A014688 -id:A014688 - cp's OEIS frontend

A337334 a(n) = pi(b(n)), where pi is the prime counting function (A000720) and b(n) = a(n-1) + b(n-1) with a(0) = b(0) = 1.

1, 1, 2, 3, 4, 5, 7, 9, 11, 14, 16, 21, 24, 30, 35, 42, 48, 58, 67, 78, 91, 103, 121, 138, 158, 181, 205, 233, 266, 298, 337, 378, 429, 480, 539, 602, 674, 751, 838, 930, 1031, 1147, 1274, 1402, 1556, 1715, 1896, 2090, 2296, 2527, 2777, 3047, 3340, 3669, 4016

Offset: 0

Ya-Ping Lu, Aug 23 2020

nonn

It can be proved that this is an increasing sequence from the theorem of Lu and Deng (see LINKS), which states "the prime gap of a prime number is less than or equal to the prime count of the prime number”, or prime(n+1) - prime(n) <= pi(prime(n)).

			a(1) = pi(b(1)) = pi(a(0) + b(0)) = pi(1 + 1) = pi(2) = 1
a(2) = pi(b(2)) = pi(a(1) + b(1)) = pi(1 + 2) = pi(3) = 2
a(3) = pi(b(3)) = pi(a(2) + b(2)) = pi(2 + 3) = pi(5) = 3
a(4) = pi(b(4)) = pi(a(3) + b(3)) = pi(3 + 5) = pi(8) = 4
a(54)= pi(b(54))= pi(a(53)+ b(53))= pi(3669+34327)=pi(37996)=4016

Ya-Ping Lu and Shu-Fang Deng, An upper bound for the prime gap, arXiv:2007.15282 [math.GM], 2020.

Cf. A000720 (pi), A014688 (prime(n)+n), A332086.

A337334 := proc(n)
    option remember;
    if n = 0 then
        1;
    else
        numtheory[pi](A061535(n)) ;
    end if;
end proc:
seq(A337334(n),n=0..20) ; # R. J. Mathar, Jun 18 2021

from sympy import primepi
a_last = 1
b_last = 1
for n in range(1, 1001):
    b = a_last + b_last
    a = primepi(b)
    print(a)
    a_last = a
    b_last = b

a(n) = pi(b(n)), where b(n) = a(n-1) + b(n-1) with a(0) = b(0) = 1.

a(n) = A000720(A061535(n)), n>=1. - R. J. Mathar, Jun 18 2021

a(0) inserted by R. J. Mathar, Jun 18 2021

A065995 a(n) = prime(prime(n) + n).

Original entry on oeis.org

5, 11, 19, 31, 53, 67, 89, 103, 131, 167, 181, 227, 251, 269, 293, 347, 383, 401, 443, 467, 491, 547, 577, 617, 673, 709, 733, 761, 787, 823, 929, 967, 1013, 1031, 1097, 1117, 1181, 1229, 1277, 1303, 1373, 1409, 1481, 1489, 1531, 1553, 1627, 1741, 1783

Offset: 1

Reinhard Zumkeller, Dec 10 2001

nonn

Popular Computing (Calabasas, CA), A Function Of Primes, Problem 206, Vol. 5 (No. 55, Oct 1977), page PC55-19. Asks for rate of growth of a(n).

N. J. A. Sloane, Table of n, a(n) for n = 1..20000 [First 1000 terms from Harry J. Smith]
David A. Corneth, PARI program

Cf. A000040, A014688.

p:=ithprime; f:=n->p(p(n)+n); [seq(f(n),n=1..1000)]; # N. J. A. Sloane, Apr 16 2015

Table[Prime[n+ Prime[n]], {n, 100}] (* Waldemar Puszkarz, Jan 24 2015 *)

a(n) = prime(prime(n) + n); \\ Harry J. Smith, Nov 06 2009

PARI
```
\\ See PARI link
```

a(n) = A000040(A014688(n)). - Omar E. Pol, Oct 22 2013

A076556 Greatest prime divisor of n-th prime + n.

Original entry on oeis.org

3, 5, 2, 11, 2, 19, 3, 3, 2, 13, 7, 7, 3, 19, 31, 23, 19, 79, 43, 13, 47, 101, 53, 113, 61, 127, 13, 5, 23, 13, 79, 163, 17, 173, 23, 17, 97, 67, 103, 71, 11, 223, 13, 79, 11, 7, 43, 271, 23, 31, 71, 97, 7, 61, 13, 29, 163, 47, 7, 31, 43, 71, 37, 5, 7, 383, 199, 5, 13, 419

Offset: 1

Zak Seidov, Oct 19 2002

Cf. A093572, A014688.

