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 1-3 of 3 results.

A001747 2 together with primes multiplied by 2.

Original entry on oeis.org

2, 4, 6, 10, 14, 22, 26, 34, 38, 46, 58, 62, 74, 82, 86, 94, 106, 118, 122, 134, 142, 146, 158, 166, 178, 194, 202, 206, 214, 218, 226, 254, 262, 274, 278, 298, 302, 314, 326, 334, 346, 358, 362, 382, 386, 394, 398, 422, 446, 454, 458, 466, 478, 482, 502
Offset: 1

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Author

N. J. A. Sloane

Keywords

Comments

When supplemented with 8, may be considered the "even primes", since these are the even numbers n = 2k which are divisible just by 1, 2, k and 2k. - Louis Zuckerman (louis(AT)trapezoid.com), Sep 12 2000
Sequence gives solutions of sigma(n) - phi(n) = n + tau(n) where tau(n) is the number of divisors of n.
Numbers n such that sigma(n) = 3*(n - phi(n)).
Except for 2, orders of non-cyclic groups k (in A060679(n)) such that x^k==1 (mod k) has only 1 solution 2<=x<=k. - Benoit Cloitre, May 10 2002
Numbers n such that A092673(n) = 2. - Jon Perry, Mar 02 2004
Except for initial terms, this sequence = A073582 = A074845 = A077017. Starting with the term 10, they are identical. - Robert G. Wilson v, Jun 15 2004
Together with 8 and 16, even numbers n such that n^2 does not divide (n/2)!. - Arkadiusz Wesolowski, Jul 16 2011
Twice noncomposite numbers. - Omar E. Pol, Jan 30 2012

Links

Crossrefs

Cf. A060679, A009530, A098764.
Equals {2} UNION {A100484}.

Programs

  • GAP
    Concatenation([2], List([1..60], n-> 2*Primes[n])); # G. C. Greubel, May 18 2019
  • Magma
    [2] cat [2*NthPrime(n): n in [1..60]]; // G. C. Greubel, May 18 2019
  • Mathematica
    Join[{2},2*Prime[Range[60]]] (* Harvey P. Dale, Jul 23 2013 *)
  • PARI
    print1(2);forprime(p=2,97,print1(", "2*p)) \\ Charles R Greathouse IV, Jan 31 2012
  • Sage
    [2]+[2*nth_prime(n) for n in (1..60)] # G. C. Greubel, May 18 2019

Formula

a(n) = A001043(n) - A001223(n+1), except for initial term.
a(n) = A116366(n-2,n-2) for n>2. - Reinhard Zumkeller, Feb 06 2006
A006093(n) = A143201(a(n+1)) for n>1. - Reinhard Zumkeller, Aug 12 2008
a(n) = 2*A008578(n). - Omar E. Pol, Jan 30 2012, and Reinhard Zumkeller, Feb 16 2012

A062234 From Bertrand's postulate: a(n) = 2*prime(n) - prime(n+1).

Original entry on oeis.org

1, 1, 3, 3, 9, 9, 15, 15, 17, 27, 25, 33, 39, 39, 41, 47, 57, 55, 63, 69, 67, 75, 77, 81, 93, 99, 99, 105, 105, 99, 123, 125, 135, 129, 147, 145, 151, 159, 161, 167, 177, 171, 189, 189, 195, 187, 199, 219, 225, 225, 227, 237, 231, 245, 251, 257, 267, 265, 273, 279
Offset: 1

Views

Author

Reinhard Zumkeller, Jun 29 2001

Keywords

Comments

The theorem that a(n) > 0 for all n is known as "Bertrand's Postulate", and was proved by Tchebycheff in 1852.
The analog for Ramanujan primes is Paksoy's theorem that 2*R(n) - R(n+1) > 0 for n > 1. See A233822. - Jonathan Sondow, Dec 16 2013

References

  • J. V. Uspensky and M. A. Heaslet, Elementary Number Theory, McGraw-Hill, NY, 1939.

Links

Crossrefs

Cf. A000040, A001223, A215808 (prime terms), A233822.
Cf. A098764, A100484, A258383 (run lengths), A258432, A258469, A257762, A258449, A257892, A257951.
When negated, forms the left edge of irregular triangle A252750, and also the leftmost column of square array A372562.

Programs

  • Haskell
    a062234 n = a062234_list !! (n-1)
a062234_list = zipWith (-) (map (* 2) a000040_list) (tail a000040_list)
-- Reinhard Zumkeller, May 31 2015
  • Maple
    a:= n-> (p-> 2*p(n)-p(n+1))(ithprime):
seq(a(n), n=1..60);  # Alois P. Heinz, Feb 09 2022
  • Mathematica
    Table[2*Prime[n]-Prime[n+1],{n,60}] (* James C. McMahon, Apr 27 2024 *)
2#[[1]]-#[[2]]&/@Partition[Prime[Range[70]],2,1] (* Harvey P. Dale, Jul 29 2024 *)
ListConvolve[{-1, 2}, Prime[Range[100]]] (* Paolo Xausa, Nov 02 2024 *)
  • PARI
    a(n) = 2*prime(n) - prime(n + 1); \\ Harry J. Smith, Aug 03 2009

Formula

a(n) = A000040(n) - A001223(n). - Zak Seidov, Sep 07 2012
a(n) = 2*A000040(n) - A000040(n+1). - Zak Seidov, May 12 2020
a(n) = A098764(n) - A000040(n). - Anthony S. Wright, Feb 19 2024

Extensions

Edited by N. J. A. Sloane, Feb 24 2023

A100021 a(n) = 3*prime(n) - prime(n+1) - 3.

Original entry on oeis.org

0, 1, 5, 7, 17, 19, 29, 31, 37, 53, 53, 67, 77, 79, 85, 97, 113, 113, 127, 137, 137, 151, 157, 167, 187, 197, 199, 209, 211, 209, 247, 253, 269, 265, 293, 293, 305, 319, 325, 337, 353, 349, 377, 379, 389, 383, 407, 439, 449, 451, 457, 473, 469, 493, 505, 517, 533, 533, 547
Offset: 1

Views

Author

Giovanni Teofilatto, Nov 20 2004

Keywords

Comments

Except for the initial term, a(n)==(1,5) mod 6.

Links

Crossrefs

Cf. A099389 for the primes arising in the sequence, A098764.

Programs

  • Maple
    f:=n->3*ithprime(n)-ithprime(n+1)-3;
  • Mathematica
    3#[[1]]-#[[2]]-3&/@Partition[Prime[Range[80]],2,1] (* Harvey P. Dale, Oct 13 2018 *)
  • PARI
    a(n) = 3*prime(n) - prime(n+1) - 3; \\ Michel Marcus, Jan 31 2023

Formula

a(n) = A098764(n) - 3.

Extensions

Corrected by Neven Juric (neven.juric(AT)apis-it.hr), rechecked by N. J. A. Sloane, Jun 13 2006
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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