A023200 -id:A023200 - cp's OEIS frontend

A049491 Numbers k such that k and k+128 are both prime.

3, 11, 23, 29, 53, 71, 83, 101, 113, 149, 179, 239, 251, 269, 281, 293, 311, 359, 419, 443, 449, 479, 491, 503, 563, 599, 641, 659, 683, 701, 809, 839, 863, 881, 911, 941, 1103, 1109, 1151, 1163, 1193, 1301, 1319, 1361, 1439, 1451, 1481, 1493, 1499, 1571

Offset: 1

Labos Elemer

			11 and 11+128 = 139 are both prime.

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

Cf. A001359, A023200, A023202.

[n: n in [0..2000] | IsPrime(n) and IsPrime(n+128)]; // Vincenzo Librandi, Feb 02 2014

Select[Prime[Range[300]],PrimeQ[#+128]&] (* Harvey P. Dale, Jan 16 2011 *)

A054902 Composite numbers n such that sigma(n)+12 = sigma(n+12).

Original entry on oeis.org

65, 170, 209, 1394, 3393, 4407, 4556, 11009, 13736, 27674, 38009, 38845, 47402, 76994, 157994, 162393, 184740, 186686, 209294, 680609, 825359, 954521, 1243574, 2205209, 3515609, 4347209, 5968502, 6539102, 6916241, 8165294, 10352294, 10595009, 10786814

Offset: 1

Labos Elemer, May 23 2000

nonn

			n = 65, sigma(65)+12 = 84+12 = 96 = sigma(65+12) = sigma(77).
n = 11009, sigma(11009)+12 = 11220+12 = 11232 = sigma(11009+12) = sigma(11021).

Donovan Johnson, Table of n, a(n) for n = 1..500

Complement of A046133 with respect to A015917.

Cf. A015913-A015917, A023200, A023203, A046133, A001359, A054799.

isok(n) = !isprime(n) && ((sigma(n)+12) == sigma(n+12)); \\ Michel Marcus, Dec 18 2013

More terms from Jud McCranie, May 24 2000

Three more terms from Michel Marcus, Dec 18 2013

A054904 x = a(n) is the smallest composite number such that sigma(x+6n) = sigma(x)+6n, where sigma = A000203.

Original entry on oeis.org

104, 65, 20, 80, 44, 125, 45, 63, 40, 99, 56, 70, 296, 125, 88, 110, 104, 145, 212, 182, 80, 170, 333, 105, 369, 185, 184, 135, 180, 301, 356, 185, 1859, 329, 176, 195, 4916, 434, 612, 287, 140, 185, 776, 255, 524, 413, 344, 205, 272, 329, 567, 215, 320, 469

Offset: 1

Labos Elemer, May 23 2000

nonn

If sigma(x+d) = sigma(x)+d and d = 6k, then composite solutions seem to be more frequent and arise sooner.

a(725) > 3*10^11 (if it exists). - Donovan Johnson, Sep 23 2013

			n = 20, 6n = 120, a(20) = 182, sigma(182)+120 = 336+120 = 456 = sigma(182+120) = sigma(302).

Donovan Johnson, Table of n, a(n) for n = 1..724

Cf. A015914-A015917, A023200-A023203, A054799, A054903-A054905.

Table[x = 4; While[Nand[CompositeQ@ x, DivisorSigma[1, x + 6 n] == DivisorSigma[1, x] + 6 n], x++]; x, {n, 54}] (* Michael De Vlieger, Feb 18 2017 *)

/* finds first 696 terms */ mx=7695851; s=vector(mx); for(j=4, mx, if(isprime(j)==0, s[j]=sigma(j))); for(n=1, 696, n6=n*6; for(x=4, 7691753, if(s[x]>0, if(s[x+n6]==s[x]+n6, write("b054904.txt", n " " x); next(2))))) /* Donovan Johnson, Sep 23 2013 */

sigma(x+6n) = sigma(x)+6n, a(n) = min(x) and it is composite.

A067775 Primes p such that p + 4 is composite.

