Emre APARI - cp's OEIS frontend

A282565 a(n) is the largest prime p < 2^n such that p + 2^n is also prime.

3, 5, 13, 29, 43, 113, 223, 509, 907, 2003, 4051, 8171, 16333, 32603, 65437, 130649, 262027, 524261, 1048507, 2097047, 4194277, 8388209, 16776481, 33554201, 67108753, 134217401, 268435273, 536869631, 1073741719, 2147483549, 4294966477, 8589934307, 17179868383

Offset: 2

Emre APARI, Feb 18 2017

nonn

			13 + 16 (2^4) = 29  (prime).
29 + 32 (2^5) = 61  (prime).
43 + 64 (2^6) = 107 (prime).

Cf. A000040, A000079.

A278693 Numbers n such that prime(n) + prime(n+1) is divisible by 2n + 1.

Original entry on oeis.org

4, 181, 6458, 6460, 40083, 40121, 10553502, 69709942, 3140421718, 3140421737, 3140421740, 3140421768, 3140421805, 3140421845

Offset: 1

Emre APARI, Nov 26 2016

The pairs of consecutive prime numbers corresponding to the terms of this sequence are 7,11; 1087,1091; 64579,64591; 64601,64609; 481001,481003; 481447,481469; 189963043,189963047; ...

The corresponding integers (prime(k) + prime(k+1))/(2*k+1) are 2, 6, 10, 10, 12, 12, 18, ...

			a(1) = 4 since prime(4) + prime(5) = 7 + 11 = 18 is divisible by 2*4 + 1 = 9.
a(2) = 181 since prime(181) + prime(182) = 1087 + 1091 = 2178 is divisible by 2*181 + 1 = 363.
a(3) = 6458 since prime(6458) + prime(6459) = 64579 + 64591 = 129170 is divisible by 2*6458 + 1 = 12917.
a(4) = 6460 since prime(6460) + prime(6461) = 64601 + 64609 = 129210 is divisible by 2*6460 + 1 = 12921.
a(5) = 40083 since prime(40083) + prime(40084) = 481001 + 481003 = 962004 is divisible by 2*40083 + 1 = 80167.
a(6) = 40121 since prime(40121) + prime(40122) = 481447 + 481469 = 962916 is divisible by 2*40121 + 1 = 80243.
a(7) = 10553502 since prime(10553502) + prime(10553503) = 189963043 + 189963047 = 379926090 is divisible by 2*10553502 + 1 = 21107005.
a(8) = 69709942 since prime(69709942) + prime(69709943) = 1394198837 + 1394198863 = 2788397700 is divisible by 2*69709942 + 1 = 139419885.
a(9) = 3140421718 since prime(3140421718) + prime(3140421719) = 75370121237 + 75370121251 = 150740242488 is divisible by 2*3140421718 + 1 = 6280843437.
a(10) = 3140421737 since prime(3140421737) + prime(3140421738) = 75370121689 + 75370121711 = 150740243400 is divisible by 2*3140421737 + 1 = 6280843475.
a(11) = 3140421740 since prime(3140421740) + prime(3140421741) = 75370121767 + 75370121777 = 150740243544 is divisible by 2*3140421740 + 1 = 6280843481.
a(12) = 3140421768 since prime(3140421768) + prime(3140421769) = 75370122439 + 75370122449 = 150740244888 is divisible by 2*3140421768 + 1 = 6280843537.
a(13) = 3140421805 since prime(3140421805) + prime(3140421806) = 75370123327 + 75370123337 = 150740246664 is divisible by 2*3140421805 + 1 = 6280843611.
a(14) = 3140421845 since prime(3140421845) + prime(3140421846) = 75370124273 + 75370124311 = 150740248584 is divisible by 2*3140421845 + 1 = 6280843691.

