A383650 Averages k of a twin prime pair such that 3*k*2^d is also the average of a twin prime pair for some divisor d of 3*k.

4, 6, 12, 18, 30, 60, 72, 108, 138, 192, 240, 270, 312, 348, 420, 432, 570, 642, 810, 822, 828, 1020, 1050, 1092, 1302, 1320, 1452, 1620, 1668, 1698, 1722, 1950, 1998, 2310, 2550, 2688, 2712, 2730, 2970, 3000, 3168, 3258, 3330, 3372, 3462, 3468, 3540, 3582, 4092

Offset: 1

Author

Juri-Stepan Gerasimov, May 04 2025

Keywords

• nonn

Comments

All terms after a(2) are abundant (A005101), this is because all primes greater than 3 are of the form 6*k +- 1, thus the average of twin primes is 6*k, and since any multiple of a perfect or abundant number is abundant itself, it means that this property holds for all n > 11. - Jakub Buczak, May 04 2025

Examples

			Average 4 of a twin prime pair is in the sequence because 3*4*2^4 = 192 is also the average of twin primes 191 and 193 for divisor d = 4 of 3*k = 3*4 = 12.
		

Links

• Jakub Buczak, Table of n, a(n) for n = 1..125

Crossrefs

Supersequence of A014574.

Cf. A000040, A383475, A005101, A173490.

Programs

Magma

[k: k in [4..4100] | not #[d: d in Divisors(3*k) | IsPrime(k-1) and IsPrime(k+1) and IsPrime(3*k*2^d-1) and IsPrime(3*k*2^d+1)] eq 0];

Formula

a(n) ~ b*n^c for some constants b and c as n tends to infinity (conjectured). - Jakub Buczak, May 04 2025

Extensions

More terms from Jakub Buczak, May 04 2025