Original entry on oeis.org

2, 5, 11, 17, 23, 29, 31, 41, 47, 53, 59, 61, 71, 73, 83, 89, 101, 107, 113, 131, 137, 139, 149, 151, 157, 167, 173, 179, 181, 191, 197, 199, 211, 227, 233, 239, 241, 251, 257, 263, 269, 271, 281, 283, 293, 311, 317, 331, 337, 347, 353, 359, 367, 373, 383, 389

Offset: 1

Benoit Cloitre, Feb 06 2002

nonn

Primes n such that n!*B(n+3) is an integer where B(k) are the Bernoulli numbers B(1) = -1/2, B(2) = 1/6, B(4) = -1/30, ..., B(2m+1) = 0 for m > 1.
If n is prime n!*B(n-1) is always an integer. Note that if Goldbach's conjecture (2n = p1 + p2 for all n >= 2) is false and K is the smallest value of n for which it fails, then for 2(K-2) = p3 + p4, the primes p3 and p4 must be taken from this list. See similar comment for A140555. - Keith Backman, Apr 06 2012
Complement of A023200 (primes p such that p + 4 is also prime) with respect to A000040 (primes). For p > 2: primes p such that there is no prime of the form r^2 + p where r is prime, subsequence of A232010. Example: the prime 7 is not in the sequence because 2^2 + 7 = 11 (prime). A232009(a(n)) = 0 for n > 1 . - Jaroslav Krizek, Nov 22 2013

Klaus Brockhaus, Table of n, a(n) for n = 1..1026
Eric Weisstein's World of Mathematics, Bernoulli Number

Cf. A049591, A140555, A232009, A232010, A232012, A023200.

A067775 = {}; Do[p = Prime@ n; If[ IntegerQ[ p! BernoulliB[p + 3]], AppendTo[A067775, p]], {n, 77}]; A067775 (* Robert G. Wilson v, Aug 19 2008 *)
Select[Prime[Range[80]], Not[PrimeQ[# + 4]] &] (* Alonso del Arte, Apr 02 2014 *)

lista(nn) = {forprime(p=1, nn, if (! isprime(p+4), print1(p, ", ")););} \\ Michel Marcus, Nov 22 2013

a(n) ~ n log n. - Charles R Greathouse IV, Nov 22 2013

New name from Klaus Brockhaus at the suggestion of Michel Marcus, Nov 22 2013

A157834 Numbers n such that 3n-2 and 3n+2 are both prime.

Original entry on oeis.org

3, 5, 7, 13, 15, 23, 27, 33, 35, 37, 43, 55, 65, 75, 77, 93, 103, 105, 117, 127, 133, 147, 153, 155, 163, 167, 205, 215, 225, 247, 253, 257, 275, 285, 287, 293, 295, 303, 313, 323, 337, 363, 365, 405, 427, 433, 435, 475, 477, 483, 495, 497, 517

Offset: 1

Kyle D. Balliet, Mar 07 2009

nonn

Barycenter of cousin primes (A029708; see also A029710, A023200, A046132), divided by 3. When p>3 and p+4 both are prime, then p = 1 (mod 6) and p+2 = 3 (mod 6). - M. F. Hasler, Jan 14 2013

			15*3 +/- 2 = 43,47 (both prime).

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

Intersection of A024893 and A153183.

[n: n in [1..1000]|IsPrime(3*n-2)and IsPrime(3*n+2)] // Vincenzo Librandi, Dec 13 2010

select(t -> isprime(3*t+2) and isprime(3*t-2), [seq(t,t=3..1000,2)]); # Robert Israel, May 28 2017

Select[Range[600],AllTrue[3#+{2,-2},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Feb 03 2019 *)

Intersection of A024893 and A153183.
a(n) = A029708(n)/3. - Zak Seidov, Aug 07 2009
a(n) = A056956(n)*2+1 = (A029710(n)+2)/3 = (A023200(n+1)+2)/3. - M. F. Hasler, Jan 14 2013

A176130 Lesser of a pair (p,p+4) of cousin primes whose arithmetic mean p+2 is a square number.

Original entry on oeis.org

7, 79, 223, 439, 1087, 13687, 56167, 74527, 91807, 95479, 149767, 184039, 194479, 199807, 263167, 314719, 328327, 370879, 651247, 804607, 1071223, 1147039, 1238767, 1306447, 1520287, 1535119, 1718719, 2442967, 2595319, 2614687

Offset: 1

Ulrich Krug (leuchtfeuer37(AT)gmx.de), Apr 09 2010

nonn

Necessarily p = 9 * (2*m - 1)^2 - 2.

			(7 + 11)/2 = 3^2, 1st term is prime(4) = 7.
(79 + 83)/2 = 9^2, 2nd term is prime(22) = 79.
m = 173 = prime(40): 21st term is p = 1071223 = prime(83637), p+2 = 3^4 * 5^2 * 23^2.
60th term is p = 27029599 = prime(1684797): p+2 = 3^2 * 1733^2.

