A046133 -id:A046133 - cp's OEIS frontend

A086139 Let p = A046133(n), that is, let p run through the list of primes such that p+12 is also prime (A046133); a(n) = number of primes in the interval p + 1 through p + 11 inclusive.

3, 3, 3, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 1, 3, 3, 2, 1, 1, 1, 1, 1, 1, 0, 0, 2, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 0, 1, 1, 1, 0, 1, 1, 1, 2, 0, 1, 2, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 0, 1, 0, 2, 2, 2, 2, 0, 1, 2, 1, 2, 0, 1, 3, 2, 0, 0, 0, 1, 1, 1, 2, 2

Offset: 1

Labos Elemer, Jul 29 2003

nonn

From Michael De Vlieger, Jul 30 2017: (Start)
a(n) = 0 for n = {24, 25, 44, 48, 53, 57, 62, 70, 82, 84, 89, 94, ...}.
a(n) = 1 for n = {9, 14, 18, 19, 20, 21, 22, 23, 28, 29, 30, 33, ...}.
a(n) = 2 for n = {4, 5, 6, 7, 8, 10, 11, 12, 13, 17, 26, 27, 31, ...}.
a(n) = 3 for n = {1, 2, 3, 15, 16, 96, 118, 183, 266, 570, 581, ...}.
(End)

			For n=1, we have p=5, the primes between 5 and 5+12=17 are 7,11,13, so a(1)=3.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A031930, A031931, A046133.

a:=[]; b:=[];
for n from 1 to 200 do if isprime(ithprime(n)+12) then
   a:=[op(a),ithprime(n)];
c:=0;
for i from 1 to 11 do if isprime(ithprime(n)+i) then c:=c+1; fi; od;
b:=[op(b),c];
fi;
od:
a; # A046133b; # this sequence

cp[x_,y_] := Count[Table[PrimeQ[i],{i,x,y}],True]; d = 12; Do[s = Prime[n]; If[PrimeQ[s+d], Print[cp[s+1,s+d-1]]], {n, 1, 1000}]
(* Second program: *)
With[{d = 12}, DeleteCases[#, -1] &@ Table[Function[p, If[PrimeQ[p + d],
Count[Range[p + 1, p + d - 1], _?PrimeQ], -1] ]@ Prime@ n, {n, 252}]]
PrimePi[#+11]-PrimePi[#+1]&/@Select[Prime[Range[400]],PrimeQ[#+12]&] (* Harvey P. Dale, Jul 30 2022 *)

Definition edited by N. J. A. Sloane, Aug 05 2017 following analysis by Michael De Vlieger, Jul 30 2017

A086140 Primes p such that three (the maximum number) primes occur between p and p+12.

Original entry on oeis.org

5, 7, 11, 97, 101, 1481, 1867, 3457, 5647, 15727, 16057, 16061, 19417, 19421, 21011, 22271, 43777, 43781, 55331, 79687, 88807, 101107, 144161, 165701, 166841, 195731, 201821, 225341, 247601, 257857, 266677, 268811, 276037, 284737, 326141, 340927

Offset: 1

Labos Elemer, Jul 29 2003

nonn

p+12 must be a prime. - Harvey P. Dale, Jun 11 2015

A086140 is the union of A022006 and A022007. By merging the two b-files I have extended the current b-file up to n=10000 (nearly n=20000 would have been possible). I add a comparison (see Links) between the frequency of prime 5-tuples and an asymptotic approximation, which is unproven but likely to be true, and based on a conjecture first published by Hardy and Littlewood in 1923. Twins, triples and quadruplets are treated as well. - Gerhard Kirchner, Dec 07 2016

			There are two types of prime 5-tuples, and both are represented in this sequence. (11, 13, 17, 19, 23) is a prime 5-tuple of the form (p, p+2, p+6, p+8, p+12), so 11 is in the sequence, and (97, 101, 103, 107, 109) is a prime 5-tuple of the form (p, p+4, p+6, p+10, p+12), so 97 is in the sequence. - _Michael B. Porter_, Dec 19 2016

Harvey P. Dale and Gerhard Kirchner, Table of n, a(n) for n = 1..10000 (first 1000 terms from Harvey P. Dale)
Gerhard Kirchner, Comparison with an assumed asymptotic distribution

Cf. A031930, A046133, A086139, A086136, A022006, A022007, A001359 (twins), A007529 (triples), A007530 (quadruplets).

cp[x_, y_] := Count[Table[PrimeQ[i], {i, x, y}], True] {d=12, k=0}; Do[s=Prime[n]; s1=Prime[n+1]; If[PrimeQ[s+d]&&Equal[cp[s+1, s+d-1], 3], k=k+1; Print[s]], {n, 1, 100000}]
(* Second program: *)
Transpose[Select[Partition[Prime[Range[30000]],5,1],#[[5]]-#[[1]] == 12&]][[1]] (* Harvey P. Dale, Jun 11 2015 *)

A015917 Numbers k such that sigma(k) + 12 = sigma(k+12).

