A014076 -id:A014076 - cp's OEIS frontend

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

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

Alexandre Wajnberg, Apr 17 2005

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

			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

Vincenzo Librandi (first 1000 terms) & Antti Karttunen, Table of n, a(n) for n = 1..10001

Cf. A006254 (complement), A246371 (a subsequence).

Cf. A005097, A008508, A014076, A047845, A053726, A064216, A144650.

[n: n in [1..220]| not IsPrime(2*n-1)]; // Vincenzo Librandi, Jan 28 2011

remove(t -> isprime(2*t-1), [$1..1000]); # Robert Israel, Apr 17 2015

Select[Range[115], !PrimeQ[2#-1] &] (* Robert G. Wilson v, Apr 18 2005 *)

select( {is_A104275(n)=!isprime(n*2-1)}, [1..115]) \\ M. F. Hasler, Aug 02 2022

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

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

[n for n in (1..250) if not is_prime(2*n-1)] # G. C. Greubel, Oct 17 2023

(define (A104275 n) (if (= 1 n) 1 (A053726 (- n 1)))) ;; More code in A053726. - Antti Karttunen, Apr 17 2015

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

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

A045751 Numbers k such that 4*k + 1 is not prime.

Original entry on oeis.org

0, 2, 5, 6, 8, 11, 12, 14, 16, 17, 19, 20, 21, 23, 26, 29, 30, 31, 32, 33, 35, 36, 38, 40, 41, 42, 44, 46, 47, 50, 51, 52, 53, 54, 55, 56, 59, 61, 62, 63, 65, 66, 68, 71, 72, 74, 75, 76, 77, 80, 81, 82, 83, 85, 86, 89, 90, 91, 92, 94, 95, 96, 98, 101, 103, 104, 106, 107, 109

Offset: 1

Felice Russo

Terms (except 0) can be written as 4xy +- (x+y) for x > 0, y > 0. - Ron R Spencer, Jul 28 2016

Numbers k such that (4*k)!/(4*k + 1) is an integer. - Peter Bala, Jan 25 2017

			Distribution of the positive terms in the following triangular array:
   2;
   *,   6;
   5,   *,  12;
   *,  11,   *,  20;
   8,   *,  19,   *,  30;
   *,  16,   *,  29,   *,  42;
  11,   *,  26,   *,  41,   *,  56;
   *,  21,   *,  38,   *,  55,   *,  72;
  14,   *,  33,   *,  52,   *,  71,   *,  90;
   *,  26,   *,  47,   *,  68,   *,  89,   *, 110;
  17,   *,  40,   *,  63,   *,  86,   *, 109,   *, 132;
etc., where * marks the noninteger values of (2*h*k + k + h)/2 with h >= k >= 1. - _Vincenzo Librandi_, Jan 14 2013

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

Cf. A014076, A153170, A095277, A153088, A153329, A153343.

[n: n in [0..220]| not IsPrime(4*n+1)]; // Vincenzo Librandi, Jan 28 2011

for n from 0 to 100 do
if irem(factorial(4*n), 4*n+1) = 0 then print(n); end if;
end do: # Peter Bala, Jan 25 2017

Select[Range[0, 200], ! PrimeQ[4 # + 1] &]

is(n)=!isprime(4*n+1) \\ Charles R Greathouse IV, Jul 29 2016

More terms from Erich Friedman

A095277 Numbers k such that 4k + 3 is composite.

Original entry on oeis.org

3, 6, 8, 9, 12, 13, 15, 18, 21, 22, 23, 24, 27, 28, 29, 30, 33, 35, 36, 38, 39, 42, 43, 45, 46, 48, 50, 51, 53, 54, 57, 58, 60, 61, 63, 64, 66, 68, 69, 71, 72, 73, 74, 75, 78, 79, 80, 81, 83, 84, 85, 87, 88, 90, 92, 93, 96, 97, 98, 99, 100, 101, 102, 103, 105, 106, 108

Offset: 1

Antti Karttunen, Jun 01 2004

Terms can be written as (4xy +- (x-y)) - 1 for x > 0, y > 0. - Ron R Spencer, Aug 01 2016

Numbers k such that (4*k)!/(4*k + 3) is an integer. - Peter Bala, Jan 25 2017

			Distribution of the positive terms in the following triangular array:
  *;
  3,  *;
  *,  8,  *;
  6,  *, 15,  *;
  *, 13,  *, 24,  *;
  9,  *, 22,  *, 35,  *;
  *, 18,  *, 33,  *, 48,  *;
etc., where * marks the noninteger values of (2*h*k + k + h-1)/2 with h >= k >= 1. - _Vincenzo Librandi_, Apr 22 2014

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

Complement of A095278. Cf. also A045751, A014076, A153170, A153088, A153329, A153343.

