A104278 -id:A104278 - cp's OEIS frontend

A040040 Average of twin prime pairs (A014574), divided by 2. Equivalently, 2a(n)-1 and 2a(n)+1 are primes.

2, 3, 6, 9, 15, 21, 30, 36, 51, 54, 69, 75, 90, 96, 99, 114, 120, 135, 141, 156, 174, 210, 216, 231, 261, 285, 300, 309, 321, 330, 405, 411, 414, 429, 441, 510, 516, 525, 531, 546, 576, 615, 639, 645, 651, 660, 714, 726, 741, 744, 804, 810, 834, 849, 861, 894

Offset: 1

N. J. A. Sloane

Intersection of A005097 and A006254. - Zak Seidov, Mar 18 2005
The only possible pairs for 2a(n)+-1 are prime/prime (this sequence), not prime/not prime (A104278), prime/notprime (A104279) and not prime/prime (A104280), ... this sequence + A104280 + A104279 + A104278 = the odd numbers.
These numbers are never k mod (2k+1) or (k+1) mod (2k+1) with 2k+1 < a(n). - Jon Perry, Sep 04 2012
Excluding the first term, all remaining terms have digital root 3, 6 or 9. - J. W. Helkenberg, Jul 24 2013
Positive numbers x such that the difference between x^2 and adjacent squares are prime (both x^2-(x-1)^2 and (x+1)^2-x^2 are prime). - Doug Bell, Aug 21 2015

T. D. Noe, Table of n, a(n) for n = 1..10001

Cf. A001359, A006512, A014574, A054735, A111046, A045753 (even terms halved), A002822 (terms divided by 3).
Cf. A221310.
Cf. A260689, A264526.

a040040 = flip div 2 . a014574  -- Reinhard Zumkeller, Nov 17 2015

P := select(isprime,[$1..1789]): map(p->(p+1)/2, select(p->member(p+2,P),P)); # Peter Luschny, Mar 03 2011

Select[Range[900], And @@ PrimeQ[{-1, 1} + 2# ] &] (* Ray Chandler, Oct 12 2005 *)

p=2; forprime(b=3, 1e4, if(b-p==2, print1((p+1)/2", ")); p=b) \\ Altug Alkan, Nov 10 2015

a(n) = A014574(n)/2 = A054735(n+1)/4 = A111046(n+1)/8.
For n > 1, a(n) = 3*A002822(n-1). - Jason Kimberley, Nov 06 2015
A260689(a(n),1) = A264526(a(n)) = 1. - Reinhard Zumkeller, Nov 17 2015
From Michael G. Kaarhus, Aug 19 2022: (Start)
a(n) = (A001359(n) + 1)/2.
a(n) = (A006512(n) - 1)/2.
For n > 1, a(n) = A167379(n-1) * 3/2. (End)

More terms from Cino Hilliard, Oct 21 2002
Title corrected by Daniel Forgues, Jun 01 2009
Edited by Daniel Forgues, Jun 21 2009
Comment corrected by Daniel Forgues, Jul 12 2009

A099047 Numbers n such that n-1 and n+1 are both composite.

Original entry on oeis.org

5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 26, 27, 29, 31, 33, 34, 35, 37, 39, 41, 43, 45, 47, 49, 50, 51, 53, 55, 56, 57, 59, 61, 63, 64, 65, 67, 69, 71, 73, 75, 76, 77, 79, 81, 83, 85, 86, 87, 89, 91, 92, 93, 94, 95, 97, 99, 101, 103, 105, 107, 109, 111, 113, 115, 116, 117

Offset: 1

Rick L. Shepherd, Nov 13 2004

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A010051, A002808, A104278.

a099047 n = a099047_list !! (n-1)
a099047_list = [m | m <- [1..],
                    a010051' (m - 1) == 0 && a010051' (m + 1) == 0]
-- Reinhard Zumkeller, Aug 04 2015

Select[Range[6! ],!PrimeQ[ #-1]&&!PrimeQ[ #+1]&] (* Vladimir Joseph Stephan Orlovsky, Apr 26 2010 *)
Select[Range[120],AllTrue[#+{1,-1},CompositeQ]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, May 10 2018 *)
Mean/@SequencePosition[Table[If[CompositeQ[n],1,0],{n,120}],{1,,1}] (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale, Aug 23 2019 *)

for(n=5,200,if(!isprime(n-1)&&!isprime(n+1),print1(n,",")))

A309120 a(n) is the least k > 1 such that n*k is adjacent to a prime.

Original entry on oeis.org

2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 4, 2, 2, 2, 4, 2, 2, 2, 2, 2, 2, 2, 4, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 4, 2, 2, 3, 6, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 6, 3, 6, 5, 2, 2, 2, 2, 4, 2, 2, 2, 4, 5, 4, 2, 2, 2, 4, 2, 2, 3, 2, 2, 2, 2, 6, 2, 2, 3, 2, 2, 2, 3, 4, 3

Offset: 1

Robert Israel, Jul 17 2019

nonn

If n is odd then a(n) is even.

a(n) exists by Dirichlet's theorem on primes in arithmetic progressions.

			a(13)=4 because 4*13+1=53 is prime but none of 2*13-1,2*13+1,3*13-1,3*13+1 are primes.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A307833, A104278, A147820.

f:= proc(m) local k;
  for k from 2 by 1+(m mod 2) do
    if isprime(k*m-1) or isprime(k*m+1) then return k fi
  od
end proc:
map(f, [$1..100]);

a[n_]:=Module[{k=2},While[Not[PrimeQ[k*n-1]||PrimeQ[k*n+1]],k++];k];
a/@Range[94] (* Ivan N. Ianakiev, Jul 18 2019 *)

a(n) = my(k=2); while (!isprime(n*k+1) && !isprime(n*k-1), k++); k; \\ Michel Marcus, Jul 19 2019

a(A104278(n)) > 2 and a(A147820(n)) = 2. - Ivan N. Ianakiev, Jul 18 2019

A130699 Numbers n for which neither 2n-3 nor 2n+3 are primes.

