A023202 -id:A023202 - cp's OEIS frontend

A086137 Number of primes between p and p+8 if p is prime, i.e., number of primes between 8+A023202(n) and A023202(n).

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

Offset: 1

Labos Elemer, Jul 29 2003

nonn

			a(n)=0,1,2 correspond to {p,p+8} prime-pairs either consecutive or pairs with various d-patterns as follows:
a(n)=0 to 89[8]97; a(n)=1 for 29[2,6]37, 53[6,2];
a(n)=2 for 101[2,4,2]109 and once to 3[2,2,4]11.

Cf. A023302, A046133, A048136, A052165.

cp[x_,y_] := Count[Table[PrimeQ[i],{i,x,y}],True] Do[s=Prime[n]; s1=Prime[n+1]; If[PrimeQ[s+d],k=k+1; Print[cp[s+1,s+d-1]]],{n,1,1000}]; k

A031926 Lower prime of a difference of 8 between consecutive primes.

Original entry on oeis.org

89, 359, 389, 401, 449, 479, 491, 683, 701, 719, 743, 761, 911, 929, 983, 1109, 1163, 1193, 1373, 1439, 1523, 1559, 1571, 1733, 1823, 1979, 2003, 2153, 2213, 2243, 2273, 2459, 2531, 2609, 2663, 2699, 2741, 2843, 2879, 2909, 3011, 3041

Offset: 1

Jeff Burch

nonn

Conjecture: The sequence is infinite and for every n, a(n+1) < a(n)^(1+1/n). Namely a(n)^(1/n) is a strictly decreasing function of n (see comment lines of the sequence A248855). - Jahangeer Kholdi and Farideh Firoozbakht, Nov 29 2014

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Index entries for primes, gaps between

Cf. A023202.

[p: p in PrimesUpTo(4000) | NextPrime(p)-p eq 8]; // Bruno Berselli, Apr 09 2013

for i from 1 to 446 do if ithprime(i+1) = ithprime(i) + 8 then print({ithprime(i)}); fi; od; # Zerinvary Lajos, Mar 19 2007
p:=ithprime; nx:=nextprime; f:=proc(d) global p,nx; local i,t0,n; t0:=[]; for n from 1 to 100000 do i:=p(n); if nx(i)-i=d then t0:=[op(t0),i]; fi; od: t0; end; f(8); # N. J. A. Sloane, Jan 17 2012

Transpose[Select[Partition[Prime[Range[500]], 2, 1], Last[#] - First[#] == 8 &]][[1]] (* Bruno Berselli, Apr 09 2013 *)

is_A031926(p)={precprime(p-1)==p-8 && isprime(p)} \\ M. F. Hasler, Apr 20 2013

q=0;forprime(p=1,5000,q+8==(q=p)&&print1(p-8",")) \\ M. F. Hasler, Apr 20 2013

A049488 Primes p such that p+16 is prime.

Original entry on oeis.org

3, 7, 13, 31, 37, 43, 67, 73, 97, 151, 157, 163, 181, 211, 223, 241, 277, 331, 337, 367, 373, 433, 463, 487, 541, 547, 571, 577, 601, 631, 643, 661, 727, 757, 811, 823, 937, 967, 997, 1033, 1087, 1093, 1171, 1201, 1213, 1291, 1303, 1423, 1471, 1483, 1543

Offset: 1

Labos Elemer

Using the Elliott-Halberstam conjecture, Goldston et al. prove that there are an infinite number of primes here. - T. D. Noe, Nov 26 2013

			7 and 7+16=23 are prime.

P. D. T. A. Elliott and H. Halberstam, A conjecture in prime number theory, Symposia Mathematica, Vol. IV (INDAM, Rome, 1968/69), pages 59-72, Academic Press, London, 1970.

Seiichi Manyama, Table of n, a(n) for n = 1..10000 (terms up to a(1000) by T. D. Noe)
D. A. Goldston, J. Pintz, and C. Y. Yildirim, Primes in Tuples I, arXiv:math/0508185 [math.NT], Aug 10 2005.
A. Granville and G. Martin, Prime number races, arXiv:math/0408319 [math.NT], Aug 24 2004.