[n: n in [0..110] |not IsPrime(4*n+3)]; // Vincenzo Librandi, Apr 22 2014\

for n from 0 to 100 do
if irem(factorial(4*n), 4*n+3) = 0 then print(n); end if;
end do: # Peter Bala, Jan 25 2017

Select[Range[150],!PrimeQ[4#+3]&] (* Harvey P. Dale, Jul 04 2011 *)

is(n)=!isprime(4*n+3) \\ Charles R Greathouse IV, Aug 01 2016

a(n) = (A091236(n) - 3)/4.

A153088 Numbers k such that 5*k - 1 is not prime.

Original entry on oeis.org

1, 2, 3, 5, 7, 8, 9, 10, 11, 13, 14, 15, 17, 19, 20, 21, 23, 24, 25, 26, 27, 29, 31, 32, 33, 34, 35, 37, 38, 39, 41, 42, 43, 44, 45, 47, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 71, 73, 74, 75, 77, 79, 80, 81, 83, 85, 86, 87, 89, 91, 92, 93

Offset: 1

Vincenzo Librandi, Jan 03 2009

			Distribution of the even terms in the following triangular array:
   2;
   *,   *;
   *,   *,  10;
   *,   *,   *,   *;
   *,   *,   *,  20,   *;
   8,   *,   *,   *,   *,  34;
   *,   *,   *,   *,   *,   *,   *;
   *,   *,  24,   *,   *,   *,   *,  58;
   *,   *,   *,   *,  42,   *,   *,   *,   *;
   *,   *,   *,  38,   *,   *,   *,   *,  80,   *;
  14,   *,   *,   *,   *,  60,   *,   *,   *,   *, 106;
etc., where * marks the noninteger values of (4*h*k + 2*k + 2*h + 2)/5 with h >= k >= 1. - _Vincenzo Librandi_, Jan 15 2013

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

Cf. A153355, A014076, A153170, A045751, A095277, A153329, A153343.

[n: n in [1..100] | not IsPrime(5*n-1)]; // Vincenzo Librandi, Oct 11 2012

# produces the sequence apart from the initial terms 1 and 2
for n from 0 to 100 do
  if irem(factorial(5*n), 5*n+4) = 0 then print(n+1); end if;
end do: # Peter Bala, Jan 25 2017

Select[Range[1, 200], !PrimeQ[5 # - 1] &] (* Vincenzo Librandi, Oct 11 2012 *)

a(n) = A153343(n) + 1. - Peter Bala, Jan 25 2017

First 29 replaced with 20, 4 replaced with 44, extended by R. J. Mathar, Jan 05 2009

Erroneous comment deleted by N. J. A. Sloane, Jun 23 2010

A153170 Numbers k such that 3*k + 2 is not prime.

Original entry on oeis.org

2, 4, 6, 8, 10, 11, 12, 14, 16, 18, 20, 21, 22, 24, 25, 26, 28, 30, 31, 32, 34, 36, 38, 39, 40, 41, 42, 44, 46, 47, 48, 50, 51, 52, 53, 54, 56, 58, 60, 61, 62, 64, 66, 67, 68, 69, 70, 71, 72, 73, 74, 76, 78, 80, 81, 82, 84, 86, 88, 90, 91, 92, 94, 95, 96, 98, 99, 100, 101, 102

Offset: 1

Vincenzo Librandi, Dec 20 2008

Contains the positive even numbers (A005843) and the odd numbers of the form 2*A059324(.) + 1. - R. J. Mathar, Nov 27 2010