Table[FactorInteger[Prime[n]+n][[-1,1]],{n,70}] (* Harvey P. Dale, Jan 19 2015 *)

A104935 Primes squared of the form k + prime(k).

Original entry on oeis.org

49, 529, 1681, 10609, 26569, 27889, 72361, 100489, 109561, 196249, 214369, 727609, 863041, 877969, 1142761, 1371241, 1471369, 1692601, 1957201, 2199289, 2601769, 2745649, 3500641, 3613801, 3798601, 3972049, 4214809, 5812921, 6405961, 7134241, 7349521

Offset: 1

Zak Seidov, Apr 25 2005

nonn

Primes squared in A014688.

Tyler Busby, Table of n, a(n) for n = 1..10000 (terms 1..106 from Zak Seidov)

Cf. A014688.

Subsequence of A104992.

lista(nn) = {vec = vector(nn, i, i + prime(i)); pp = select(i->(issquare(i) && isprime(sqrtint(i))), vec); print(pp);} \\ Michel Marcus, Oct 09 2013

a(12) - a(16) from Michel Marcus, Oct 09 2013

a(17) onward from Zak Seidov, Mar 13 2014

A104992 Squares of the form n+prime(n).

Original entry on oeis.org

16, 49, 529, 676, 1225, 1521, 1681, 1764, 3249, 4096, 5929, 9604, 10404, 10609, 11664, 12321, 19600, 24336, 25921, 26569, 27889, 33856, 34225, 34596, 46656, 51984, 71289, 72361, 91204, 100489, 101124, 104976, 106929, 109561, 110889, 111556, 121104, 125316, 131769, 135424, 136161, 141376, 152881, 156816, 163216, 166464, 179776, 188356

Offset: 1

Zak Seidov, Apr 25 2005

nonn

Zak Seidov, Table of n, a(n) for n = 1..300

Squares in A014688.

Select[Table[n+Prime[n],{n,20000}],IntegerQ[Sqrt[#]]&] (* Harvey P. Dale, Sep 21 2024 *)

lista(nn) = {vec = vector(nn, i, i + prime(i)); pp = select(i->(issquare(i)), vec); print(pp);} \\ Michel Marcus, Oct 09 2013

A106588 Difference between n-th prime squared and n-th perfect square.

Original entry on oeis.org

3, 5, 16, 33, 96, 133, 240, 297, 448, 741, 840, 1225, 1512, 1653, 1984, 2553, 3192, 3397, 4128, 4641, 4888, 5757, 6360, 7345, 8784, 9525, 9880, 10665, 11040, 11869, 15168, 16137, 17680, 18165, 20976, 21505, 23280, 25125, 26368, 28329, 30360

Offset: 1

Alexandre Wajnberg, May 10 2005

			a(5) = 96 because 121 (fifth prime^2) - 25 (fifth square) = 96.

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A000290, A001248, A014688, A075526, A076368.

[NthPrime(n)^2 - n^2: n in [1..50]]; // G. C. Greubel, Sep 07 2021

Mathematica
```
Table[Prime[n]^2 - n^2, {n, 50}]
```

a(n) = prime(n)^2 - n^2; \\ Michel Marcus, Sep 08 2021

[nth_prime(n)^2 - n^2 for n in (1..50)] # G. C. Greubel, Sep 07 2021

a(n) = prime(n)^2 - n^2.

Extended by Ray Chandler, May 13 2005

A115888 Palindromes equal to the sum of a prime number with its index.

Original entry on oeis.org

3, 5, 8, 11, 101, 242, 383, 424, 454, 545, 585, 606, 666, 676, 757, 949, 2552, 3443, 3663, 4664, 5445, 6006, 6886, 9339, 10001, 10601, 11411, 12321, 15551, 15651, 17871, 17971, 18281, 21412, 22622, 22922, 24642, 24942, 25752, 26762, 28582

Offset: 1

Giovanni Resta, Feb 06 2006

			666 = prime(103)+103.

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

Cf. A014688, A115884.

Subsequence of A002113.

