A306261 -id:A306261 - cp's OEIS frontend

A306247 Numbers k such that 2k - p is not a prime where p is any prime divisor of 4k^2 - 1.

1, 2, 3, 6, 9, 14, 15, 19, 21, 26, 29, 30, 34, 36, 40, 48, 49, 51, 54, 61, 63, 64, 69, 74, 75, 79, 82, 84, 86, 89, 90, 95, 96, 99, 103, 106, 110, 111, 112, 114, 119, 120, 135, 139, 141, 146, 147, 149, 151, 152, 153, 154, 156, 161, 166, 169, 173, 174, 179, 180, 184, 186, 187, 190, 194

Offset: 1

Juri-Stepan Gerasimov, Jan 31 2019

Primes in a(n): 2, 3, 19, 29, 61, 79, 89, 103, 139, 149, 151, 173, 179, ...

			1 is a term because 4*1^2 - 1 = 3 and 2*1 - 3 = -1 (nonprime);
2 is a term because 4*2^2 - 1 = 15 and 2*2 - 15 = -11 (nonprime);
3 is a term because 4*3^2 - 1 = 35 and 2*3 - 35 = -29 (nonprime);
6 is a term because 4*6^2 - 1 = 143 = 11*13 and 2*6 - 11 = 1 (nonprime), 2*6 - 13 = -1 (nonprime);
9 is a term because 4*9^2 - 1 = 323 = 17*19 and 2*9 - 17 = 1 (nonprime), 2*9 - 19 = -1 (nonprime).

Includes A040040.

Cf. A306261.

filter:= proc(n) andmap(`not` @ isprime, map(p -> 2*n-p, numtheory:-factorset(4*n^2-1))) end proc:
select(filter, [$1..300]); # Robert Israel, Jan 31 2019

Select[Range@ 200, AllTrue[2 # - FactorInteger[4 #^2 - 1][[All, 1]], ! PrimeQ@ # &] &] (* Michael De Vlieger, Feb 03 2019 *)

isok(k) = {my(pf = factor(4*k^2-1)[,1]); #select(x->isprime(2*k-x), pf) == 0;} \\ Michel Marcus, Mar 02 2019

A306261(a(n)) > 1 for n >= 4.

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

A306615 Least positive k such that 2n - p is prime where p is some prime divisor of n^k - 1, or 0 if no such k exists.

Original entry on oeis.org

0, 0, 0, 1, 2, 1, 1, 2, 2, 1, 1, 1, 1, 2, 1, 1, 2, 1, 6, 2, 1, 1, 2, 2, 1, 1, 1, 1, 2, 1, 1, 2, 4, 2, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 1, 2, 3, 2, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 3, 10, 3, 3, 2, 1, 1, 2, 2, 1, 1, 1, 2, 10, 1, 1, 2, 1, 3, 2, 1, 6, 2, 2, 1, 1, 1, 1, 1, 2

Offset: 1

Juri-Stepan Gerasimov, Feb 28 2019

nonn

Conjecture: a(n) >= 1 for n >= 4.
Records: a(4) = 1, a(5) = 2, a(19) = 6, a(62) = 10, a(166) = 18, ...
For n >= 4, a(n) < b(n) where b(n) is the smallest m > 1 such that q(2n - q) is some semiprime divisor of n^m - 1, or 0 if no such m exists: 0, 0, 0, 2, 6, 2, 10, 4, 6, 6, 6, 2, 11, 22, 22, 7, 4, 2, 30, 35, 18, 30, 20, 42, 9, 40, 8, 13, 26, 2, 42, 12, 20, 10, 52, 21, 3, 36, 42, 11, 26, 2, 24, 82, 21, 12, 44, 88, 39, 8, 32, 25, 88, 24, 30, 25, 20, 96, 88, 2, 54, 220, 48, 6, ... (from Goldbach's problem).

			a(4) = 1 because 4^1 - 1 = 3 where 3 is some prime divisor of 3 and 2*4 - 3 = 5 is prime;
a(5) = 2 because 5^2 - 1 = 24 where 3 is some prime divisor of 24 and 2*5 - 3 = 7 is prime.

Cf. A306261.

Table[If[n < 4, 0, Block[{k = 1}, While[NoneTrue[FactorInteger[n^k - 1][[All, 1]], PrimeQ[2 n - #] &], k++]; k]], {n, 104}] (* Michael De Vlieger, Mar 11 2019 *)

isok(n,k) = {my(pf=factor(n^k-1, 2*n)[,1]); for (j=1, #pf, if (isprime(2*n-pf[j]), return (1)););}
a(n) = {if (n < 4, return(0)); my(k=1); while (!isok(n, k), k++); k;} \\ Michel Marcus, Mar 02 2019

cp's OEIS Frontend

A306247 Numbers k such that 2k - p is not a prime where p is any prime divisor of 4k^2 - 1.

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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.

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A306615 Least positive k such that 2n - p is prime where p is some prime divisor of n^k - 1, or 0 if no such k exists.

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Comments

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Mathematica

PARI