L. E. Dickson, History of the Theory of numbers, vol. 2: Diophantine Analysis, Dover Publications 2005.
H. Pieper, Zahlen aus Primzahlen. Eine Einfuehrung in die Zahlentheorie. VEB Deutscher Verlag der Wissenschaften, 2. Aufl., 1984.
A. Warusfel, Les nombres et leurs mystères, Edition du Seuil, Paris 1980.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A023200, A046132, A174454.

Select[Range[1617]^2 - 2, And @@ PrimeQ[# + {0, 4}] &] (* Amiram Eldar, Dec 24 2019 *)

isok(n) = isprime(n) && isprime(n+4) && issquare(n+2) \\ Michel Marcus, Jul 22 2013

forstep(n=3,1e4,2,if(isprime(n^2-2)&&isprime(n^2+2),print1(n^2-2", "))) \\ Charles R Greathouse IV, Jul 23 2013

Edited by D. S. McNeil, Nov 18 2010

A178228 Numbers k such that (k^3 - 2, k^3 + 2) is a pair of cousin primes (see A178227).

Original entry on oeis.org

129, 189, 369, 435, 549, 555, 561, 819, 1245, 1491, 1719, 1779, 1839, 1875, 1935, 2175, 2289, 2415, 2451, 2595, 2709, 2769, 3141, 3441, 4401, 4611, 4851, 5655, 5775, 6075, 6099, 6795, 6969, 7125, 7239, 7365, 8109, 8139, 8325, 8361, 8385, 8535, 8685, 9591, 9765

Offset: 1

Ulrich Krug (leuchtfeuer37(AT)gmx.de), May 23 2010

nonn

Necessarily k is an odd multiple of 3, Least significant digit of k is e = 1, 5 or 9 (3^3 - 2, 7^3 + 2 are multiples of 5).

			189 is a term since 189^3 - 2 = 6751267 = prime(460792), 189^3 + 2 = 6751271 = prime(460793).
12471 is a term since 12471^3 - 2 = 1939562763109 = prime(i), i = 71166976775, 12471^3 + 2 = 1939562763113 = prime(i+1).

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A023200, A046132, A090121, A164834, A172494, A174370, A176130, A176229, A178227.

Select[Range[10^4], And @@ PrimeQ[#^3 + {-2, 2}] &] (* Amiram Eldar, Dec 24 2019 *)

for(n=1,10000,my(p1=n^3-2,p2=n^3+2);if(isprime(p1)&&isprime(p2)&&ispower((p1+p2)/2,3),print1(n,", "))) \\ Hugo Pfoertner, Dec 24 2019

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

a(1) and a(21) inserted by Amiram Eldar, Dec 24 2019

A187757 Number of ways to write n=x+y (x,y>0) with 6x-1, 6x+1, 6y+1 and 6y+5 all prime.

Original entry on oeis.org

0, 1, 2, 3, 2, 2, 2, 4, 3, 2, 2, 3, 4, 4, 2, 3, 2, 6, 6, 5, 4, 2, 6, 5, 4, 4, 2, 6, 4, 4, 4, 3, 5, 7, 5, 5, 3, 4, 9, 5, 6, 4, 5, 6, 4, 5, 5, 6, 7, 6, 6, 3, 7, 7, 6, 6, 4, 6, 6, 5, 6, 4, 7, 6, 7, 2, 3, 7, 7, 7, 5, 3, 5, 5, 7, 8, 5, 8, 8, 4, 5, 4, 10, 10, 6, 6, 2, 9, 6, 9, 7, 1, 8, 4, 5, 7, 3, 9, 5, 3

Offset: 1

Zhi-Wei Sun, Jan 03 2013

Conjecture: a(n)>0 for all n>1.

This has been verified for n up to 10^9. It implies that there are infinitely many twin primes and also infinitely many cousin primes, since the interval [m!+2,m!+m] of length m-2 contains no prime for any integer m>1.

			a(92)=1 since 92=40+52 with 6*40-1, 6*40+1, 6*52+1 and 6*52+5 all prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..20000
Zhi-Wei Sun, Conjectures involving primes and quadratic forms, arXiv:1211.1588.

Cf. A001359, A006512, A023200, A046132, A219157, A218867, A219185, A220455.

a[n_]:=a[n]=Sum[If[PrimeQ[6k-1]==True&&PrimeQ[6k+1]==True&&PrimeQ[6(n-k)+1]==True&&PrimeQ[6(n-k)+5]==True,1,0],{k,1,n-1}]
Do[Print[n," ",a[n]],{n,1,100}]

A194098 Decimal expansion of sum of reciprocals of cousin primes.

