cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

A014688 a(n) = n-th prime + n.

Original entry on oeis.org

3, 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

Views

Author

Mohammad K. Azarian

Keywords

Comments

Conjecture: this sequence contains an infinite number of primes (A061068), yet contains arbitrarily long "prime deserts" such as the 11 composites in A014688 between a(6) = 19 and a(18) = 79 and the 17 composites in A014688 between a(48) = 271 and a(66) = 383. - Jonathan Vos Post, Nov 22 2004
Does an n exist such that n*prime(n)/(n+prime(n)) is an integer? - Ctibor O. Zizka, Mar 04 2008. The answer to Zizka's question is easily seen to be No: such an integer k would be positive and less than prime(n), but then k*(n + prime(n)) = prime(n)*n would be impossible. - Robert Israel, Apr 20 2015
Complement of A064427. - Jaroslav Krizek, Oct 28 2009
According to a theorem of Lu and Deng (see LINKS), there exists at least one prime number p such that a(n)-n < p <= a(n); equivalently pi(a(n)) - pi(a(n)-n) >= 1 (see A332086). For example, prime number 3 is in the range of (2,3], 5 in (3,5], 7 in (5,8], and 29 & 31 in (23,32]. - Ya-Ping Lu, Sep 02 2020

Links

Crossrefs

Cf. A000040, A078916, A093570, A093571, A076556, A061068, A332086.
See also A065995.

Programs

Formula

a(n) = n + A000040(n) = n + A008578(n+1) = n + A158611(n+2). - Jaroslav Krizek, Aug 31 2009
a(n) = A090178(n+1) - 1 = (n+1)-th noncomposite number + n for n >= 2. a(n) = A167136(n+1). a(1) = 3, a(n) = a(n-1) + A008578(n+1) - A008578(n) + 1 for n >= 2. a(1) = 3, a(n) = a(n-1) + A001223(n-1) + 1 for n >= 3. - Jaroslav Krizek, Oct 28 2009
a(n) = 2*OR(p,n) - XOR(p,n), for n-th prime p. - Gary Detlefs, Oct 26 2013
a(n) = A078916(n) - n. - Zak Seidov, Nov 10 2013

Extensions

More terms from Vasiliy Danilov (danilovv(AT)usa.net), Jul 1998
Corrected for changes of offsets of A008578 and A158611 by Jaroslav Krizek, Oct 28 2009

A174008 n-th prime plus n-th even nonnegative nonprime.

Original entry on oeis.org

2, 7, 11, 15, 21, 25, 31, 35, 41, 49, 53, 61, 67, 71, 77, 85, 93, 97, 105, 111, 115, 123, 129, 137, 147, 153, 157, 163, 167, 173, 189, 195, 203, 207, 219, 223, 231, 239, 245, 253, 261, 265, 277, 281, 287, 291, 305, 319, 325, 329, 335, 343, 347, 359, 367, 375
Offset: 1

Views

Author

Juri-Stepan Gerasimov, Mar 05 2010

Keywords

Comments

Apart from the first term, same as A078916 = prime(n) + 2n.

Examples

			a(1)=2 because 2+0=2; a(2)=7 because 3+4=7.

Crossrefs

Cf. A000040, A001477, A078916, A163300.

Programs

  • Maple
    Contribution from R. J. Mathar, Apr 14 2010: (Start)
A163300 := proc(n) option remember ; if n = 1 then 0; else for a from procname(n-1)+2 by 2 do if not isprime(a) then return a ; end if; end do; end if; end proc:
A174008 := proc(n) ithprime(n)+A163300(n) ; end proc: seq(A174008(n),n=1..80) ; (End)

Formula

a(n)=A000040(n)+A163300(n).
a(n) ~ n log n.

A078917 Primes of the form prime(k) + 2*k.

Original entry on oeis.org

7, 11, 31, 41, 53, 61, 67, 71, 97, 137, 157, 163, 167, 173, 223, 239, 277, 281, 347, 359, 367, 383, 401, 433, 439, 443, 449, 503, 521, 569, 601, 643, 673, 761, 769, 809, 821, 829, 877, 883, 941, 953, 1031, 1063, 1093, 1109, 1153, 1163, 1217, 1223, 1277, 1307
Offset: 1

Views

Author

Reinhard Zumkeller, Dec 13 2002

Keywords

Links

Crossrefs

Cf. A078916, A061068, A000040.

Programs

  • Magma
    [a: n in [1..200] | IsPrime(a) where a is NthPrime(n)+2*n ]; // Vincenzo Librandi, Dec 09 2011
  • Mathematica
    Select[Table[Prime[n]+2n,{n,68000}],PrimeQ] (* Vincenzo Librandi, Dec 09 2011 *)

A174009 Numbers n such that A174008(k)=n-th prime.

