A306247 -id:A306247 - cp's OEIS frontend

A306261 Least k > 0 such that 2n - p is prime where p is some prime divisor of 4n^2 - (2k-1)^2 for n >= 4.

1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 3, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 3, 2, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 3, 1, 2, 3, 1, 1, 1, 1, 3, 1, 1, 1, 1, 3, 4, 1, 1, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 3, 1, 1, 1, 1, 2, 2, 1, 1, 2, 1

Offset: 4

Juri-Stepan Gerasimov, Feb 01 2019

nonn

Conjecture: a(n) exists for n >= 4.

The conjecture holds up to 10^6. Records: a(4) = 1, a(6) = 2, a(34) = 3, a(75) = 4, a(154) = 9, a(1027) = 10, a(1097) = 11, a(1477) = 14, a(1552) = 17, a(5179) = 18, a(10684) = 29, a(70201) = 32, a(79861) = 43, a(519632) = 45, a(1018804) = 46, a(1713031) = 47, .... - Charles R Greathouse IV, Feb 17 2019

			a(4) = 1 because 4*4^2 - (2*1-1)^2 = 63 = 3^2*7 and 2*4 - 3 = 5 is prime;
a(5) = 1 because 4*5^2 - (2*1-1)^2 = 99 = 3^2*11 and 2*5 - 3 = 7 is prime;
a(6) = 2 because 4*6^2 - (2*1-1)^2 = 143 = 11*13 and 2*6 - 11 = 1 is not a prime, 2*6 - 13 = -1 is not a prime, but 4*6^2 -(2*2-1)^2 = 135 = 3^3*5 and 2*6 - 5 = 7 is prime.

Cf. A020481, A306247.

a(n)=for(k=1,2*n,my(f=factor(4*n^2-(2*k-1)^2)[,1]);for(i=1,#f,if(isprime(2*n-f[i]),return(k)))); "does not exist" \\ Charles R Greathouse IV, Feb 17 2019

A306196 Irregular triangle read by rows where row n lists the primes 2n - k, with 1 < k < 2n-1, and if k is composite also 2n - p has to be prime for some prime divisor p of k.

Original entry on oeis.org

2, 3, 2, 3, 5, 3, 5, 7, 2, 5, 7, 2, 3, 5, 7, 11, 3, 5, 7, 11, 13, 3, 5, 7, 11, 13, 2, 3, 5, 7, 11, 13, 17, 2, 3, 5, 7, 11, 13, 17, 19, 2, 3, 5, 7, 11, 13, 17, 19, 2, 3, 5, 7, 11, 13, 17, 19, 23, 3, 5, 11, 13, 17, 23, 2, 7, 11, 13, 17, 19, 23, 2, 3, 5, 11, 13, 17, 19, 23, 29

Offset: 2

Juri-Stepan Gerasimov, Jan 28 2019

Conjectures:
(i) 1 <= A035026(n) <= (n-th row length of this triangle) for n >= 2;
(ii) a(n,1) < A171637(n,1) for n >= 4.
Numbers m such that m-th row length of this triangle is equal to A000720(m): 1, 2, 11, 13, 25, 56, 60, ...

			Row 2 = [2] because 2*2 = 2 + 2;
Row 3 = [3] because 2*3 = 3 + 3;
Row 4 = [2,3,5] because 2*4 - 2 = 6 = 2*3 and 2*4 = 3 + 5;
Row 5 = [3,5,7] because 2*5 = 3 + 7 = 5 + 5.
The table starts:
  2;
  3;
  2,  3,  5;
  3,  5,  7;
  2,  5,  7;
  2,  3,  5,  7, 11;
  3,  5,  7, 11, 13;
  3,  5,  7, 11, 13;
  2,  3,  5,  7, 11, 13, 17;
  2,  3,  5,  7, 11, 13, 17, 19;
  2,  3,  5,  7, 11, 13, 17, 19;
  2,  3,  5,  7, 11, 13, 17, 19, 23;
  3,  5, 11, 13, 17, 23;
  2,  7, 11, 13, 17, 19, 23;
  2,  3,  5, 11, 13, 17, 19, 23, 29;

Supersequence of A171637.

Cf. A000720, A020481, A035026, A306247, A306261.

isok(k,n) = {if (isprime(2*n-k), pf = factor(k)[,1]; for (j=1, #pf, if (isprime(2*n-pf[j]), return (1));););}
row(n) = {my(v = []); for (k=1, 2*n, if (isok(k,n), v = concat(v, 2*n-k))); vecsort(v);} \\ Michel Marcus, Mar 02 2019

cp's OEIS Frontend

A306261 Least k > 0 such that 2n - p is prime where p is some prime divisor of 4n^2 - (2k-1)^2 for n >= 4.

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