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.

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A104275 Numbers k such that 2k-1 is not prime.

Original entry on oeis.org

1, 5, 8, 11, 13, 14, 17, 18, 20, 23, 25, 26, 28, 29, 32, 33, 35, 38, 39, 41, 43, 44, 46, 47, 48, 50, 53, 56, 58, 59, 60, 61, 62, 63, 65, 67, 68, 71, 72, 73, 74, 77, 78, 80, 81, 83, 85, 86, 88, 89, 92, 93, 94, 95, 98, 101, 102, 103, 104, 105, 107, 108, 109, 110, 111, 113
Offset: 1

Views

Author

Alexandre Wajnberg, Apr 17 2005

Keywords

Comments

Same as A053726 except for the first term of this sequence.
Numbers k such that A064216(k) is not prime. - Antti Karttunen, Apr 17 2015
Union of 1 and terms of the form (u+1)*(v+1) + u*v with 1 <= u <= v. - Ralf Steiner, Nov 17 2021

Examples

			a(1) = 1 because 2*1-1=1, not prime.
a(2) = 5 because 2*5-1=9, not prime (2, 3 and 4 give 3, 5 and 7 which are primes).
From _Vincenzo Librandi_, Jan 15 2013: (Start)
As a triangular array (apart from term 1):
   5;
   8,  13;
  11,  18,  25;
  14,  23,  32,  41;
  17,  28,  39,  50,  61;
  20,  33,  46,  59,  72,  85;
  23,  38,  53,  68,  83,  98, 113;
  26,  43,  60,  77,  94, 111, 128, 145;
  29,  48,  67,  86, 105, 124, 143, 162, 181;
  32,  53,  74,  95, 116, 137, 158, 179, 200, 221; etc.
which is obtained by (2*h*k + k + h + 1) with h >= k >= 1. (End)
The above array, which contains the same terms as A053726 but in different order and with some duplicates, has its own entry A144650. - _Antti Karttunen_, Apr 17 2015

Links

Crossrefs

Cf. A006254 (complement), A246371 (a subsequence).
Cf. A005097, A008508, A014076, A047845, A053726, A064216, A144650.

Programs

  • Magma
    [n: n in [1..220]| not IsPrime(2*n-1)]; // Vincenzo Librandi, Jan 28 2011
  • Maple
    remove(t -> isprime(2*t-1), [$1..1000]); # Robert Israel, Apr 17 2015
  • Mathematica
    Select[Range[115], !PrimeQ[2#-1] &] (* Robert G. Wilson v, Apr 18 2005 *)
  • PARI
    select( {is_A104275(n)=!isprime(n*2-1)}, [1..115]) \\ M. F. Hasler, Aug 02 2022
  • Python
    from sympy import isprime
def ok(n): return not isprime(2*n-1)
print(list(filter(ok, range(1, 114)))) # Michael S. Branicky, May 08 2021
  • Python
    from sympy import primepi
def A104275(n):
    if n <= 2: return ((n-1)<<2)+1
    m, k = n-1, (r:=primepi(n-1)) + n - 1 + (n-1>>1)
    while m != k:
        m, k = k, (r:=primepi(k)) + n - 1 + (k>>1)
    return r+n-1 # Chai Wah Wu, Aug 02 2024
  • SageMath
    [n for n in (1..250) if not is_prime(2*n-1)] # G. C. Greubel, Oct 17 2023
  • Scheme
    (define (A104275 n) (if (= 1 n) 1 (A053726 (- n 1)))) ;; More code in A053726. - Antti Karttunen, Apr 17 2015

Formula

a(n) = A047845(n-1) + 1.
For n > 1, a(n) = A053726(n-1) = n + A008508(n-1). - Antti Karttunen, Apr 17 2015
a(n) = (A014076(n)+1)/2. - Robert Israel, Apr 17 2015

Extensions

More terms from Robert G. Wilson v, Apr 18 2005

A053726 "Flag numbers": number of dots that can be arranged in successive rows of K, K-1, K, K-1, K, ..., K-1, K (assuming there is a total of L > 1 rows of size K > 1).

Original entry on oeis.org

5, 8, 11, 13, 14, 17, 18, 20, 23, 25, 26, 28, 29, 32, 33, 35, 38, 39, 41, 43, 44, 46, 47, 48, 50, 53, 56, 58, 59, 60, 61, 62, 63, 65, 67, 68, 71, 72, 73, 74, 77, 78, 80, 81, 83, 85, 86, 88, 89, 92, 93, 94, 95, 98, 101, 102, 103, 104, 105, 107, 108, 109, 110, 111, 113, 116
Offset: 1

