A046138 -id:A046138 - cp's OEIS frontend

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

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

A023202 Primes p such that p + 8 is also prime.

Original entry on oeis.org

3, 5, 11, 23, 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, 1319, 1373, 1439

Offset: 1

David W. Wilson

All terms > 3 are congruent to 5 mod 6 (observation by Zak Seidov in SeqFan). Thus each corresponding p + 8 is congruent to 1 mod 6. - Rick L. Shepherd, Mar 25 2023

Matt C. Anderson, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe, corrected by Sean A. Irvine and Georg Fischer)
A. Granville and G. Martin, Prime number races, arXiv:math/0408319 [math.NT], 2004.
Maxie D. Schmidt, New Congruences and Finite Difference Equations for Generalized Factorial Functions, arXiv:1701.04741 [math.CO], 2017.
Eric Weisstein's World of Mathematics, Twin Primes

Disjoint union of A007530, A031926, A049437, A049438.

Cf. A046134, A049436, A046138, A015915.

Filtered([1..1500], k-> IsPrime(k) and IsPrime(k+8)); # G. C. Greubel, Feb 07 2020

[n: n in [0..1500] | IsPrime(n) and IsPrime(n+8)]; // Vincenzo Librandi, Nov 20 2010

select(n-> isprime(n) and isprime(n+8), [`$`(1..1500)]); # G. C. Greubel, Feb 07 2020

Select[Range[1500], PrimeQ[#] && PrimeQ[#+8]&] (* Vladimir Joseph Stephan Orlovsky, Aug 29 2008 *)
Select[Prime[Range[250]],PrimeQ[#+8]&] (* Harvey P. Dale, Dec 24 2020 *)

is(n)=isprime(n)&&isprime(n+8) \\ Charles R Greathouse IV, Jul 01 2013

[n for n in (1..1500) if is_prime(n) and is_prime(n+8)] # G. C. Greubel, Feb 07 2020

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

A049436 p, p+8 and either p+2 or p+6 or both are all primes.

Original entry on oeis.org

3, 5, 11, 23, 29, 53, 59, 71, 101, 131, 149, 173, 191, 233, 263, 269, 431, 563, 569, 593, 599, 653, 821, 1013, 1031, 1061, 1223, 1229, 1283, 1289, 1319, 1451, 1481, 1601, 1613, 1619, 1871, 2081, 2129, 2333, 2339, 2381, 2543, 2549, 2711, 2789, 2963, 3251

Offset: 1

Labos Elemer

nonn

			3 is here because 5, 7 and 11 are primes; 5 is here because 7, 11 and 13 are primes.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

A031926, A023202, A049437, A049438, A046134, A046138, A007530.

Select[Prime[Range[500]],PrimeQ[#+8]&&AnyTrue[#+{2,6},PrimeQ]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jan 04 2017 *)

A144842 Numbers k such that the three numbers k+3, k-3 and k+5 are all prime.

Original entry on oeis.org

8, 14, 26, 56, 104, 134, 176, 194, 236, 266, 566, 596, 656, 824, 1016, 1226, 1286, 1484, 1604, 1616, 1874, 2084, 2336, 2546, 2966, 3254, 3326, 3464, 3536, 3764, 3914, 3926, 4016, 4214, 4256, 4646, 4796, 5006, 5276, 5474, 5654, 5846, 5864, 6266, 6356, 6566

Offset: 1

Giovanni Teofilatto, Sep 22 2008

Subset of A087695. - R. J. Mathar, Sep 24 2008

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Prime Triplet.

Cf. A046138, A087695, A086801, A144840.

Select[Range[7000], And @@ PrimeQ[# + {-3, 3, 5}] &] (* Amiram Eldar, Apr 14 2022 *)

from sympy import isprime
def ok(n): return n > 4 and isprime(n-3) and isprime(n+3) and isprime(n+5)
print(list(filter(ok, range(6567)))) # Michael S. Branicky, Aug 14 2021

a(n) = A046138(n) + 3. - R. J. Mathar, Sep 24 2008

Definition edited and extended by R. J. Mathar, Sep 24 2008

A185718 For positive n with prime decomposition n = Product_{j=1..m} (p_j^k_j) define A_n = Sum_{j=1..m} (p_j*k_j) and B_n = Sum_{j=1..m} (p_j^k_j). This sequence gives those n for which A_n and B_n are both prime and B_n = A_n + 2 (i.e., form a twin prime pair).

