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A256602 Primes of form 12*k + 1 and not a twin prime.

37, 97, 157, 277, 337, 373, 397, 409, 457, 541, 577, 613, 673, 709, 733, 757, 769, 853, 877, 937, 997, 1009, 1069, 1117, 1129, 1201, 1213, 1237, 1249, 1297, 1381, 1549, 1597, 1657, 1693, 1741, 1753, 1777, 1801, 1861, 1993, 2017, 2053, 2137, 2161, 2221, 2281, 2293, 2377

Offset: 1

Neri Gionata, Apr 06 2015

nonn

A142793 is a subsequence.

			37 = 12*3+1 and 35 is not prime, so 37 belongs to the sequence.

Cf. A142793 (primes congruent to 37 mod 60).

is(n)=isprime(n) && n%12==1 && !isprime(n+2) && !isprime(n-2) \\ Charles R Greathouse IV, Apr 06 2015

a(n) ~ 4n log n. - Charles R Greathouse IV, Apr 06 2015

A255230 Integers n such that n^2 = 2x(y-x), where x and y are consecutive terms in A014574.

Original entry on oeis.org

4, 12, 48, 120, 468, 1260, 720, 2448, 10080, 12060, 15912, 7560, 18480, 7392, 9660, 27720, 33480, 14400, 25080, 36708, 10092, 34188, 42120, 83400, 29820, 20040, 67320, 114408, 206628, 67368, 72192, 102648, 152928, 51732, 59880, 152700, 106440, 100980, 171480

Offset: 1

Neri Gionata, Feb 18 2015

nonn

n is a term if n^2 = 2*x*(y-x), where x and y are the averages of two consecutive twin prime pairs.

			48^2 = 2*192*(198-192), and 192 and 198 are consecutive terms in A014574, so 48 is in the sequence.

Cf. A014574.

lista(nn) = {p=2; last = 0; forprime (q=3, nn, if (q-p==2, if (! last, last = p+1, new = p+1; val = new^2-last^2 - (new-last)^2; if (issquare(val), print1(sqrtint(val), ", ")); last = new; );); p=q;);} \\ Michel Marcus, Feb 18 2015

More terms from Michel Marcus, Feb 18 2015

A255933 a(n) is the largest integer m such that s/(m!-1) is an integer, where s is the sum of all previous terms; a(1)=1.

Original entry on oeis.org

1, 2, 2, 3, 2, 3, 2, 3, 2, 3, 4, 2, 2, 2, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 4, 5, 2, 2, 2, 3, 2, 3, 4, 2, 2, 2, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 4, 2, 2, 5, 2, 3, 2, 3, 4, 2, 2, 2, 2, 3, 2

Offset: 1

Neri Gionata, Mar 11 2015

nonn

For all n>1, a(n) exists and is at least 2, since 2 gives a denominator (2!-1) = 1, thus an integer.
The sequence of partial sums is: 1,3,5,8,10,13,15,18,20,23,27,29,31,33,35,38,...
The record values occur at n=1,2,4,11,49,286,1997,...

a(5) = 2 since (1+2+2+3)/(n!-1) = 8/(2!-1) = 8, an integer.
a(6) = 3 since (1+2+2+3+2)/(n!-1) = 10/(3!-1) = 2, an integer.

lista(nn) = {v = [1]; s = 1; print1(s, ", "); for (n=2, nn, k = 2; while(k!-1 <= s, k++); until (type(s/(k!-1)) == "t_INT", k--); s += k; print1(k, ", "); v = concat(v, k););} \\ Michel Marcus, Mar 11 2015

A255229 Integers n such that n^2 - 1 is the difference of the squares of twin primes.

Original entry on oeis.org

5, 7, 11, 13, 17, 31, 41, 43, 49, 77, 83, 101, 109, 119, 133, 179, 203, 263, 277, 283, 307, 311, 329, 353, 377, 407, 413, 419, 431, 437, 463, 473, 493, 577, 581, 619, 629, 703, 757, 791, 811, 907, 911, 913, 991, 1001, 1037, 1061, 1103, 1121, 1249, 1289, 1337, 1373, 1441, 1457, 1487, 1523, 1597, 1651, 1781

Offset: 1

Neri Gionata, Feb 18 2015

nonn

			31^2 - 1 = 241^2 - 239^2, and (239, 241) is a twin prime pair, so 31 is in the sequence.

Cf. A088486 (corresponding lesser twin primes), A111046.

lst={};f[n_]:=Sqrt[Prime[n]^2-NextPrime[Prime[n],-1]^2+1];
Do[If[Prime[n]-NextPrime[Prime[n],-1]==2&&IntegerQ[f[n]],AppendTo[lst,f[n]]],{n,3,10^5}];lst (* Ivan N. Ianakiev, Mar 30 2015 *)

lista(nn) = {forprime(p=3, nn, q = precprime(p-1); if (((p-q) == 2) && issquare(d=p^2-q^2+1), print1(sqrtint(d), ", ")); ); } \\ Michel Marcus, Feb 18 2015

cp's OEIS Frontend

User: Neri Gionata

A256602 Primes of form 12*k + 1 and not a twin prime.

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A255230 Integers n such that n^2 = 2x(y-x), where x and y are consecutive terms in A014574.

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Extensions

A255933 a(n) is the largest integer m such that s/(m!-1) is an integer, where s is the sum of all previous terms; a(1)=1.

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Maple

PARI

A255229 Integers n such that n^2 - 1 is the difference of the squares of twin primes.

Views

Author

Keywords

Examples

Crossrefs

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Mathematica

PARI

cp's OEIS Frontend

User: Neri Gionata

A256602 Primes of form 12*k + 1 and not a twin prime.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

Formula

A255230 Integers n such that n^2 = 2*x*(y-x), where x and y are consecutive terms in A014574.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

Extensions

A255933 a(n) is the largest integer m such that s/(m!-1) is an integer, where s is the sum of all previous terms; a(1)=1.

Views

Author

Keywords

Comments

Programs

Maple

PARI

A255229 Integers n such that n^2 - 1 is the difference of the squares of twin primes.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

PARI

A255230 Integers n such that n^2 = 2x(y-x), where x and y are consecutive terms in A014574.