Original entry on oeis.org

1, 1, 9, 7, 0, 4, 4, 9

Offset: 1

Kausthub Gudipati, Aug 15 2011

The value is obtained by summing cousin prime pairs with values less than 2^42 (which yields 1.10633...) and a logarithmic extrapolation of Brun's constant A065421.

The estimate by [Park-Lee] is 1.197054+-7e-6. - R. J. Mathar, Feb 09 2013

Yeonyong Park, Heonsoo Lee, On the several differences between primes, J. Appl. Math. & Computing 13 (2003) vol 1-2, pp 37-51.

Prime Curios!, Sum of reciprocals of cousin primes
Marek Wolf, On the Twin and Cousin Primes, (1996) IFTUWr 909/96
Marek Wolf, Generalized Brun's constants, (1997) IFTUWr 910/97
Eric E. Weisstein, MathWorld: Cousin primes

Equals 1.1970449... = (1/7+1/11)+(1/13+1/17)+.. = Sum_{n>=2} (1/A023200(n) + 1/A046132(n)).

A220455 Number of ways to write n=x+y (x>0, y>0) with 3x-2, 3x+2 and 2xy+1 all prime.

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 0, 2, 3, 2, 1, 2, 1, 1, 4, 4, 1, 2, 2, 3, 3, 2, 2, 5, 1, 4, 1, 1, 5, 4, 1, 2, 5, 5, 3, 8, 3, 6, 5, 5, 4, 4, 2, 4, 5, 3, 1, 8, 3, 4, 4, 1, 2, 8, 6, 3, 4, 5, 4, 4, 7, 1, 3, 6, 5, 7, 3, 3, 8, 2, 4, 5, 2, 6, 10, 7, 1, 5, 5, 6, 8, 6, 4, 5, 5, 7, 5, 4, 4, 11, 4, 5, 5, 5, 6, 6, 3, 1, 12, 8

Offset: 1

Zhi-Wei Sun, Dec 15 2012

nonn

Conjecture: a(n)>0 for all n>7.
This has been verified for n up to 10^8. It implies that there are infinitely many cousin primes.
Conjecture verified for n up to 10^9. - Mauro Fiorentini, Aug 06 2023
Zhi-Wei Sun also made some other similar conjectures, e.g., he conjectured that any integer n>17 can be written as x+y (x>0, y>0) with 2x-3, 2x+3 and 2xy+1 all prime, and each integer n>28 can be written as x+y (x>0, y>0) with 2x+1, 2y-1 and 2xy+1 all prime.
Both conjectures verified for n up to 10^9. - Mauro Fiorentini, Aug 06 2023

			a(25)=1 since 25=13+12 with 3*13-2, 3*13+2 and 2*13*12+1=313 all prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Conjectures involving primes and quadratic forms, arXiv:1211.1588 [math.NT], 2012-2017.

Cf. A220431, A023200, A046132, A218867, A219055, A220419, A220413, A220272, A219842, A219864, A219923.

a[n_]:=a[n]=Sum[If[PrimeQ[3k-2]==True&&PrimeQ[3k+2]==True&&PrimeQ[2k(n-k)+1]==True,1,0],{k,1,n-1}]
Do[Print[n," ",a[n]],{n,1,1000}]
apQ[{a_,b_}]:=AllTrue[{3a-2,3a+2,2a*b+1},PrimeQ]; Table[Count[Flatten[ Permutations/@ IntegerPartitions[n,{2}],1],?(apQ[#]&)],{n,100}] (* The program uses the AllTrue function from Mathematica version 10 *) (* _Harvey P. Dale, Jun 09 2018 *)

cp's OEIS Frontend

A049491 Numbers k such that k and k+128 are both prime.

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A054902 Composite numbers n such that sigma(n)+12 = sigma(n+12).

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A054904 x = a(n) is the smallest composite number such that sigma(x+6n) = sigma(x)+6n, where sigma = A000203.

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A067775 Primes p such that p + 4 is composite.

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A157834 Numbers n such that 3n-2 and 3n+2 are both prime.

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Magma

Maple

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A176130 Lesser of a pair (p,p+4) of cousin primes whose arithmetic mean p+2 is a square number.

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A178228 Numbers k such that (k^3 - 2, k^3 + 2) is a pair of cousin primes (see A178227).

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