Original entry on oeis.org

40, 88, 184, 424, 808, 1048, 1384, 1528, 1864, 2104, 2184, 3080, 4504, 4744, 5224, 5928, 6440, 6568, 7224, 8104, 8360, 8840, 9784, 10264, 10472, 11480, 11544, 11848, 12808, 12904, 14136, 14840, 14968, 16280, 16648, 18664, 19608, 20344, 21080, 22040, 23240, 23704, 24440, 24648, 24920, 26008, 26584, 27384, 27608, 27688, 28264, 28952, 29240

Offset: 1

Bobby Browning, Feb 10 2011

nonn

Assuming the twin prime conjecture, my advisor and I are able to prove there are infinitely many of these pairs. In other words, there are infinitely many n such that A_n and B_n are prime and B_n = A_n + 2.
From Bobby Browning, Feb 14 2011: (Start)
8*A046138 is a subsequence of A185718 for the following reasons:
i) the n in A185718 for which A_n and B_n form a twin prime pair are of the form n=2^3*p_1*p_2*...p_k.
ii) the A046138 sequence consists of primes p such that p+6 and p+8 form a twin prime pair.
iii) so if p is a prime such that p+6 and p+8 form a twin prime pair and n = 2^3*p then A_n = p+6 and B_n = p+8. Thus, the integers such that n = 2^3*p are a subsequence of A185718. (End)

			a(1) = 40  = 2^3*5^1, with a = 11 and b = 13.
a(2) = 88  = 2^3*11^1 with a = 17 and b = 19.
a(3) = 184 = 2^3*23^1 with a = 29 and b = 31.

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A001414 (A_n), A008475 (B_n).

okQ[n_] := Module[{p, e, a, b}, {p, e} = Transpose[FactorInteger[n]]; a = Plus @@ (p*e); b = Plus @@ (p^e); b == a + 2 && PrimeQ[a] && PrimeQ[b]]; Select[Range[30000], okQ]

sopfr(n)=my(f=factor(n));sum(i=1,#f[,1],f[i,1]*f[i,2]);
forstep(n=8,1e5,16,if(issquarefree(n/8)&&isprime(k=sopfr(n))&isprime(k+2), print1(n", ")))

A360758 Numbers k for which k' - 1 and k' + 1 are twin primes, where the prime denotes the arithmetic derivative.

Original entry on oeis.org

4, 8, 9, 35, 36, 64, 65, 68, 77, 81, 112, 160, 161, 185, 188, 208, 209, 221, 225, 236, 335, 341, 371, 377, 428, 437, 441, 485, 515, 576, 596, 611, 671, 707, 731, 736, 756, 767, 779, 783, 792, 851, 868, 899, 917, 952, 965, 972, 1007, 1028, 1067, 1115, 1152, 1157

Offset: 1

Marius A. Burtea, Mar 01 2023

nonn

Numbers that have an arithmetic derivative equal to the average of twin prime pairs (A014574).
If p is in A022005 then m = 5*p is a term. Indeed, p is prime and m' = (5*p)' = p + 5 and m' - 1 = p + 4 and m' + 1 = p + 6 which are twin prime numbers.
If p is in A046138 then m = 7*p is a term. Indeed, p is prime and m' = (7*p)' = p + 7 and m' - 1 = p + 6 and m' + 1 = p + 8 which are twin prime numbers.
If p is in A212492 then m = 11*p is a term. Indeed, p is prime and m' = (11*p)' = p + 11 and m' - 1 = p + 10 and m' + 1 = p + 12 which are twin prime numbers.

			4' =  4, 4' - 1 =  4 - 1 =  3, 4' + 1 =  4 + 1 =  5, so 4 is a term.
8' = 12, 8' - 1 = 12 - 1 = 11, 8' + 1 = 12 + 1 = 13, so 8 is a term.
9' =  6, 9' - 1 =  6 - 1 =  5, 9' + 1 =  6 + 1 =  7, so 9 is a term.

Cf. A003415, A014574, A022005, A046138, A212492, A077800.

f:=func; [p:p in [2..1200]| IsPrime(Floor(f(p))-1) and IsPrime(Floor(f(p))+1)];

d[1] = 0; d[n_] := n * Plus @@ ((Last[#]/First[#]) & /@ FactorInteger[n]); Select[Range[1200], And @@ PrimeQ[d[#] + {-1, 1}] &] (* Amiram Eldar, Mar 01 2023 *)

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

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A023202 Primes p such that p + 8 is also 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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A049436 p, p+8 and either p+2 or p+6 or both are all primes.

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A144842 Numbers k such that the three numbers k+3, k-3 and k+5 are all prime.

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A185718 For positive n with prime decomposition n = Product_{j=1..m} (p_j^k_j) define A_n = Sum_{j=1..m} (p_j*k_j) and B_n = Sum_{j=1..m} (p_j^k_j). This sequence gives those n for which A_n and B_n are both prime and B_n = A_n + 2 (i.e., form a twin prime pair).

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Mathematica

PARI

A360758 Numbers k for which k' - 1 and k' + 1 are twin primes, where the prime denotes the arithmetic derivative.

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Keywords

Comments

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Magma

Mathematica