A141098 -id:A141098 - cp's OEIS frontend

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

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}]

A141096 Even numbers not representable as the sum of two coprime nonprime numbers.

Original entry on oeis.org

4, 6, 8, 12, 14, 18, 20, 24, 30, 32, 38, 42, 48, 54, 60, 72, 80, 84, 90, 108, 110, 132, 138, 140, 150, 180

Offset: 1

T. D. Noe, Jun 02 2008

Numbers k such that A141095(k/2) = 0.
180 is the last term.
This sequence is a subsequence of A141098.

Cf. A141095, A141098.

t = Table[Length[Select[Range[n/2], ! PrimeQ[#] && ! PrimeQ[n - #] && GCD[#, n - #] == 1 &]], {n, 2, 2000, 2}]; Flatten[2*Position[t, 0]] (* T. D. Noe, Dec 05 2013 *)

A352587 Even numbers 2m such that A352612(2m) = A103131(2m).

Original entry on oeis.org

2, 4, 6, 10, 16, 18, 20, 28, 60, 84, 228, 240, 280, 366, 420, 468, 484, 604, 684, 942, 990, 1152, 1170, 1196, 1440, 2064, 5292, 5954, 8968, 9176, 13242, 13680, 14160, 15190, 24524, 28764, 29422, 30558, 30646, 34804, 35190, 38164, 44642, 56772, 62790, 93024

Offset: 1

Craig J. Beisel, Mar 21 2022

nonn

Any counterexample to the Goldbach conjecture must have this form.

Conjecture: For all a(n) > 18, a(n) is never equal to 2*q^x where q is prime and x is an integer x > 0. In other words, the product of its totatives is never congruent to -1 (mod 2m).

			For a(1) we have A352612(228) == -(59)(85) (mod 228) == 1 (mod 228) == A103131(228). Therefore A352612(228) == A103131(228) and 228 belongs to the sequence.

Craig J. Beisel, Table of n, a(n) for n = 1..56

Cf. A103131, A141098, A352612.

for(n=1,150000, prod_t=1; prod_p=1; prod_r=1; for(k=3, 2*n-3, if(gcd(k,2*n)==1, prod_t=prod_t*k; ); if(gcd(k,2*n)==1 && isprime(k), prod_p=prod_p*k*(2*n-k); ); if(gcd(k,2*n)==1 && !isprime(k) && !isprime(2*n-k), prod_r=prod_r*k; ); ); if(-prod_t%(2*n)==(-prod_p*prod_r)%(2*n), print1(2*n,","); ); );

A352612 (n-1)prod(-p^2 where 2 <= p <= n-2 is prime and relatively prime to n)prod(k where both k and (n-k) are composite and relatively prime to n) (mod n).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 4, 7, 4, 9, 10, 11, 4, 9, 11, 1, 16, 17, 4, 1, 17, 17, 22, 23, 4, 1, 26, 1, 28, 29, 4, 17, 29, 25, 1, 25, 36, 11, 35, 39, 40, 25, 4, 7, 41, 27, 46, 23, 4, 31, 1, 23, 52, 1, 51, 55, 1, 49, 58, 1, 4, 37, 59, 55, 1, 49, 66, 33, 65, 31, 70, 25, 4

Offset: 1

Craig J. Beisel, Mar 23 2022

nonn

The convention for the empty product here is 1. The second product exists for all numbers greater than 210. See A141098.

Conjecture: For odd n, if a(n) == -1 (mod n) then n must be a prime power.

			For n=6 there are no prime totatives between 2 and 4 and there are also no composite totative pairs which add to 6 so both products do not exist and a(6)=n-1=5.
For n=25 these products exist and are given -44618574^2*12096 == 4 (mod 25). Therefore, a(25)=4.

Cf. A103131, A141098, A352587.

a(n)= {prod_p=1; prod_r=1; for(k=2, n-2, if(gcd(k,n)==1, if(isprime(k), prod_p=prod_p*k*(n-k); ); if(!isprime(k) && !isprime(n-k), prod_r=prod_r*k; );); ); (-prod_p*prod_r)%n; }

A232721 Numbers not representable as the sum of two coprime nonprime numbers.

Original entry on oeis.org

1, 3, 4, 6, 8, 12, 14, 18, 20, 24, 30, 32, 38, 42, 48, 54, 60, 72, 80, 84, 90, 108, 110, 132, 138, 140, 150, 180

Offset: 1

Irina Gerasimova, Nov 28 2013

Numbers n such that A185279(n) = 0. 1 and 3 together with A141096.

Cf. A062303, A018252, A141095, A141098.

t = Table[Length[Select[Range[n/2], ! PrimeQ[#] && ! PrimeQ[n - #] && GCD[#, n - #] == 1 &]], {n, 2000}]; Flatten[Position[t, 0]] (* T. D. Noe, Dec 05 2013 *)

cp's OEIS Frontend

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

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A141096 Even numbers not representable as the sum of two coprime nonprime numbers.

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A352587 Even numbers 2m such that A352612(2m) = A103131(2m).

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A352612 (n-1)*prod(-p^2 where 2 <= p <= n-2 is prime and relatively prime to n)*prod(k where both k and (n-k) are composite and relatively prime to n) (mod n).

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A232721 Numbers not representable as the sum of two coprime nonprime numbers.

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A352612 (n-1)prod(-p^2 where 2 <= p <= n-2 is prime and relatively prime to n)prod(k where both k and (n-k) are composite and relatively prime to n) (mod n).