A023303 -id:A023303 - cp's OEIS frontend

A086135 Numbers n such that n and n+10 are both prime but are non-consecutive; which means that at least one prime is between n and n+10; it is not identical with A023303 because here the terms of A031928 are missing.

3, 7, 13, 19, 31, 37, 43, 61, 73, 79, 97, 103, 127, 157, 163, 223, 229, 271, 307, 349, 373, 379, 433, 439, 457, 499, 607, 643, 673, 733, 751, 853, 877, 937, 967, 1009, 1087, 1093, 1213, 1279, 1291, 1297, 1423, 1429, 1483, 1489, 1543, 1549, 1597, 1609, 1657, 1777

Offset: 1

Labos Elemer, Jul 28 2003

nonn

			First deviation from A023303 = {3,7,13,19,31,37,43,61,73,79,97,103,127,139,157,..} is due to the absence of 139=A031928(1).

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Do[s=Prime[n]; s1=Prime[n+1]; If[PrimeQ[s+d]&& !Equal[s1-s, d], Print[s]], {n, 1, 1000}]
Select[Prime[Range[300]],PrimeQ[#+10]&&NextPrime[#]!=(#+10)&] (* Harvey P. Dale, Oct 25 2020 *)

Complement of a=A031928 with respect to b=A023303: [b]&[nota]: this and A031928 are disjoint, but A031928 is a proper subset of A023303.

A023331 Primes that remain prime through 5 iterations of function f(x) = 2x + 3.

Original entry on oeis.org

47, 67, 1427, 6047, 12097, 16057, 44507, 76537, 92387, 100057, 132707, 209647, 263387, 354247, 407947, 465407, 477727, 507757, 775237, 787477, 788687, 865327, 907367, 955457, 1015517, 1121387, 1178197, 1302277, 1360367, 1524247, 1623707

Offset: 1

David W. Wilson

nonn

Primes p such that 2*p+3, 4*p+9, 8*p+21, 16*p+45 and 32*p+93 are also prime. - Vincenzo Librandi, Aug 04 2010

Subsequence of A023303. - Michel Marcus, Jun 25 2014

Vincenzo Librandi, Table of n, a(n) for n = 1..148

Cf. A023303 (through 4 iterations).

[p: p in PrimesUpTo(2000000) | IsPrime(2*p+3) and IsPrime(4*p+9) and IsPrime(8*p+21) and IsPrime(16*p+45) and IsPrime(32*p+93)]; // Vincenzo Librandi, Aug 04 2010

Select[Prime[Range[20000]], And@@PrimeQ[NestList[2 # + 3 &, #, 5]] &] (* Vincenzo Librandi, Jun 25 2014 *)

A086138 Number of primes between p and p+10 if both p and (p+10) are prime, i.e., number of primes somewhere between 10+A023203(n) and A023203(n).

Original entry on oeis.org

3, 2, 2, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 0, 1, 1, 0, 2, 1, 0, 1, 0, 2, 0, 1, 1, 1, 0, 0, 1, 1, 2, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 2, 2, 0, 1, 1, 1, 0, 0, 0, 2, 1, 0, 0, 1, 0, 1, 1, 2, 0, 2, 1, 0, 2, 1, 1, 1, 1, 1, 0, 1, 0, 0, 1, 0, 1, 2, 0, 1, 2, 0, 0, 0, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 2

Offset: 1

Labos Elemer, Jul 29 2003

nonn

			a(n)=0,1,2,3 correspond to {p,p+10} prime-pairs either consecutive ones or those with various d-patterns like as follows: a(n)=0 to cases like 139[10]149; a(n)=2 to 7[4,2,4]17 etc.; a(n)=3 to one case 3[2,2,4,2]13 and a(n)=2 to cases like 31[6,4]37 or 43[4,6]53.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A023303, A031928, A031929.

cp[x_,y_] := Count[Table[PrimeQ[i],{i,x,y}],True] Do[s=Prime[n]; s1=Prime[n+1]; If[PrimeQ[s+d],k=k+1; Print[cp[s+1,s+d-1]]],{n,1,1000}]; k; d=10
Count[Range[#+1,#+9],?PrimeQ]&/@Select[Prime[Range[500]],PrimeQ[#+10]&] (* _Harvey P. Dale, Jan 17 2025 *)

forprime(p=2,1e5,if(isprime(p+10), print1(isprime(p+2)+isprime(p+4)+isprime(p+6)+isprime(p+8)", "))) \\ Charles R Greathouse IV, May 15 2013

Definition clarified by Harvey P. Dale, Jan 17 2025

A244086 Primes p such that p remains prime through 7 iterations of function f(x) = 2*x + 3.

Original entry on oeis.org

477727, 507757, 30596497, 33145687, 36180527, 61192997, 141339217, 148590307, 193394347, 297180617, 374066267, 395534747, 398001547, 419795137, 488716897

Offset: 1

Vincenzo Librandi, Jun 24 2014

Cf. similar sequences with k iterations: A067076 (k=1), A023242 (k=2), A023273 (k=3), A023303 (k=4), A023331 (k=5), A175160 (k=6), this sequence (k=7).

[p: p in PrimesUpTo(600000000) | forall{i: i in [1..7] | IsPrime(2^i*p+3*(2^i-1))}];

Select[Prime[Range[600000]], And@@PrimeQ[NestList[2 # + 3 &, #, 7]] &]

Inappropriate MAGMA code simplified and other terms added from Bruno Berselli, Jun 24 2014

cp's OEIS Frontend

A086135 Numbers n such that n and n+10 are both prime but are non-consecutive; which means that at least one prime is between n and n+10; it is not identical with A023303 because here the terms of A031928 are missing.

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A023331 Primes that remain prime through 5 iterations of function f(x) = 2x + 3.

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A086138 Number of primes between p and p+10 if both p and (p+10) are prime, i.e., number of primes somewhere between 10+A023203(n) and A023203(n).

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A244086 Primes p such that p remains prime through 7 iterations of function f(x) = 2*x + 3.

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Magma

Mathematica

Extensions