Numbers k such that (3*k)!/(3*k + 2) is an integer. - Peter Bala, Jan 25 2017

			Distribution of the odd terms in the following triangular array:
  *;
  *,   *;
  *,  11,   *;
  *,   *,   *,   *;
  *,   *,  25,   *,   *;
  *,  21,   *,   *,  47,   *;
  *,   *,   *,   *,   *,   *,   *;
  *,   *,  39,   *,   *,  73,   *,   *;
  *,  31,   *,   *,  69,   *,   *, 107,   *;
  *,   *,   *,   *,   *,   *,   *,   *,   *,   *;
  *,   *,  53,   *,   *,  99,   *,   *, 145,   *,   *;
  *,  41,   *,   *,  91,   *,   *, 141,   *,   *, 191,   *;
etc., where * marks the noninteger values of (4*h*k + 2*k + 2*h - 1)/3 with h >= k >= 1. - _Vincenzo Librandi_, Jan 15 2013

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

Cf. A024893, A014076, A045751, A095277, A153088, A153329, A153343.

[n: n in [1..110] | not IsPrime(3*n + 2)]; // Vincenzo Librandi, Oct 11 2012

for n from 0 to 100 do
if irem(factorial(3*n), 3*n+2) = 0 then print(n); end if;
end do: # Peter Bala, Jan 25 2017

Select[Range[1, 200], !PrimeQ[3*# + 2] &] (* Vincenzo Librandi, Oct 11 2012 *)

for(n=1,200,if(!isprime(3*n+2), print1(n,", "))) \\  Joerg Arndt, Nov 27 2010

Edited by N. J. A. Sloane, Jun 23 2010

A203069 Lexicographically earliest sequence of distinct positive numbers such that a(n-1)+a(n) is odd and composite.

Original entry on oeis.org

1, 8, 7, 2, 13, 12, 3, 6, 9, 16, 5, 4, 11, 10, 15, 18, 17, 22, 23, 26, 19, 14, 21, 24, 25, 20, 29, 28, 27, 30, 33, 32, 31, 34, 35, 40, 37, 38, 39, 36, 41, 44, 43, 42, 45, 46, 47, 48, 51, 54, 57, 58, 53, 52, 59, 56, 49, 50, 55, 60, 61, 62, 63, 66, 67, 68, 65

Offset: 1

Zak Seidov, Dec 28 2011

nonn

Inspired by an idea of Eric Angelini on the Sequence Fans list on Dec 28 2011.
Comments from N. J. A. Sloane, Aug 16 2021: (Start)
It is conjectured that this is a permutation of the positive integers. Is there a proof? The terms are distinct, by definition, and the sequence is clearly infinite. But does every number appear?
In the first 100000 terms, the only differences a(i)-a(i-1) that occur are -11, -9, -7, -5, -3, -1, 1, 3, 5, 7, 9, 11 (see A346610).
Also a(n) is surprisingly close to n - see A346611. (End)

			a(1)=1; the smallest possible even number m such that 1+m is composite is m=8, hence a(2)=8;
the smallest possible odd number m such that 8+m is composite is m=7, hence a(3)=7;
the smallest possible even number m such that 7+m is composite is m=2, hence a(4)=2.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Éric Angelini, A December quiz with a non-prime array, SeqFan list, Dec 28 2011.
N. J. A. Sloane, Table of n, a(n) for n = 1..100000
N. J. A. Sloane, Maple program
Index entries for sequences that are permutations of the natural numbers

Cf. A010051, A249918 (inverse), A014076, A055266, A346610 (first differences), A346611.

See A346609 for the successive odd nonprimes that arise.

import Data.List (delete)
a203069 n = a203069_list !! (n-1)
a203069_list = 1 : f 1 [2..] where
   f u vs = g vs where
     g (w:ws) | odd z && a010051' z == 0 = w : f w (delete w vs)
              | otherwise = g ws
              where z = u + w
-- Reinhard Zumkeller, Jan 14 2015

Maple
```
(See link)
```

Clear[used];used={1};oc[n_]:=Module[{k=If[OddQ[n],2,1]},While[ !CompositeQ[ n+k]||MemberQ[used,k],k+=2];Flatten[AppendTo[used,k]];k] (* Harvey P. Dale, Aug 16 2021 *)

@cached_function
def A203069(n):
    if n == 1: return 1
    used = set(A203069(i) for i in [1..n-1])
    works = lambda an: (A203069(n-1)+an) % 2 == 1 and len(divisors((A203069(n-1)+an))) > 2
    return next(k for k in PositiveIntegers() if k not in used and works(k)) # D. S. McNeil, Dec 28 2011

Revised by N. J. A. Sloane, Aug 15 2021 at the suggestion of Harvey P. Dale.

