cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A001359 Lesser of twin primes.

Original entry on oeis.org

3, 5, 11, 17, 29, 41, 59, 71, 101, 107, 137, 149, 179, 191, 197, 227, 239, 269, 281, 311, 347, 419, 431, 461, 521, 569, 599, 617, 641, 659, 809, 821, 827, 857, 881, 1019, 1031, 1049, 1061, 1091, 1151, 1229, 1277, 1289, 1301, 1319, 1427, 1451, 1481, 1487, 1607
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

Also, solutions to phi(n + 2) = sigma(n). - Conjectured by Jud McCranie, Jan 03 2001; proved by Reinhard Zumkeller, Dec 05 2002
The set of primes for which the weight as defined in A117078 is 3 gives this sequence except for the initial 3. - Rémi Eismann, Feb 15 2007
The set of lesser of twin primes larger than three is a proper subset of the set of primes of the form 3n - 1 (A003627). - Paul Muljadi, Jun 05 2008
It is conjectured that A113910(n+4) = a(n+2) for all n. - Creighton Dement, Jan 15 2009
I would like to conjecture that if f(x) is a series whose terms are x^n, where n represents the terms of sequence A001359, and if we inspect {f(x)}^5, the conjecture is that every term of the expansion, say a_n * x^n, where n is odd and at least equal to 15, has a_n >= 1. This is not true for {f(x)}^k, k = 1, 2, 3 or 4, but appears to be true for k >= 5. - Paul Bruckman (pbruckman(AT)hotmail.com), Feb 03 2009
A164292(a(n)) = 1; A010051(a(n) - 2) = 0 for n > 1. - Reinhard Zumkeller, Mar 29 2010
From Jonathan Sondow, May 22 2010: (Start)
About 15% of primes < 19000 are the lesser of twin primes. About 26% of Ramanujan primes A104272 < 19000 are the lesser of twin primes.
About 46% of primes < 19000 are Ramanujan primes. About 78% of the lesser of twin primes < 19000 are Ramanujan primes.
A reason for the jumps is in Section 7 of "Ramanujan primes and Bertrand's postulate" and in Section 4 of "Ramanujan Primes: Bounds, Runs, Twins, and Gaps". (End)
Primes generated by sequence A040976. - Odimar Fabeny, Jul 12 2010
Primes of the form 2*n - 3 with 2*n - 1 prime n > 2. Primes of the form (n^2 - (n-2)^2)/2 - 1 with (n^2 - (n-2)^2)/2 + 1 prime so sum of two consecutive odd numbers/2 - 1. - Pierre CAMI, Jan 02 2012
Conjecture: For any integers n >= m > 0, there are infinitely many integers b > a(n) such that the number Sum_{k=m..n} a(k)*b^(n-k) (i.e., (a(m), ..., a(n)) in base b) is prime; moreover, when m = 1 there is such an integer b < (n+6)^2. - Zhi-Wei Sun, Mar 26 2013
Except for the initial 3, all terms are congruent to 5 mod 6. One consequence of this is that no term of this sequence appears in A030459. - Alonso del Arte, May 11 2013
Aside from the first term, all terms have digital root 2, 5, or 8. - J. W. Helkenberg, Jul 24 2013
The sequence provides all solutions to the generalized Winkler conjecture (A051451) aside from all multiples of 6. Specifically, these solutions start from n = 3 as a(n) - 3. This gives 8, 14, 26, 38, 56, ... An example from the conjecture is solution 38 from twin prime pairs (3, 5), (41, 43). - Bill McEachen, May 16 2014
Conjecture: a(n)^(1/n) is a strictly decreasing function of n. Namely a(n+1)^(1/(n+1)) < a(n)^(1/n) for all n. This conjecture is true for all a(n) <= 1121784847637957. - Jahangeer Kholdi and Farideh Firoozbakht, Nov 21 2014
a(n) are the only primes, p(j), such that (p(j+m) - p(j)) divides (p(j+m) + p(j)) for some m > 0, where p(j) = A000040(j). For all such cases m=1. It is easy to prove, for j > 1, the only common factor of (p(j+m) - p(j)) and (p(j+m) + p(j)) is 2, and there are no common factors if j = 1. Thus, p(j) and p(j+m) are twin primes. Also see A067829 which includes the prime 3. - Richard R. Forberg, Mar 25 2015
Primes prime(k) such that prime(k)! == 1 (mod prime(k+1)) with the exception of prime(991) = 7841 and other unknown primes prime(k) for which (prime(k)+1)*(prime(k)+2)*...*(prime(k+1)-2) == 1 (mod prime(k+1)) where prime(k+1) - prime(k) > 2. - Thomas Ordowski and Robert Israel, Jul 16 2016
For the twin prime criterion of Clement see the link. In Ribenboim, pp. 259-260 a more detailed proof is given. - Wolfdieter Lang, Oct 11 2017
Conjecture: Half of the twin prime pairs can be expressed as 8n + M where M > 8n and each value of M is a distinct composite integer with no more than two prime factors. For example, when n=1, M=21 as 8 + 21 = 29, the lesser of a twin prime pair. - Martin Michael Musatov, Dec 14 2017
For a discussion of bias in the distribution of twin primes, see my article on the Vixra web site. - Waldemar Puszkarz, May 08 2018
Since 2^p == 2 (mod p) (Fermat's little theorem), these are primes p such that 2^p == q (mod p), where q is the next prime after p. - Thomas Ordowski, Oct 29 2019, edited by M. F. Hasler, Nov 14 2019
The yet unproved "Twin Prime Conjecture" states that this sequence is infinite. - M. F. Hasler, Nov 14 2019
Lesser of the twin primes are the set of elements that occur in both A162566, A275697. Proof: A prime p will only have integer solutions to both (p+1)/g(p) and (p-1)/g(p) when p is the lesser of a twin prime, where g(p) is the gap between p and the next prime, because gcd(p+1,p-1) = 2. - Ryan Bresler, Feb 14 2021
From Lorenzo Sauras Altuzarra, Dec 21 2021: (Start)
J. A. Hervás Contreras observed the subsequence 11, 311, 18311, 1518311, 421518311... (see the links), which led me to conjecture the following statements.
I. If i is an integer greater than 2, then there exist positive integers j and k such that a(j) equals the concatenation of 3k and a(i).
II. If k is a positive integer, then there exist positive integers i and j such that a(j) equals the concatenation of 3k and a(i).
III. If i, j, and r are positive integers such that i > 2 and a(j) equals the concatenation of r and a(i), then 3 divides r. (End)