Original entry on oeis.org

5, 7, 11, 17, 19, 29, 31, 41, 47, 59, 61, 65, 67, 71, 89, 97, 101, 127, 137, 139, 151, 167, 170, 179, 181, 199, 209, 211, 227, 229, 239, 251, 257, 269, 271, 281, 337, 347, 367, 389, 397, 409, 419, 421, 431, 449, 467, 479, 487, 491, 509, 557, 587

Offset: 1

Robert G. Wilson v

nonn

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

Cf. A000203.

Union of A046133 and A054902.

Select[Range[600],DivisorSigma[1,#]+12==DivisorSigma[1,#+12]&] (* Harvey P. Dale, Oct 14 2012 *)

is(n)=sigma(n)+12==sigma(n+12) \\ Charles R Greathouse IV, Mar 09 2014

A156104 Primes p such that p+36 is also prime.

Original entry on oeis.org

5, 7, 11, 17, 23, 31, 37, 43, 47, 53, 61, 67, 71, 73, 101, 103, 113, 127, 131, 137, 157, 163, 191, 193, 197, 227, 233, 241, 257, 271, 277, 281, 311, 313, 317, 331, 337, 347, 353, 373, 383, 397, 421, 431, 443, 463, 467, 487, 521, 541, 557, 563, 571, 577, 607

Offset: 1

Vincenzo Librandi, Feb 08 2009

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

Cf. A156112.

Cf. sequences of the type p+n are primes: A001359 (n=2), A023200 (n=4), A023201 (n=6), A023202 (n=8), A023203 (n=10), A046133 (n=12), A153417 (n=14), A049488 (n=16), A153418 (n=18), A153419 (n=20), A242476 (n=22), A033560 (n=24), A252089 (n=26), A252090 (n=28), A049481 (n=30), A049489 (n=32), A252091 (n=34), this sequence (n=36); A062284 (n=50), A049490 (n=64), A156105 (n=72), A156107 (n=144).

[p: p in PrimesUpTo(1000) | IsPrime(p + 36)]; // Vincenzo Librandi, Oct 31 2012

Select[Prime[Range[1000]], PrimeQ[(#+ 36)]&] (* Vincenzo Librandi, Oct 31 2012 *)

A015915 Numbers k such that sigma(k) + 8 = sigma(k+8).

Original entry on oeis.org

3, 5, 11, 23, 27, 29, 53, 59, 71, 89, 101, 131, 149, 173, 191, 233, 263, 269, 359, 389, 401, 431, 449, 479, 491, 563, 569, 593, 599, 653, 683, 701, 719, 743, 761, 821, 911, 929, 983, 1013, 1031, 1061, 1109, 1163, 1193, 1223, 1229, 1283, 1289

Offset: 1

Robert G. Wilson v

nonn

Different from A023202. Below 1000000 four composites were found [27, 1615, 1885, 218984] satisfying the "sigma(k) + 8 = sigma(k+8)" relation, together with more than 8000 primes. - Labos Elemer, May 23 2000

			sigma(27) + 8 = 48 = sigma(27+8), so 27 is in the sequence.

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

Cf. A000203, A015913-A015917, A023200-A023203, A046133, A001359, A054799.

Composite solutions are in A059118.

Select[Range[1300],DivisorSigma[1,#]+8==DivisorSigma[1,#+8]&] (* Harvey P. Dale, Jul 16 2011 *)

is(n)=sigma(n)+8==sigma(n+8) \\ Charles R Greathouse IV, Mar 11 2014

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

A137796 Prime numbers p such that p + 12 and p - 12 are primes.

Original entry on oeis.org

17, 19, 29, 31, 41, 59, 71, 101, 139, 151, 179, 211, 239, 251, 269, 281, 409, 421, 431, 479, 491, 619, 631, 739, 809, 941, 1009, 1021, 1051, 1289, 1291, 1439, 1459, 1471, 1499, 1511, 1571, 1609, 1709, 1721, 1789, 1889, 1901, 1999, 2099, 2141, 2281, 2411

Offset: 1

Vladimir Joseph Stephan Orlovsky, Apr 28 2008

nonn

			17 + 12 = 29 (a prime), 17 - 12 = 5 (a prime);
19 + 12 = 31 (a prime), 19 - 12 = 7 (a prime).

Daniel Starodubtsev, Table of n, a(n) for n = 1..10000

Cf. A092216, A046133. Note that this is different from A137873.

isA092216 := proc(n) RETURN(isprime(n) and isprime(n-12) ) ; end: isA046133 := proc(n) RETURN(isprime(n) and isprime(n+12) ) ; end: isA137796 := proc(n) RETURN(isA092216(n) and isA046133(n)) ; end: for i from 1 to 400 do if isA137796(ithprime(i)) then printf("%d,",ithprime(i)) ; fi ; od: # R. J. Mathar, May 03 2008

a=12; Select[Table[Prime[n],{n,10^3}], PrimeQ[ #-a] && PrimeQ[ #+a] &]

lista(nn) = forprime(p=2, nn, if (isprime(p-12) && isprime(p+12), print1(p, ", "))); \\ Michel Marcus, Oct 04 2015

A092216 INTERSECT A046133. - R. J. Mathar, May 03 2008

Corrected and extended by R. J. Mathar, May 03 2008

A092216 Primes of the form p + 12 where p is a prime.