Views

Author

Dan Asimov, asimovd(AT)aol.com, Apr 09 2003

Keywords

Comments

Numbers of the form F(K, L) = KL+(K-1)(L-1), K, L > 1, i.e. 2KL - (K+L) + 1, sorted and duplicates removed.
If K=1, L=1 were allowed, this would contain all positive integers.
Positive numbers > 1 but not of the form (odd primes plus one)/2. - Douglas Winston (douglas.winston(AT)srupc.com), Sep 11 2003
In other words, numbers n such that 2n-1, or equally, A064216(n) is a composite number. - Antti Karttunen, Apr 17 2015
Note: the following comment was originally applied in error to the numerically similar A246371. - Allan C. Wechsler, Aug 01 2022
From Matthijs Coster, Dec 22 2014: (Start)
Also area of (over 45 degree) rotated rectangles with sides > 1. The area of such rectangles is 2ab - a - b + 1 = 1/2((2a-1)(2b-1)+1).
Example: Here a = 3 and b = 5. The area = 23.
*
***
*****
*****
*****
***
*
(End)
The smallest integer > k/2 and coprime to k, where k is the n-th odd composite number. - Mike Jones, Jul 22 2024
Numbers k such that A193773(k-1) > 1. - Allan C. Wechsler, Oct 22 2024

Links

Crossrefs

Essentially same as A104275, but without the initial one.
A144650 sorted into ascending order, with duplicates removes.
Cf. A006254 (complement, apart from 1, which is in neither sequence).
Cf. also A000720, A008508, A064216, A071904, A199593, A250474.
Differs from its subsequence A246371 for the first time at a(8) = 20, which is missing from A246371.

Programs

  • PARI
    select( {is_A053726(n)=n>4 && !isprime(n*2-1)}, [1..115]) \\ M. F. Hasler, Aug 02 2022
  • Python
    from sympy import isprime
def ok(n): return n > 1 and not isprime(2*n-1)
print(list(filter(ok, range(1, 117)))) # Michael S. Branicky, May 08 2021
  • Python
    from sympy import primepi
def A053726(n):
    if n == 1: return 5
    m, k = n, (r:=primepi(n)) + n + (n>>1)
    while m != k:
        m, k = k, (r:=primepi(k)) + n + (k>>1)
    return r+n # Chai Wah Wu, Aug 02 2024
  • Scheme
    ;; with Antti Karttunen's IntSeq-library.
(define A053726 (MATCHING-POS 1 1 (lambda (n) (and (> n 1) (not (prime? (+ n n -1)))))))
;; Antti Karttunen, Apr 17 2015
  • Scheme
    ;; with Antti Karttunen's IntSeq-library.
(define (A053726 n) (+ n (A000720 (A071904 n))))
;; Antti Karttunen, Apr 17 2015

Formula

a(n) = A008508(n) + n + 1.
From Antti Karttunen, Apr 17 2015: (Start)
a(n) = n + A000720(A071904(n)). [The above formula reduces to this. A000720(k) gives number of primes <= k, and A071904 gives the n-th odd composite number.]
a(n) = A104275(n+1). (End)
a(n) = A116922(A071904(n)). - Mike Jones, Jul 22 2024
a(n) = A047845(n+1)+1. - Amiram Eldar, Jul 30 2024

Extensions

More terms from Douglas Winston (douglas.winston(AT)srupc.com), Sep 11 2003

A163301 a(n) = Sum_{x=n-th even nonprime..n-th odd nonprime} -x*(-1)^x.

Original entry on oeis.org

1, 3, 5, 7, 8, 8, 10, 10, 11, 13, 14, 14, 15, 15, 17, 17, 18, 20, 20, 21, 22, 22, 23, 23, 23, 24, 26, 28, 29, 29, 29, 29, 29, 29, 30, 31, 31, 33, 33, 33, 33, 35, 35, 36, 36, 37, 38, 38, 39, 39, 41, 41, 41, 41, 43, 45, 45, 45, 45, 45, 46, 46, 46, 46, 46, 47, 49, 50, 50, 52, 52
Offset: 1

Views

Author

Juri-Stepan Gerasimov, Jul 26 2009

Keywords

Comments

Here n-th even nonprime = A163300(n), n-th odd nonprime = A014076(n) and A163300 U A014076 = A141468.
Seems to be essentially the same as A008508. - R. J. Mathar, May 30 2025

Examples

			a(1) = -0*(-1)^0 - 1*(-1)^1 = 0 + 1 = 1;
a(2) = -4*(-1)^4 - 5*(-1)^5 - 6*(-1)6 - 7*(-1)^7 - 8*(-1)^8 - 9*(-1)^9 = -4 + 5 - 6 + 7 - 8 + 9 = 3.

Crossrefs

Cf. A014076, A141468, A163300.

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:
A014076 := proc(n) if n = 1 then 1; 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:
A001057 := proc(n) (1-(-1)^n*(2*n+1))/4; end proc:
A163301 := proc(n) A001057( A014076(n)) - A001057(A163300(n)-1) ; end proc: seq(A163301(n),n=1..120) ; # R. J. Mathar, May 21 2010

Formula

a(n) = Sum_{x=A163300(n)..A014076(n)}-x*(-1)^x.
a(n) = A001057( A014076(n)) - A001057(A163300(n)-1). - R. J. Mathar, May 21 2010

Extensions

Corrected from a(39) onwards by R. J. Mathar, May 21 2010
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