Select[Total/@Table[{n,Prime[n]},{n,3200}],PalindromeQ] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jun 03 2017 *)

ispal(n) = my(e=digits(n));e == Vecrev(e) \\ A002113
for(k=1,10^6, b=k+prime(k);if(ispal(b),print1(b,", "))) \\ Alexandru Petrescu, Jun 15 2022

A135682 a(n)=n if n=1 or if n=prime. Otherwise, n=4 if n is even and n=7 if n is odd.

Original entry on oeis.org

1, 2, 3, 4, 5, 4, 7, 4, 7, 4, 11, 4, 13, 4, 7, 4, 17, 4, 19, 4, 7, 4, 23, 4, 7, 4, 7, 4, 29, 4, 31, 4, 7, 4, 7, 4, 37, 4, 7, 4, 41, 4, 43, 4, 7, 4, 47, 4, 7, 4, 7, 4, 53, 4, 7, 4, 7, 4, 59, 4, 61, 4, 7, 4, 7, 4, 67, 4, 7, 4, 71, 4, 73

Offset: 1

Mohammad K. Azarian, Dec 01 2007

nonn

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A000027; A000040; A135679; A135680; A014681; A014682; A014683; A014684; A014685; A014686; A014687; A014688; A014689; A014690; A135681.

a[n_] := If[PrimeQ[n] || n == 1, n, If[EvenQ[n], 4, 7] ]; Table[a[n], {n,1,25}] (* G. C. Greubel, Oct 26 2016 *)

A135684 a(n)=11 if n is a prime number. Otherwise, a(n)=n.

Original entry on oeis.org

1, 11, 11, 4, 11, 6, 11, 8, 9, 10, 11, 12, 11, 14, 15, 16, 11, 18, 11, 20, 21, 22, 11, 24, 25, 26, 27, 28, 11, 30, 11, 32, 33, 34, 35, 36, 11, 38, 39, 40, 11, 42, 11, 44, 45, 46, 11, 48, 49, 50, 51, 52, 11, 54, 55, 56, 57, 58, 11

Offset: 1

Mohammad K. Azarian, Dec 01 2007

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A000027; A000040; A135679; A135680; A014681; A014682; A014683; A014684; A014685; A014686; A014687; A014688; A014689; A014690; A135681; A135682; A135683.

[IsPrime(n) select 11 else n: n in [1..70]]; // Vincenzo Librandi, Feb 22 2013

Table[If[PrimeQ[n], 11, n], {n, 70}] (* Vincenzo Librandi, Feb 22 2013 *)

A176628 a(n) = prime(n) - n*(-1)^prime(n).

Original entry on oeis.org

1, 5, 8, 11, 16, 19, 24, 27, 32, 39, 42, 49, 54, 57, 62, 69, 76, 79, 86, 91, 94, 101, 106, 113, 122, 127, 130, 135, 138, 143, 158, 163, 170, 173, 184, 187, 194, 201, 206, 213, 220, 223, 234, 237, 242, 245, 258, 271, 276, 279, 284, 291, 294, 305, 312, 319, 326

Offset: 1

Juri-Stepan Gerasimov, Apr 22 2010

nonn

Unit together with n-th odd prime + n + 1.

			a(1) = prime(1) - 1*(-1)^prime(1) = 2-1 = 1.

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

[1]cat[NthPrime(n) +n: n in [2..60]]; // G. C. Greubel, Jul 01 2021

np[n_]:=Module[{npr=Prime[n]},npr-n (-1)^npr];Array[np,60] (* Harvey P. Dale, Oct 08 2011 *)

[1]+[nth_prime(n) +n for n in (2..60)] # G. C. Greubel, Jul 01 2021

a(n+1) = A014688(n+1).

Entries checked by R. J. Mathar, Apr 27 2010

cp's OEIS Frontend

A337334 a(n) = pi(b(n)), where pi is the prime counting function (A000720) and b(n) = a(n-1) + b(n-1) with a(0) = b(0) = 1.

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A065995 a(n) = prime(prime(n) + n).

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PARI

PARI

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A076556 Greatest prime divisor of n-th prime + n.

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Mathematica

A104935 Primes squared of the form k + prime(k).

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PARI

Extensions

A104992 Squares of the form n+prime(n).

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Mathematica

PARI

A106588 Difference between n-th prime squared and n-th perfect square.

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Magma

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PARI

Sage

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A115888 Palindromes equal to the sum of a prime number with its index.

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Examples

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Programs

Mathematica

PARI

A135682 a(n)=n if n=1 or if n=prime. Otherwise, n=4 if n is even and n=7 if n is odd.

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