Original entry on oeis.org

17, 19, 23, 29, 31, 41, 43, 53, 59, 71, 73, 79, 83, 101, 109, 113, 139, 149, 151, 163, 179, 191, 193, 211, 223, 239, 241, 251, 263, 269, 281, 283, 293, 349, 359, 379, 401, 409, 421, 431, 433, 443, 461, 479, 491, 499, 503, 521, 569, 599, 613, 619, 631, 643, 653

Offset: 1

Douglas Winston (douglas.winston(AT)srupc.com), Apr 02 2004

nonn

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

Cf. A000040, A006512, A046132, A046117, A092402, A092146.

f[n_]:=n+12; lst={};Do[r=Prime[n];p=f[r];If[PrimeQ[p],AppendTo[lst,p]],{n,6!}];lst (* Vladimir Joseph Stephan Orlovsky, Jul 04 2009 *)
Select[Prime[Range[5,120]],PrimeQ[#-12]&]  (* Harvey P. Dale, Feb 18 2011 *)

a(n) = 12 + A046133(n). - R. J. Mathar, Jun 21 2010

A252089 Primes p such that p + 26 is prime.

Original entry on oeis.org

3, 5, 11, 17, 41, 47, 53, 71, 83, 101, 113, 131, 137, 167, 173, 197, 251, 257, 281, 311, 347, 353, 383, 431, 461, 521, 587, 593, 617, 647, 683, 701, 743, 761, 797, 827, 857, 881, 911, 941, 971, 983, 1013, 1061, 1091, 1097, 1103, 1187, 1223, 1277, 1301, 1373

Offset: 1

Vincenzo Librandi, Dec 14 2014

			17 is in this sequence because 17+26 = 43 is prime.
431 is in this sequence because 431+26 = 457 is prime.

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. sequences of the type p+n are primes: A001359 (n=2), A023200 (n=4), A023201 (n=6), A023202 (n=8), A023203 (n=10), A046133 (n=12), A153417 (n=14), A049488 (n=16), A153418 (n=18), A153419 (n=20), A242476 (n=22), A033560 (n=24), this sequence (n=26), A252090 (n=28), A049481 (n=30), A049489 (n=32), A252091 (n=34), A156104 (n=36); A062284 (n=50), A049490 (n=64), A156105 (n=72), A156107 (n=144).

[NthPrime(n): n in [1..250] | IsPrime(NthPrime(n)+26)];

Select[Prime[Range[200]], PrimeQ[# + 26] &]

A054903 Composite numbers n such that sigma(n)+6 = sigma(n+6), where sigma=A000203.

Original entry on oeis.org

104, 147, 596, 1415, 4850, 5337, 370047, 1630622, 35020303, 120221396, 3954451796, 742514284703

Offset: 1

Labos Elemer, May 23 2000

Complement of A023201 with respect to A015914.
Intersection of A015914 and A018252.
Below 1000000 there are only 7 such composite numbers, compared with more than 16000 primes.
a(13) > 10^13. - Giovanni Resta, Jul 11 2013

			n=104, sigma(104)+6 = 210+6 = 216 = sigma(104+6) = sigma(110).
a(4) = 1415 = 5*283, 1415+6 = 1421 = 7*7*29:
sigma(1415) = 1+5+283+1415 = 1704,
sigma(1421) = 1+7+29+49+203+1421 = 1710 = sigma(1415)+6.

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 104, p. 37, Ellipses, Paris 2008.

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

forcomposite(n=9,1e7,if(sigma(n)+6==sigma(n+6),print1(n", "))) \\ Charles R Greathouse IV, Feb 14 2013

More terms from Jud McCranie, May 25 2000
New definition from Reinhard Zumkeller, Jan 27 2009
Edited by N. J. A. Sloane, Jan 31 2009 at the suggestion of R. J. Mathar.
a(12) from Giovanni Resta, Jul 11 2013

cp's OEIS Frontend

A086139 Let p = A046133(n), that is, let p run through the list of primes such that p+12 is also prime (A046133); a(n) = number of primes in the interval p + 1 through p + 11 inclusive.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Extensions

A086140 Primes p such that three (the maximum number) primes occur between p and p+12.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A015917 Numbers k such that sigma(k) + 12 = sigma(k+12).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

A156104 Primes p such that p+36 is also prime.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

A015915 Numbers k such that sigma(k) + 8 = sigma(k+8).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

Extensions

A137796 Prime numbers p such that p + 12 and p - 12 are primes.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A092216 Primes of the form p + 12 where p is a prime.

Views

Author

Keywords

Links