Cf. A001359, A023200, A023202.

Select[Range[1000], PrimeQ[#] && PrimeQ[#+16]&] (* Vladimir Joseph Stephan Orlovsky, Dec 25 2008 *)
Select[Prime[Range[250]],PrimeQ[#+16 ]&] (* Harvey P. Dale, Oct 30 2015 *)

select(p->isprime(p+16),primes(100)) \\ Charles R Greathouse IV, Jul 08 2013

A049437 Primes p such that p+2 and p+8 are also primes but p+6 is not.

Original entry on oeis.org

3, 29, 59, 71, 149, 269, 431, 569, 599, 1031, 1061, 1229, 1289, 1319, 1451, 1619, 2129, 2339, 2381, 2549, 2711, 2789, 3299, 3539, 4019, 4049, 4091, 4649, 4721, 5099, 5441, 5519, 5639, 5741, 5849, 6269, 6359, 6569, 6701, 6959, 7211

Offset: 1

Labos Elemer

nonn

p+4 is not prime here except for p=3.

			p=29 is the smallest prime so that p, p+2 and p+8 are consecutive primes.

Iain Fox, Table of n, a(n) for n = 1..10000 (first 1000 terms from R. J. Mathar)

Cf. A007530, A023202, A031926, A046134, A046138, A049436, A049438, A046138.

Subsequence of A001359. - R. J. Mathar, Feb 10 2013

[p: p in PrimesUpTo(8000)| IsPrime(p+2) and IsPrime(p+8) and not IsPrime(p+6) ] // Vincenzo Librandi, Jan 28 2011

select(p -> isprime(p) and isprime(p+2) and isprime(p+8) and not isprime(p+6), [3, seq(i,i=5..10000, 6)]); # Robert Israel, Nov 20 2017

{3}~Join~Select[Partition[Prime@ Range[10^3], 3, 1], Differences@ # == {2, 6} &][[All, 1]] (* Michael De Vlieger, Nov 20 2017 *)

lista(nn) = forprime(p=3, nn, if(isprime(p+2) && isprime(p+8) && !isprime(p+6), print1(p, ", "))) \\ Iain Fox, Nov 20 2017

A054905 Smallest composite x such that sigma(x) + 2n = sigma(x + 2n).

Original entry on oeis.org

434, 305635357, 104, 27, 195556, 65, 12, 39, 20, 56, 916, 80, 212282, 57, 44, 106645, 52, 125

Offset: 1

Labos Elemer May 23 2000

a(19) > 4293000000, if it exists. - Jud McCranie, May 25 2000

a(19) > 10^11, if it exists. - Charles R Greathouse IV, Oct 26 2022

			a(5) corresponds to n=3+2=5, d=2n=10 and the smallest composite integer is 195556. The next solution is 1152136225.

Mauro Fiorentini, Le soluzioni dell’equazione s(n + k) = s(n) + k, con n composto fino a 109 e k sino a 1000.

Cf. A015913, A015914, A015915, A015916, A015917, A054799.

Cf. A023200, A023201, A023202, A023203,

a(n)=forcomposite(x=3,10^66,if(sigma(x)+2*n==sigma(x+2*n),return(x)));
for(n=1,66,print1(a(n),", ")); \\ Joerg Arndt, Nov 15 2014

a19(lim,startAt=39)=startAt=ceil(startAt); my(v=vectorsmall(38),i=(startAt-1)%38); forfactored(n=startAt,lim\1+38, my(t=sigma(n)); if(i++>38,i=1); if(t==v[i]+38, return(n[1]-38)); v[i]=if(vecsum(n[2][,2])>1,t,0)) \\ Charles R Greathouse IV, Oct 25 2022

Description corrected by Jud McCranie, May 25 2000

A049489 Primes p such that p + 32 is also prime.