Cf. A000040, A001043.

isok(k) = denominator((prime(k)+prime(k+1))/(2*k+1)) == 1; \\ Michel Marcus, Nov 26 2016

n=0; p=2; forprime(q=3,1e9, n++; if((p+q)%(2*n+1)==0, print1(n", ")); p=q) \\ Charles R Greathouse IV, Nov 28 2016

a(8)-a(14) from Charles R Greathouse IV, Nov 28 2016

A272245 Cubes of the form prime(n)+n.

Original entry on oeis.org

8, 27, 1000, 2744, 4096, 46656, 68921, 274625, 941192, 1295029, 1481544, 1906624, 14886936, 34328125, 35937000, 45882712, 50243409, 63521199, 64000000, 67917312, 76225024, 95443993, 112678587, 142236648, 143877824, 174676879, 198155287, 203297472, 216000000

Offset: 1

Emre APARI, Apr 23 2016

nonn

The cube root of the first 10 terms are: 2,3,10,14,16,36,41,65,98,109.

			prime(147) + 147 = 853 + 147 = 1000; which is 10^3.

Giovanni Resta, Table of n, a(n) for n = 1..500

Cf. A000578, A014688, A030078, A076147, A053149.

lista(nn) = {for (n=1, nn, if (ispower(p=n+prime(n), 3), print1(p, ", ")););} \\ Michel Marcus, Apr 23 2016

a(n) = A014688(A076147(n)). - Michel Marcus, Apr 23 2016

a(20)-a(29) from Giovanni Resta, Apr 23 2016

A271575 Primes p such that p+10, p+100 and p+1000 are all prime.

Original entry on oeis.org

13, 31, 97, 163, 181, 283, 409, 499, 709, 787, 811, 877, 1087, 1399, 1423, 1609, 1777, 1801, 1879, 2347, 2677, 2719, 3457, 3517, 3919, 4273, 4483, 5701, 6043, 6121, 6211, 6481, 6691, 7573, 8941, 9733, 9739, 10069, 10093, 10159, 10243, 10789, 11161, 11251, 11689, 12799, 12907

Offset: 1

Emre APARI, Apr 10 2016

nonn

Number of terms < 10^k: 0, 3, 12, 37, 159, 789, 3960, 21708, 129910, ..., . - Robert G. Wilson v, Jun 20 2018

			p=13; p+10=23 (is prime), p+100=113 (is prime), p+1000=1013 (is prime).

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000040, A002385, A015916, A023203.

select(t -> isprime(t+1000) and isprime(t+100) and isprime(t+10) and isprime(t), [seq(i,i=7..20000, 6)]); # Robert Israel, Jun 20 2018

Select[Prime[Range[10000]], PrimeQ[# + 10] && PrimeQ[# + 100] && PrimeQ[# + 1000] &] (* Robert Price, Apr 10 2016 *)

lista(nn) = forprime(p=2, nn, if (isprime(p+10) && isprime(p+100) && isprime(p+1000), print1(p, ", "))); \\ Michel Marcus, Apr 10 2016

A271549 Primes p such that p+10^2, p+10^3, p+10^5, p+10^7, p+10^11, p+10^13 and p+10^17 are all prime.

Original entry on oeis.org

1399, 2157763, 13034041, 38208649, 38502313, 41518651, 42745111, 48154147, 49435063, 53872447, 58981513, 75194563, 83037247, 86139409, 101533963, 106287019, 140778403, 144593431, 155554237, 166083133, 166650193, 189371671, 199865893, 201738379, 224472877, 240133753, 271331773

Offset: 1

Emre APARI, Apr 10 2016

nonn

The exponents of 10 are all prime (2,3,5,7,11,13,17).

			p = 1399:
p+10^2  = 1499 (is prime).
p+10^3  = 2399 (is prime).
p+10^5  = 101399 (is prime).
p+10^7  = 10001399 (is prime).
p+10^11 = 100000001399 (is prime).
p+10^13 = 10000000001399 (is prime).
p+10^17 = 100000000000001399 (is prime).

Cf. A000040, A002385, A015916, A023203, A271575.