References

  • Milton Abramowitz and Irene A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 870.
  • T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 6.
  • Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §3.2 Prime Numbers, p. 81.
  • Paulo Ribenboim, The New Book of Prime Number Records, Springer-Verlag NY 1996, pp. 259-260.
  • Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See pp. 192-197.
  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
  • James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, pages 111-112.

Links

Crossrefs

Subsequence of A003627.
Cf. A001223, A006512 (greater of twin primes), A014574, A001097, A077800, A002822, A040040, A054735, A067829, A082496, A088328, A117078, A117563, A074822, A071538, A007508, A146214, A350246, A350247.
Cf. A104272 (Ramanujan primes), A178127 (lesser of twin Ramanujan primes), A178128 (lesser of twin primes if it is a Ramanujan prime).
Cf. A010051, A000040.

Programs

  • Haskell
    a001359 n = a001359_list !! (n-1)
a001359_list = filter ((== 1) . a010051' . (+ 2)) a000040_list
-- Reinhard Zumkeller, Feb 10 2015
  • Magma
    [n: n in PrimesUpTo(1610) | IsPrime(n+2)];  // Bruno Berselli, Feb 28 2011
  • Maple
    select(k->isprime(k+2),select(isprime,[$1..1616])); # Peter Luschny, Jul 21 2009
A001359 := proc(n)
   option remember;
   if n = 1
      then 3;
   else
      p := nextprime(procname(n-1)) ;
      while not isprime(p+2) do
         p := nextprime(p) ;
      end do:
      p ;
   end if;
end proc: # R. J. Mathar, Sep 03 2011
  • Mathematica
    Select[Prime[Range[253]], PrimeQ[# + 2] &] (* Robert G. Wilson v, Jun 09 2005 *)
a[n_] := a[n] = (p = NextPrime[a[n - 1]]; While[!PrimeQ[p + 2], p = NextPrime[p]]; p); a[1] = 3; Table[a[n], {n, 51}]  (* Jean-François Alcover, Dec 13 2011, after R. J. Mathar *)
nextLesserTwinPrime[p_Integer] := Block[{q = p + 2}, While[NextPrime@ q - q > 2, q = NextPrime@ q]; q]; NestList[nextLesserTwinPrime@# &, 3, 50] (* Robert G. Wilson v, May 20 2014 *)
Select[Partition[Prime[Range[300]],2,1],#[[2]]-#[[1]]==2&][[All,1]] (* Harvey P. Dale, Jan 04 2021 *)
q = Drop[Prepend[p = Prime[Range[100]], 2], -1];
Flatten[q[[#]] & /@ Position[p - q, 2]] (* Horst H. Manninger, Mar 28 2021 *)
  • PARI
    A001359(n,p=3) = { while( p+2 < (p=nextprime( p+1 )) || n-->0,); p-2}
/* The following gives a reasonably good estimate for any value of n from 1 to infinity; compare to A146214. */
A001359est(n) = solve( x=1,5*n^2/log(n+1), 1.320323631693739*intnum(t=2.02,x+1/x,1/log(t)^2)-log(x) +.5 - n)
/* The constant is A114907; the expression in front of +.5 is an estimate for A071538(x) */ \\  M. F. Hasler, Dec 10 2008
  • Python
    from sympy import primerange, isprime
print([n for n in primerange(1, 2001) if isprime(n + 2)]) # Indranil Ghosh, Jul 20 2017