A056653 Composite numbers together with 1 but excluding 4.

Original entry on oeis.org

1, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 64, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 81, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95

Offset: 1

Robert G. Wilson v, Aug 30 2000

These are also the numbers k such that k divides (k-1)!.

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

Cf. A018252, A002808, A226198, A014076 (odd terms).

for n from 1 to 100 do
  if irem(factorial(n-1),n) = 0 then print(n) end if;
end do: # Peter Bala, Jan 24 2017

Select[ Range[ 1, 100 ], Mod[ (# - 1)!, # ] == 0 & ]
Join[{1},Select[Range[5,100],CompositeQ]] (* Harvey P. Dale, Jun 14 2024 *)

from sympy import composite
def A056653(n): return composite(n) if n>1 else 1 # Chai Wah Wu, Jul 31 2024

Edited by Vladeta Jovovic, Apr 30 2003

A153329 Numbers k such that 5*k + 1 is not prime.

Original entry on oeis.org

0, 1, 3, 4, 5, 7, 9, 10, 11, 13, 15, 16, 17, 18, 19, 21, 22, 23, 24, 25, 27, 28, 29, 31, 32, 33, 34, 35, 37, 39, 40, 41, 43, 44, 45, 46, 47, 49, 51, 52, 53, 55, 57, 58, 59, 60, 61, 63, 64, 65, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 81, 82, 83, 85, 87

Offset: 1

Vincenzo Librandi, Dec 23 2008

Numbers k such that (5*k)!/(5*k + 1) is an integer. - Peter Bala, Jan 25 2017

			Distribution of the even terms in the following triangular array:
   *;
   *,  *;
   4,  *,  *;
   *,  *,  *, 16;
   *,  *,  *,  *, 24;
   *,  *, 18,  *,  *,  *;
   *,  *,  *,  *,  *,  *,  *;
  10,  *,  *,  *,  *, 44,  *,  *;
   *,  *,  *, 34,  *,  *,  *,  *, 72;
   *,  *,  *,  *, 46,  *,  *,  *,  *, 88;
   *,  *, 32,  *,  *,  *,  *, 78,  *,  *,  *;
etc., where * marks the noninteger values of (4*h*k + 2*k + 2*h)/5 with h >= k >= 1. - _Vincenzo Librandi_, Jan 17 2013

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

Cf. A024894, A014076, A153170, A045751, A095277, A153088, A153343.

[n: n in [0..150] | not IsPrime(5*n + 1)]; // Vincenzo Librandi, Jan 12 2013

for n from 0 to 100 do
if irem(factorial(5*n), 5*n+1) = 0 then print(n); end if;
end do: # Peter Bala, Jan 25 2017

Select[Range[0, 200], !PrimeQ[5*# + 1]&] (* Vincenzo Librandi, Jan 12 2013 *)

Erroneous comment deleted by N. J. A. Sloane, Jun 23 2010

0 added by Arkadiusz Wesolowski, Aug 03 2011

A153343 Numbers k such that 5*k + 4 is not prime.

Original entry on oeis.org

0, 1, 2, 4, 6, 7, 8, 9, 10, 12, 13, 14, 16, 18, 19, 20, 22, 23, 24, 25, 26, 28, 30, 31, 32, 33, 34, 36, 37, 38, 40, 41, 42, 43, 44, 46, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 70, 72, 73, 74, 76, 78, 79, 80, 82, 84, 85, 86, 88

Offset: 1

Vincenzo Librandi, Dec 24 2008

Apart from a(0) = 0 and a(1) = 1 this sequence comprises those numbers k such that (5*k)!/(5*k + 4) is an integer. - Peter Bala, Jan 25 2017

			Distribution of the odd terms in the following triangular array:
   1;
   *,   *;
   *,   *,   9;
   *,   *,   *,   *;
   *,   *,   *,  19,   *;
   7,   *,   *,   *,   *,  33;
   *,   *,   *,   *,   *,   *,   *;
   *,   *,  23,   *,   *,   *,   *,  57;
   *,   *,   *,   *,  41,   *,   *,   *,   *;
   *,   *,   *,  37,   *,   *,   *,   *,  79,   *;
  13,   *,   *,   *,   *,  59,   *,   *,   *,   *,  105;
etc., where * marks the noninteger values of (4*h*k + 2*k + 2*h - 3)/5 with h >= k >= 1. - _Vincenzo Librandi_, Jan 17 2013

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

Cf. A024897, A014076, A153170, A045751, A095277, A153088, A153343.