Original entry on oeis.org

1, 4, 5, 11, 13, 16, 18, 19, 20, 25, 33, 37, 38, 39, 40, 48, 52, 59, 60, 69, 72, 73, 76, 79, 84, 85, 86, 87, 96, 98, 104, 110, 117, 122, 135, 136, 140, 142, 145, 151, 153, 160, 162, 173, 179, 183, 186, 191, 192, 199, 200, 206, 214, 218, 221, 226, 232, 234, 239, 242
Offset: 1

Views

Author

Juri-Stepan Gerasimov, Mar 05 2010

Keywords

Comments

n-th prime in sequence A174008.

Examples

			a(1)=1 because A174008(1)=2=1st prime;
a(2)=4 because A174008(2)=7=4th prime;
a(3)=5 because A174008(3)=11=5th prime;
a(4)=11 because A174008(7)=31=11th prime.

Crossrefs

Cf. A000040, A001477, A078916, A163300, A174008.

Programs

  • Maple
    From R. J. Mathar, Apr 28 2010: (Start)
A163300 := proc(n) option remember ; if n = 1 then 0; else for a from procname(n-1)+2 by 2 do if not isprime(a) then return a ; end if; end do; end if; end proc:
A174008 := proc(n) ithprime(n)+A163300(n) ; end proc:
A174009 := proc(k) p := A174008(k) ; if isprime(p) then printf("%d,", numtheory[pi](p) ) ; end if; return ; end proc:
seq(A174009(k),k=1..400 ) ; (End)

Extensions

More terms from R. J. Mathar, Apr 28 2010

A174010 Primes p of the form p = A000040(k) - A163300(k) for some k (includes duplicates).

Original entry on oeis.org

2, 3, 3, 5, 13, 17, 29, 31, 31, 37, 41, 47, 53, 67, 71, 71, 79, 79, 83, 89, 97, 97, 107, 107, 127, 131, 151, 181, 197, 211, 229, 241, 257, 257, 269, 271, 281, 283, 283, 311, 353, 373, 389, 401, 409, 409, 419, 419, 431, 449, 463, 479, 491, 499, 547, 563, 577, 577
Offset: 1

Views

Author

Juri-Stepan Gerasimov, Mar 05 2010

Keywords

Comments

Primes of form k-th prime minus k-th even nonnegative nonprime.
Essentially the same as A144419.

Examples

			a(1)=2 because 2-0=2; a(2)=3 because 17-14=3; a(3)=3 because 19-16=3; a(4)=5 because 23-18=5; a(5)=13 because 37-24=13.

Crossrefs

Cf. A144419, A001477, A078916, A163300, A174008.

Programs

  • Maple
    A163300 := proc(n) if n <= 2 then op(n,[0,4]) ; else for a from procname(n-1)+2 by 2 do if not isprime(a) then return a; end if; end do; end if; end proc:
for n from 1 to 400 do p := ithprime(n) -A163300(n) ; if isprime(p) then printf("%d,",p) ; end if; end do: # R. J. Mathar, May 02 2010

Extensions

Corrected (83 inserted) by R. J. Mathar, May 02 2010

A338467 a(n+1) = prime(n) + 2*n - a(n). a(1) = 1.

Original entry on oeis.org

1, 3, 4, 7, 8, 13, 12, 19, 16, 25, 24, 29, 32, 35, 36, 41, 44, 49, 48, 57, 54, 61, 62, 67, 70, 77, 76, 81, 82, 85, 88, 101, 94, 109, 98, 121, 102, 129, 110, 135, 118, 143, 122, 155, 126, 161, 130, 175, 144, 181, 148, 187, 156, 191, 168, 199, 176, 207, 180, 215
Offset: 1

Views

Author

Carole Dubois, Jan 31 2021

Keywords

Examples

			a(1) + a(2) - 2*1 = 1st prime; 1 + 3 - 2*1 = 2.
a(13) + a(14) - 2*13 = 13th prime; 32 + 35 - 2*13 = 41.

Crossrefs

Cf. A000040, A001223, A036467, A078916.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n=1, 1,
      ithprime(n-1)-a(n-1)+2*n-2)
    end:
seq(a(n), n=1..60);  # Alois P. Heinz, Jan 31 2021
  • Mathematica
    a[1] = 1; a[n_] := a[n] = Prime[n - 1] + 2*(n - 1) - a[n - 1]; Array[a, 60] (* Amiram Eldar, Feb 01 2021 *)
  • PARI
    a(n) = if (n==1, 1, prime(n-1) + 2*(n-1) - a(n-1)); \\ Michel Marcus, Jan 31 2021
  • Python
    from sympy import prime
S=[1]
nomb=100
for n in range(1,nomb):
    derterm=S[-1]
    terme= prime(n)-derterm+2*(len(S))
    S.append(terme)
print(S)

Formula

a(n+1) = A078916(n) - a(n). - Michel Marcus, Jan 31 2021
Showing 1-6 of 6 results.

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