Formula

a(n) = A077800(2n-1).
A001359 = { n | A071538(n-1) = A071538(n)-1 }; A071538(A001359(n)) = n. - M. F. Hasler, Dec 10 2008
A001359 = { prime(n) : A069830(n) = A087454(n) }. - Juri-Stepan Gerasimov, Aug 23 2011
a(n) = prime(A029707(n)). - R. J. Mathar, Feb 19 2017

A006512 Greater of twin primes.

Original entry on oeis.org

5, 7, 13, 19, 31, 43, 61, 73, 103, 109, 139, 151, 181, 193, 199, 229, 241, 271, 283, 313, 349, 421, 433, 463, 523, 571, 601, 619, 643, 661, 811, 823, 829, 859, 883, 1021, 1033, 1051, 1063, 1093, 1153, 1231, 1279, 1291, 1303, 1321, 1429, 1453, 1483, 1489, 1609
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

Also primes that are the sum of two primes (which is possible only if 2 is one of the primes). - Cino Hilliard, Jul 02 2004, edited by M. F. Hasler, Nov 14 2019
The set of greater of twin primes larger than five is a proper subset of the set of primes of the form 3n + 1 (A002476). - Paul Muljadi, Jun 05 2008
Smallest prime > n-th isolated composite. - Juri-Stepan Gerasimov, Nov 07 2009
Subsequence of A175075. Union of a(n) and sequence A175080 is A175075. - Jaroslav Krizek, Jan 30 2010
A164292(a(n))=1; A010051(a(n)+2)=0 for n > 1. - Reinhard Zumkeller, Mar 29 2010
Omega(n) = Omega(n-2); d(n) = d(n-2). - Juri-Stepan Gerasimov, Sep 19 2010
Aside from the first term, all subsequent terms have digital root 1, 4, or 7. - J. W. Helkenberg, Jul 24 2013
Also primes p with property that the sum of the successive gaps between primes <= p is a prime number. - Robert G. Wilson v, Dec 19 2014
The phrase "x is an element of the {primes, positive integers} and there {exist no, exist} elements a,b of {1 and primes, primes}: a+b=x" determines A133410, A067829, A025584, A006512, A166081, A014092, A014091 and A038609 for the first few hundred terms with only de-duplication or omitting/including 3, 4 and 6 in the case of A166081/A014091 and one case of omitting/including 3 given 1 isn't prime. - Harry G. Coin, Nov 25 2015
The yet unproved Twin Prime Conjecture states that this sequence is infinite. - M. F. Hasler, Nov 14 2019

References

  • See A001359 for further references and links.
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Subsequence of A139690.
Bisection of A077800.
Cf. A001097, A001359, A014574, A067829, A002476.

Programs

  • Haskell
    a006512 = (+ 2) . a001359 -- Reinhard Zumkeller, Feb 10 2015
  • Magma
    [n: n in PrimesUpTo(1610)|IsPrime(n-2)]; // Bruno Berselli, Feb 28 2011
  • Maple
    for i from 1 to 253 do if ithprime(i+1) = ithprime(i) + 2 then print({ithprime(i+1)}); fi; od; # Zerinvary Lajos, Mar 19 2007
P := select(isprime,[$1..1609]): select(p->member(p-2,P),P); # Peter Luschny, Mar 03 2011
A006512 := proc(n)
    2+A001359(n) ;
end proc: # R. J. Mathar, Nov 26 2014
  • Mathematica
    Select[Prime[Range[254]], PrimeQ[# - 2] &] (* Robert G. Wilson v, Jun 09 2005 *)
Transpose[Select[Partition[Prime[Range[300]], 2, 1], Last[#] - First[#] == 2 &]][[2]] (* Harvey P. Dale, Nov 02 2011 *)
Cases[Prime[Range[500]] + 2, ?PrimeQ] (* _Fred Patrick Doty, Aug 23 2017 *)
  • PARI
    select(p->isprime(p-2),primes(1000))
  • PARI
    a(n)=p=3; while(p+2 < (p=nextprime(p+1)) || n-->0, ); p
vector(100, n, a(n)) \\ Altug Alkan, Dec 04 2015
  • Python
    from sympy import primerange, isprime
print([n for n in primerange(1, 2001) if isprime(n - 2)]) # Indranil Ghosh, Jul 20 2017