Select[Prime[Range[10^9]], PrimeQ[# + 10^2] && PrimeQ[# + 10^3] && PrimeQ[# + 10^5] &&  PrimeQ[# + 10^7] && PrimeQ[# + 10^11] &&  PrimeQ[# + 10^13] && PrimeQ[# + 10^17] &] (* Robert Price, Apr 10 2016 *)

lista(nn) = forprime(p=2, nn, if (isprime(p+10^2) && isprime(p+10^3) && isprime(p+10^5) && isprime(p+10^7) && isprime(p+10^11) && isprime(p+10^13) && isprime(p+10^17), print1(p, ", "))); \\ Altug Alkan, Apr 10 2016

More terms from Altug Alkan, Apr 10 2016

A267799 a(n) = (1 + 2^n + 3^n)/2.

Original entry on oeis.org

3, 7, 18, 49, 138, 397, 1158, 3409, 10098, 30037, 89598, 267769, 801258, 2399677, 7190838, 21556129, 64635618, 193841317, 581392878, 1743916489, 5231225178, 15692626957, 47075783718, 141223156849, 423661081938, 1270966468597, 3812865851358, 11438530445209, 34315457117898, 102946102918237

Offset: 1

Emre APARI, Apr 07 2016

			a(3) = (1 + 2^3 + 3^3)/2 = 18.

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (6,-11,6).

Cf. A000079, A000244, A001550, A007689.

Table[(1 + 2^n + 3^n)/2, {n, 30}] (* Michael De Vlieger, Apr 07 2016 *)

Vec(x*(3-11*x+9*x^2)/((1-x)*(1-2*x)*(1-3*x)) + O(x^50)) \\ Colin Barker, Apr 07 2016

a(n) = (1 + 2^n + 3^n)/2 \\ Charles R Greathouse IV, Apr 07 2016

a(n) = (A007689(n)+1)/2.
From Colin Barker, Apr 07 2016: (Start)
a(n) = 6*a(n-1)-11*a(n-2)+6*a(n-3) for n>3.
G.f.: x*(3-11*x+9*x^2) / ((1-x)*(1-2*x)*(1-3*x)).
(End)
a(n) = A001550(n)/2, for n >= 1. - Altug Alkan, Apr 08 2016

A271050 Positive integer k such that k^2 = p^2 + q^2 - 1 where p and q are consecutive primes.

Original entry on oeis.org

13, 17, 37, 157, 307, 437, 1451, 6487, 60773, 421133, 1445957, 2064493, 15247789, 177075397, 16509255853, 4270704255979, 56635565799013, 124750634536736711, 628179811369719907, 81815870181890275241, 13861061749008806276269, 91566796731172246399571

Offset: 1

Emre APARI, Mar 29 2016

nonn

Prime terms of this sequence are listed in A167276. - Altug Alkan, Mar 30 2016

			         7^2 + 11^2 - 1 = 169 (13^2, k is prime),
        11^2 + 13^2 - 1 = 289 (17^2, k is prime),
        23^2 + 29^2 - 1 = 1369 (37^2, k is prime),
      109^2 + 113^2 - 1 = 24649 (157^2, k is prime),
      211^2 + 223^2 - 1 = 94249 (307^2, k is prime),
      307^2 + 311^2 - 1 = 190969 (437^2, k is semiprime),
    1021^2 + 1031^2 - 1 = 2105401 (1451^2, k is prime),
  42967^2 + 42979^2 - 1 = 3693357529 (60773^2, k is prime).

Cf. A001248, A069484, A160054 (the corresponding primes p), A167276.

p = 2; q = 3; lst = {}; While[p < 10^15, If[ IntegerQ@ Sqrt[p^2 + q^2 - 1], AppendTo[lst, Sqrt[p^2 + q^2 - 1]];
Print[Sqrt[p^2 + q^2 - 1]]]; p = q; q = NextPrime@ q] (* Robert G. Wilson v, Mar 30 2016 *)

list(nn) = {p = 2; forprime(q=3, nn, if (issquare(s = q^2+p^2-1), print1(sqrtint(s), ", ")); p = q;);} \\ Michel Marcus, Mar 29 2016

More terms from Jinyuan Wang, Jan 09 2021

A270972 Primes p such that p-2, p^2-2 and p^3-2 are all prime.