Original entry on oeis.org

6, 9, 12, 15, 18, 21, 24, 26, 27, 30, 33, 36, 39, 42, 44, 45, 48, 51, 54, 57, 59, 60, 61, 63, 66, 69, 72, 75, 78, 79, 81, 84, 86, 87, 90, 93, 96, 99, 102, 103, 105, 106, 108, 109, 111, 114, 117, 120, 123, 125, 126, 128, 129, 131, 132, 135, 138, 141, 144, 146

Offset: 2

W. Neville Holmes, Jul 11 2007

			Not 5 because 7 and 13 are prime, but 6 because neither 9 nor 15 are primes.

Cf. A104278.

Select[Range[200],NoneTrue[2#+{3,-3},PrimeQ]&] (* The program uses the NoneTrue function from Mathematica version 10 *) (* Harvey P. Dale, Nov 12 2014 *)

isok(n) = !isprime(2*n-3) && !isprime(2*n+3) \\ Michel Marcus, Jul 11 2013

Missing term 24 added by Michel Marcus, Jul 11 2013

A296304 Numbers whose absolute difference from a square is never a prime.

Original entry on oeis.org

0, 169, 289, 625, 784, 1024, 1444, 1849, 2116, 2209, 3364, 3481, 3600, 3721, 3844, 4489, 5041, 5184, 5329, 5929, 6400, 7225, 7744, 8464, 8649, 8836, 10201, 10404, 10609, 10816, 11449, 11664, 11881, 12100, 13924, 14884, 15129, 15376, 16129, 16900, 17689, 18769

Offset: 1

Jon E. Schoenfield, Dec 10 2017

nonn

0 and the squares of numbers k such that 2k+1 and 2k-1 are not primes; i.e., 0 and the squares of the terms of A104278.

			The absolute difference between any square j^2 and 169 is |j^2 - 169| = |(j-13)*(j+13)| = |j-13|*|j+13|, which cannot be a prime unless one of the two factors |j-13| and |j+13| is 1, i.e., j is -14, -12, 12, or 14; however, in each case, the other factor is nonprime (-27, -25, 25, or 27, respectively), so |j^2 - 169| is not a prime for any integer j. Thus, 169 is in the sequence.
49 - 6^2 = 49 - 36 = 13 (a prime), so 49 is not in the sequence.

Muniru A Asiru, Table of n, a(n) for n = 1..20000

Cf. A104278.

Cf. A292990 (Numbers whose absolute difference from a triangular number is never a prime).

o := [];; for n in [1..10^4] do if not IsPrime(2*n-1) and not IsPrime(2*n+1) then Add(o,n^2); fi; od;
sequence := Concatenation([0],o); # Muniru A Asiru, Jan 01 2018

Join[{0}, Select[Range[200], CompositeQ[2# + 1] && CompositeQ[2# - 1]&]^2] (* Jean-François Alcover, Dec 21 2017 *)

a(1) = 0; for n > 1, A104278(n-1)^2.

A172462 Numbers k such that 2k-3, 2k-1, 2k+1 and 2k+3 are composite.

Original entry on oeis.org

59, 60, 61, 72, 93, 102, 103, 108, 109, 123, 144, 149, 150, 151, 161, 162, 163, 171, 207, 213, 236, 237, 257, 258, 264, 265, 266, 267, 268, 276, 291, 312, 313, 318, 333, 334, 348, 357, 389, 390, 391, 396, 401, 402, 408, 417, 422, 423, 424, 434, 435, 436, 446

Offset: 1

Juri-Stepan Gerasimov, Feb 03 2010

Almost all numbers are in this sequence, by the Prime Number Theorem.

			a(1)=59 because 2*59-1=117, 2*59+1=119, 2*59-3=115 and 2*59+3=121 are all composite.

Cf. A104278.

a := proc (n): if isprime(2*n-3) = false and isprime(2*n-1) = false and isprime(2*n+1) = false and isprime(2*n+3) = false then n else end if end proc: seq(a(n), n = 1 .. 500); # Emeric Deutsch, Feb 15 2010

Corrected and extended by Emeric Deutsch, Feb 15 2010

Comment from Charles R Greathouse IV, Mar 25 2010

cp's OEIS Frontend

A040040 Average of twin prime pairs (A014574), divided by 2. Equivalently, 2*a(n)-1 and 2*a(n)+1 are primes.

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A099047 Numbers n such that n-1 and n+1 are both composite.

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A309120 a(n) is the least k > 1 such that n*k is adjacent to a prime.

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Maple

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A130699 Numbers n for which neither 2n-3 nor 2n+3 are primes.

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Mathematica

PARI

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A296304 Numbers whose absolute difference from a square is never a prime.

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Programs

GAP

Mathematica

Formula

A172462 Numbers k such that 2k-3, 2k-1, 2k+1 and 2k+3 are composite.

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Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Extensions

A040040 Average of twin prime pairs (A014574), divided by 2. Equivalently, 2a(n)-1 and 2a(n)+1 are primes.