A092402 Primes of the form p+8 where p is a prime.

Original entry on oeis.org

11, 13, 19, 31, 37, 61, 67, 79, 97, 109, 139, 157, 181, 199, 241, 271, 277, 367, 397, 409, 439, 457, 487, 499, 571, 577, 601, 607, 661, 691, 709, 727, 751, 769, 829, 919, 937, 991, 1021, 1039, 1069, 1117, 1171, 1201, 1231, 1237, 1291, 1297, 1327, 1381, 1447
Offset: 1

Views

Author

Douglas Winston (douglas.winston(AT)srupc.com), Mar 22 2004

Keywords

Links

Crossrefs

Select primes from A000040 + 8.
Cf. A067829 A067830 A067831.

Programs

  • Mathematica
    Select[Prime[Range[5,2000]],PrimeQ[#-8]&] (* Vincenzo Librandi, Jul 14 2012 *)

A133410 Least prime p such that p-6*n is prime.

Original entry on oeis.org

2, 11, 17, 23, 29, 37, 41, 47, 53, 59, 67, 71, 79, 83, 89, 97, 101, 107, 113, 127, 127, 131, 137, 149, 149, 157, 163, 167, 173, 179, 191, 191, 197, 211, 211, 223, 223, 227, 233, 239, 251, 251, 257, 263, 269, 277, 281, 293, 293, 307, 307, 311, 317, 331, 331, 337
Offset: 0

Views

Author

Pierre CAMI, Nov 25 2007

Keywords

Comments

If duplicates are omitted, this is the sequence of primes p such that all p - phi(k) - 1 are composite for 1 <= phi(k)-1 < p. - Michel Lagneau, Sep 14 2012
If duplicates are omitted, the given entries equal A025584 (p: p-2 is not a prime) except A025584 includes 3 (since 1 is not prime). - Harry G. Coin, Nov 29 2015

Links

Crossrefs

Cf. A025584, A067829 (complement w.r.t. primes), A133387.

Programs

  • Maple
    Primes:= select(isprime,{2,seq(i,i=3..10^4,2)}):
seq(min(Primes intersect map(`+`,Primes,6*n)),n=0..1000); # Robert Israel, Nov 30 2015
  • Mathematica
    a={};Do[i=6*n+1; While[Not[PrimeQ[i]&&PrimeQ[i-6*n]],i++ ];AppendTo[a,i],{n,0,60}]; a (* Stefan Steinerberger, Nov 26 2007 *)
Table[Module[{p=NextPrime[6n]},While[!PrimeQ[p-6n],p=NextPrime[p]];p],{n,0,60}] (* Harvey P. Dale, Apr 07 2025 *)
  • PARI
    a(n) = {k=1; while(k, if(ispseudoprime(prime(k)-6*n), return(prime(k))); k++)} \\ Altug Alkan, Dec 04 2015

Extensions

More terms from Stefan Steinerberger, Nov 26 2007

A098933 Primes of the form p+14, where p is a prime.

Original entry on oeis.org

17, 19, 31, 37, 43, 61, 67, 73, 97, 103, 127, 151, 163, 181, 193, 211, 241, 271, 277, 283, 307, 331, 367, 373, 397, 433, 457, 463, 523, 571, 577, 601, 607, 613, 631, 661, 673, 691, 733, 757, 787, 811, 823, 853, 877, 967, 991, 997, 1033, 1063, 1117, 1123, 1201
Offset: 1

Views

Author

Douglas Winston (douglas.winston(AT)srupc.com), Oct 20 2004

Keywords

Crossrefs

Cf. A000040 A006512 A067829 A094743 A046132 A067830 A046117 A067831 A092402 A092146 A092216.

Programs

  • PARI
    isok(n) = isprime(n) && isprime(n - 14) \\ Michel Marcus, Jul 17 2013
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