Original entry on oeis.org

5, 11, 29, 41, 47, 71, 107, 131, 149, 167, 179, 191, 197, 239, 251, 281, 317, 347, 389, 401, 431, 467, 491, 509, 569, 587, 599, 641, 659, 677, 701, 719, 797, 821, 827, 887, 977, 1019, 1031, 1061, 1091, 1097, 1181, 1217, 1259, 1289, 1367, 1427, 1439, 1451

Offset: 1

Labos Elemer

			29 and 29 + 32 = 61 are both prime.

Karl V. Keller, Jr., Table of n, a(n) for n = 1..10000

Cf. A001359, A023200, A023202, A049488.

[p: p in PrimesUpTo(2000) | IsPrime(p+32)]; // Vincenzo Librandi, Apr 22 2015

Primes:= select(isprime,{seq(i,i=3..10000,2)}):
sort(convert(Primes intersect map(`-`,Primes,32),list)); # Robert Israel, Dec 20 2015

Select[Range[2000], PrimeQ[#] && PrimeQ[# + 32] &] (* Vincenzo Librandi, Apr 22 2015 *)
Select[Prime[Range[300]],PrimeQ[#+32]&] (* Harvey P. Dale, Oct 14 2017 *)

isok(n) = isprime(n) && isprime(n+32); \\ Michel Marcus, Dec 31 2013

Name improved by Bruno Berselli, Apr 22 2015

A049490 a(n) and a(n)+64 both prime.

Original entry on oeis.org

3, 7, 19, 37, 43, 67, 73, 103, 109, 127, 163, 193, 199, 229, 283, 337, 367, 379, 397, 439, 457, 499, 523, 577, 613, 619, 709, 733, 757, 823, 877, 883, 907, 919, 967, 997, 1033, 1039, 1087, 1117, 1123, 1129, 1153, 1213, 1237, 1297, 1303, 1423, 1429, 1447

Offset: 1

Labos Elemer

nonn

			19 and 19+64=83 both prime.

Zak Seidov, Table of n, a(n) for n = 1..1000

Cf. A001359, A023200, A023202.

Select[Prime[Range[300]],PrimeQ[#+64]&] (* Harvey P. Dale, Mar 21 2011 *)

select(p -> isprime(p+64),primes(1000)) \\ Edward Jiang, Sep 05 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 *)

A049438 p, p+6 and p+8 are all primes (A046138) but p+2 is not.

Original entry on oeis.org

23, 53, 131, 173, 233, 263, 563, 593, 653, 1013, 1223, 1283, 1601, 1613, 2333, 2543, 2963, 3323, 3533, 3761, 3911, 3923, 4013, 4211, 4253, 4643, 4793, 5003, 5273, 5471, 5843, 5861, 6263, 6353, 6563, 6653, 6863, 7121, 7451, 7481, 7541, 7583

Offset: 1

Labos Elemer

R. J. Mathar, Table of n, a(n) for n = 1..1000

Subsequence of A031924. - R. J. Mathar, Jun 15 2013

Cf. A007530, A023202, A031926, A046134, A046138, A049436, A049437.

Select[Prime@ Range[10^3], MatchQ[Boole@ PrimeQ@ {# + 2, # + 6, # + 8}, {0, 1, 1}] &] (* Michael De Vlieger, Feb 05 2017 *)

isok(p) = isprime(p) && !isprime(p+2) && isprime(p+6) && isprime(p+8); \\ Michel Marcus, Dec 13 2013

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

cp's OEIS Frontend

A086137 Number of primes between p and p+8 if p is prime, i.e., number of primes between 8+A023202(n) and A023202(n).

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A031926 Lower prime of a difference of 8 between consecutive primes.

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A049488 Primes p such that p+16 is prime.

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A049437 Primes p such that p+2 and p+8 are also primes but p+6 is not.

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A054905 Smallest composite x such that sigma(x) + 2n = sigma(x + 2n).

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PARI

PARI

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A049489 Primes p such that p + 32 is also prime.

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A049490 a(n) and a(n)+64 both prime.

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