A230699 -id:A230699 - cp's OEIS frontend

A230852 Smallest k<32^n such that 32^n+k is the smallest of four consecutive primes in arithmetic progression or 0 if no solution.

0, 0, 0, 0, 0, 59, 0, 0, 205, 229, 167, 353, 1595, 4459, 6407, 6215, 14995, 4559, 4697, 11399, 365, 10199, 19327, 39103, 3185, 13649, 15787, 2693, 21455, 24929, 32209, 30509, 13421, 5389, 36947, 12869, 27277, 38389, 973, 69199, 58835, 165629, 52597, 25463, 17923, 38629, 90263, 17153, 48143, 2171, 1255

Offset: 1

Pierre CAMI, Oct 31 2013

nonn

Conjecture: if n>8 there is always a solution.

			3*2^6+59=251, and 251, 257, 263, 269 are four consecutive primes in arithmetic progression 6 so a(6)=59.
3*2^9+205=1741, and 1741, 1747, 1753, 1759 are four consecutive primes in arithmetic progression 6 so a(9)=205.

Pierre CAMI, Table of n, a(n) for n = 1..110

Cf. A054800, A230699.

cpap4(p)=my(q=nextprime(p+1),g=q-p);nextprime(q+1)-q==g&&nextprime(p+2*g+1)==p+3*g
a(n)=forprime(p=3<Charles R Greathouse IV, Oct 31 2013

A231576 Sequence of pairs k,g such that k is the smallest odd number and k2^n-1-g, k2^n-1, k*2^n-1+g are three consecutive primes in arithmetic progression.

Original entry on oeis.org

3, 2, 53, 12, 33, 6, 69, 6, 19, 6, 2193, 12, 93, 6, 113, 6, 87, 6, 413, 12, 1165, 12, 143, 6, 237, 6, 47, 6, 315, 18, 779, 6, 631, 30, 797, 6, 735, 12, 567, 18, 397, 6, 351, 24, 195, 18, 39, 36, 2719, 6, 971, 6, 1369, 30, 635, 18, 1501, 12, 593, 72, 2053, 6

Offset: 1

Pierre CAMI, Nov 11 2013

nonn

			3*2^1-1-2=3, 3*2^1-1=5, 3*2^1-1+2=7, so first pair = 3,2 (the only one with g=2).
53*2^2-1-12=199, 53*2^2-1=211, 53*2^2-1+12=223, so second pair = 53,12.

Pierre CAMI, Table of n, a(n) for n = 1..1300
Pierre CAMI, PFGW Script

Cf. A228452, A228454, A230699, A230852.

A227856 Sequence of pairs k,g with k<32^n the smallest such that 32^n+k, 32^n+k+g, 32^n+k+2*g are three consecutive primes in arithmetic progression starting at n=5 as there is not any solution for n<5.

Original entry on oeis.org

55, 6, 7, 12, 173, 6, 173, 6, 205, 6, 229, 6, 113, 6, 203, 6, 95, 6, 475, 6, 163, 6, 119, 12, 377, 18, 1045, 6, 133, 12, 551, 24, 131, 12, 259, 6, 1105, 42, 539, 6, 1487, 18, 1295, 12, 5, 12, 289, 36, 311, 36, 269, 6, 2833, 6, 1813, 18, 835, 6, 319, 6, 587, 6, 239, 30, 1225, 6, 1825, 12, 973, 12, 89, 30, 551, 12, 1805, 30, 1039, 18, 1219, 6

Offset: 5

Pierre CAMI, Nov 01 2013

nonn

The ratio k/n^2 is in average near 0.8 and < 7 for n<701.
The ratio g/n^2 is in average near 0.5 and < 4 for n<701.
If 3*2^n+k > 10^22 the numbers are probable primes.

			3*2^5+55=151, 3*2^5+55+6=157 3*2^5+55*2*6=163
151, 157, 163 three consecutive primes in arithmetic progression 6, so first pair is 55, 6

Pierre CAMI, Table of n, a(n) for n = 5..1396

Cf. A230699, A230852

cp's OEIS Frontend

A230852 Smallest k<32^n such that 32^n+k is the smallest of four consecutive primes in arithmetic progression or 0 if no solution.

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A231576 Sequence of pairs k,g such that k is the smallest odd number and k2^n-1-g, k2^n-1, k*2^n-1+g are three consecutive primes in arithmetic progression.

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A227856 Sequence of pairs k,g with k<32^n the smallest such that 32^n+k, 32^n+k+g, 32^n+k+2*g are three consecutive primes in arithmetic progression starting at n=5 as there is not any solution for n<5.

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cp's OEIS Frontend

A230852 Smallest k<3*2^n such that 3*2^n+k is the smallest of four consecutive primes in arithmetic progression or 0 if no solution.

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A231576 Sequence of pairs k,g such that k is the smallest odd number and k*2^n-1-g, k*2^n-1, k*2^n-1+g are three consecutive primes in arithmetic progression.

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A227856 Sequence of pairs k,g with k<3*2^n the smallest such that 3*2^n+k, 3*2^n+k+g, 3*2^n+k+2*g are three consecutive primes in arithmetic progression starting at n=5 as there is not any solution for n<5.

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A230852 Smallest k<32^n such that 32^n+k is the smallest of four consecutive primes in arithmetic progression or 0 if no solution.

A231576 Sequence of pairs k,g such that k is the smallest odd number and k2^n-1-g, k2^n-1, k*2^n-1+g are three consecutive primes in arithmetic progression.

A227856 Sequence of pairs k,g with k<32^n the smallest such that 32^n+k, 32^n+k+g, 32^n+k+2*g are three consecutive primes in arithmetic progression starting at n=5 as there is not any solution for n<5.