18)=2 because 36 = 9+27 = 15+21.

T. D. Noe, Table of n, a(n) for n=1..1000

Cf. A005171, A118081, A141095, A141097, A141099.

Table[cnt=0; Do[If[ !PrimeQ[i] && !PrimeQ[2n-i], cnt++ ], {i,3,n,2}]; cnt, {n,100}]

a(n) = 1 - floor(n/2) + Sum_{i=3..n} c(i) * c(2n-i), n>1, where c = A005171. - Wesley Ivan Hurt, Dec 27 2013

A141097 Number of unordered pairs of coprime composite numbers that sum to 2n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 2, 0, 1, 3, 0, 1, 0, 0, 3, 3, 0, 0, 3, 0, 3, 2, 0, 3, 4, 0, 1, 2, 1, 4, 5, 0, 0, 3, 1, 5, 5, 0, 4, 6, 0, 5, 2, 0, 7, 6, 0, 0, 9, 2, 8, 8, 0, 6, 4, 1, 8, 4, 1, 9, 9, 1, 4, 10, 2, 8, 11, 0, 3, 11, 3, 10, 4, 2, 12, 8, 1, 4, 13, 2

Offset: 1

T. D. Noe, Jun 02 2008

nonn

See A141095 for pairs of coprime nonprime numbers. It appears that a(n) > 0 except for the 43 values of 2n given in A141098. Roberts says that A. M. Vaidya proved that a(n) > 0 for all sufficiently large n.

			a(17)=1 because 34 = 9+25.

Joe Roberts, "Lure of the Integers", The Mathematical Association of America, 1992, p. 190.

T. D. Noe, Table of n, a(n) for n=1..10000
Carlos Rivera, Conjecture 33: The Goldbach Temptation

Cf. A141099, A141100.

Table[cnt=0; Do[If[GCD[2n-i,i]==1 && !PrimeQ[i] && !PrimeQ[2n-i], cnt++ ], {i,3,n,2}]; cnt, {n,100}]

cp's OEIS Frontend

A141095 Number of unordered pairs of coprime nonprime numbers that sum to 2n.

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A141100 Number of unordered pairs of odd composite numbers that sum to 2n.

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A141097 Number of unordered pairs of coprime composite numbers that sum to 2n.

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