Original entry on oeis.org

19, 8629, 9721, 12109, 13831, 15331, 17029, 17989, 25849, 33151, 56209, 70999, 73039, 78541, 92461, 97369, 97609, 103069, 103969, 147139, 174469, 179719, 203341, 217369, 221401, 242059, 249541, 269431, 277549, 283009, 285559, 324619, 333451, 346669, 393079, 404269, 409261, 424891, 440551, 488689

Offset: 1

Emre APARI, Mar 27 2016

nonn

Subsequence of A006512. - Altug Alkan, Mar 27 2016

			p=19; p-2 = 17 (is prime), p^2-2 = 359 (is prime), p^3-2 = 6857 (is prime).

Cf. A000040, A001248, A006512, A030078, A049001, A040976, A153481.

[p: p in PrimesUpTo(500000) | IsPrime(p-2) and IsPrime(p^2-2) and IsPrime(p^3-2)]; // Vincenzo Librandi, Mar 28 2016

Select[Prime@ Range@ 42000, Function[k, AllTrue[k^# & /@ Range@ 3 - 2, PrimeQ]]] (* Michael De Vlieger, Mar 27 2016, Version 10 *)

lista(nn) = {forprime(p=5, nn, if(isprime(p-2) && isprime(p^2-2) && isprime(p^3-2), print1(p, ", ")));} \\ Altug Alkan, Mar 27 2016

More terms from Altug Alkan, Mar 27 2016

A262782 a(n) = sum_{k=1..n} 3^prime(k).

Original entry on oeis.org

9, 36, 279, 2466, 179613, 1773936, 130914099, 1293175566, 95436354393, 68725813719276, 686399210003223, 450970305101000586, 36923966682271786989, 365180934076808864616, 26953995293034312152403, 19410199662973054208949126, 14149796291401707558973760193, 141323271117050318101857059796

Offset: 1

Emre APARI, Mar 24 2016

			a(2) = 3^prime(1) + 3^prime(2) = 3^2+3^3=36.

Cf. A057901.

Accumulate@ Array[3^Prime@ # &, {18}] (* Michael De Vlieger, Mar 24 2016 *)

a(n) = sum(k=1, n, 3^prime(k)); \\ Altug Alkan, Mar 24 2016

More terms from Michael De Vlieger, Mar 24 2016

A262778 a(n) = 10^n + prime(n).

Original entry on oeis.org

12, 103, 1005, 10007, 100011, 1000013, 10000017, 100000019, 1000000023, 10000000029, 100000000031, 1000000000037, 10000000000041, 100000000000043, 1000000000000047, 10000000000000053, 100000000000000059, 1000000000000000061

Offset: 1

Emre APARI, Mar 24 2016

nonn

			a(4) = 10^4+7=10007.

Cf. A000040, A011557.

a(n) = 10^n + prime(n); \\ Altug Alkan, Mar 24 2016

cp's OEIS Frontend

User: Emre APARI

A282565 a(n) is the largest prime p < 2^n such that p + 2^n is also prime.

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A278693 Numbers n such that prime(n) + prime(n+1) is divisible by 2n + 1.

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A272245 Cubes of the form prime(n)+n.

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A271575 Primes p such that p+10, p+100 and p+1000 are all prime.

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A271549 Primes p such that p+10^2, p+10^3, p+10^5, p+10^7, p+10^11, p+10^13 and p+10^17 are all prime.

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A267799 a(n) = (1 + 2^n + 3^n)/2.

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A271050 Positive integer k such that k^2 = p^2 + q^2 - 1 where p and q are consecutive primes.

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A270972 Primes p such that p-2, p^2-2 and p^3-2 are all prime.

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