[n: n in [0..150] | not IsPrime(5*n + 4)]; // Vincenzo Librandi, Jan 12 2013

# produces the sequence apart from the initial terms 0 and 1
for n from 0 to 100 do
if irem(factorial(5*n), 5*n+4) = 0 then print(n); end if;
end do: # Peter Bala, Jan 25 2017

Select[Range[0, 200], !PrimeQ[5*# + 4]&] (* Vincenzo Librandi, Jan 12 2013 *)

Erroneous comment deleted by N. J. A. Sloane, Jun 23 2010

0 added by Arkadiusz Wesolowski, Aug 03 2011

A093183 Number of consecutive runs of just 1 odd nonprime congruent to 1 mod 4 below 10^n.

Original entry on oeis.org

0, 3, 74, 1114, 13437, 151311, 1642197, 17405273, 181925434, 1883327626, 19364371468, 198115934511, 2019328584101

Offset: 1

Enoch Haga, Mar 30 2004

Split the odd nonprime sequence A014076 into two subsequences A091113 and A091236 with nonprimes labeled 1 mod 4 or 3 mod 4. Add count of nonprimes to sequence if just 1 nonprime congruent to 1 mod 4 occurs before interruption of a nonprime congruent to 3 mod 4.

Otherwise said: count the nonprimes congruent to 1 mod 4 such that the next larger and next smaller odd nonprime is congruent to 3 mod 4. - M. F. Hasler, Sep 30 2018

			a(3) = 74 because 74 single nonprime runs occur below 10^3, each run interrupted by a nonprime congruent to 3 mod 4.
Below 10^2 = 100, there are only a(2) = 3 isolated odd nonprimes congruent to 1 mod 4: 33, 57 and 93. (Credits: _Peter Munn_, SeqFan list.) - _M. F. Hasler_, Sep 30 2018

Cf. A091113, A091236, A093184-A093188, A093397-A093399, A092636.

A014076 := proc(n)
    option remember;
    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:
isA091113 := proc(n)
    option remember;
    if modp(n,4) = 1 and not isprime(n) then
        true;
    else
        false;
    end if;
end proc:
isA091236 := proc(n)
    option remember;
    if modp(n,4) = 3 and not isprime(n) then
        true;
    else
        false;
    end if;
end proc:
ct := 0 :
n := 1 :
for i from 2 do
    odnpr := A014076(i) ;
    prev := A014076(i-1) ;
    nxt := A014076(i+1) ;
    if isA091113(odnpr) and isA091236(prev) and isA091236(nxt) then
        ct := ct+1 ;
    end if;
    if odnpr< 10^n and nxt >= 10^n then
        print(n,ct) ;
        n := n+1 ;
    end if;
end do: # R. J. Mathar, Oct 02 2018

A091113 = Select[4 Range[0, 10^5] + 1, ! PrimeQ[#] &];
A091236 = Select[4 Range[0, 10^5] + 3, ! PrimeQ[#] &];
lst = {}; Do[If[Length[s = Select[A091113,Between[{A091236[[i]], A091236[[i + 1]]}]]] == 1, AppendTo[lst, s]], {i, Length[A091236] - 1}]; Table[Count[Flatten[lst], x_ /; x < 10^n], {n, 5}]  (* Robert Price, May 30 2019 *)

a(9)-a(13) from Bert Dobbelaere, Dec 19 2018

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

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A045751 Numbers k such that 4*k + 1 is not prime.

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A095277 Numbers k such that 4k + 3 is composite.

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A153088 Numbers k such that 5*k - 1 is not prime.

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A153170 Numbers k such that 3*k + 2 is not prime.

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A203069 Lexicographically earliest sequence of distinct positive numbers such that a(n-1)+a(n) is odd and composite.

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