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 1-6 of 6 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

A077005 Smallest k such that prime(n) divides k*prime(n-1) + 1, n > 1.

Original entry on oeis.org

1, 3, 4, 3, 7, 13, 10, 6, 5, 16, 31, 31, 22, 12, 9, 10, 31, 56, 18, 37, 66, 21, 15, 85, 76, 52, 27, 55, 85, 118, 33, 23, 70, 15, 76, 131, 136, 42, 29, 30, 91, 172, 97, 148, 100, 88, 93, 57, 115, 175, 40, 121, 226, 43, 44, 45, 136, 231, 211, 142, 88, 22, 78, 157, 238, 71, 281
Offset: 2

Views

Author

Amarnath Murthy, Oct 26 2002

Keywords

Comments

a(n) = inverse of (prime(n)-prime(n-1)) mod prime(n). This is the least k such that prime(n)|k*((prime(n)-prime(n-1))-1). Since prime(n)|k*prime(n), it must divide (k*prime(n-1)+1), so k = a(n). Also, a(n) = prime(n) - (x*prime(n)+1)/prime(n-1) for the least such x. - David James Sycamore, Oct 05 2018

Examples

			a(4) = 3 as prime(5) = 11 divides 3*7 + 1, where 7 = prime(4).

Crossrefs

Cf. A069830.

Programs

  • Mathematica
    sk[a_,b_]:=Module[{k=1},While[!Divisible[k*a+1,b],k++];k]; sk@@@ Partition[ Prime[Range[70]],2,1] (* Harvey P. Dale, Jun 23 2013 *)
  • PARI
    a(n) = {my(k = 1, p = prime(n-1), q = prime(n)); while ((k*p+1) % q, k++); k;} \\ Michel Marcus, Aug 14 2018

Formula

a(n) = prime(n) - A069830(n - 1). - Emmanuel Vantieghem, Aug 12 2018 [Corrected by Georg Fischer, Sep 21 2024]

Extensions

More terms from Ralf Stephan, Oct 31 2002
More terms from Ray Chandler, Oct 24 2003

A087454 Multiplicative inverse of the n-th prime prime(n) modulo prime(n-1).

Original entry on oeis.org

1, 2, 3, 2, 6, 10, 9, 5, 4, 15, 26, 28, 21, 11, 8, 9, 30, 51, 17, 36, 61, 20, 14, 78, 73, 51, 26, 54, 82, 105, 32, 22, 69, 14, 75, 126, 131, 41, 28, 29, 90, 163, 96, 145, 99, 83, 88, 56, 114, 172, 39, 120, 217, 42, 43, 44, 135, 226, 208, 141, 85, 21, 77, 156, 235, 68, 276
Offset: 2

Views

Author

Lekraj Beedassy, Oct 23 2003

Keywords

Examples

			We have a(7) = 10 because prime(7)*10 = 170 = 1 [mod 13] = 1 [mod prime(6)].

Links

Crossrefs

Cf. A069830, A077005.

Programs

  • Magma
    [InverseMod(NthPrime(n), NthPrime(n-1)): n in [2..70]]; // G. C. Greubel, Aug 09 2019
  • Maple
    seq(`mod`(1/ithprime(n), ithprime(n-1)), n = 2..70); # G. C. Greubel, Aug 09 2019
  • Mathematica
    Table[PowerMod[Prime[n], -1, Prime[n - 1]], {n, 2, 68}] (* Geoffrey Critzer, May 16 2015 *)
  • PARI
    a(n)=my(p=prime(n-1),g=nextprime(p+1)-p);lift(Mod(1/g,p)) \\ Charles R Greathouse IV, Aug 23 2011
  • Sage
    [nth_prime(n).inverse_mod(nth_prime(n-1)) for n in (2..70)] # G. C. Greubel, Aug 09 2019

Extensions

Corrected and extended by Ray Chandler, Oct 24 2003

A073603 Smallest multiple of n-th prime which is == 1 mod (n+1)-st prime.

Original entry on oeis.org

4, 6, 15, 56, 66, 52, 153, 323, 552, 435, 186, 370, 861, 1505, 2068, 2597, 1770, 671, 3551, 2556, 949, 4898, 6142, 1068, 2425, 5151, 8240, 5778, 3052, 1017, 12446, 14934, 9453, 18626, 11175, 3926, 4239, 20375, 24048, 25777, 16110, 3439, 18336
Offset: 1

Views

Author

Amarnath Murthy, Aug 04 2002

Keywords

Comments

If n is in A001359, a(n) = prime(n)*(prime(n)+1)/2. - Robert Israel, Apr 21 2021

Examples

			a(4) = 56 which is 8*7 and 56 ==1 (mod 11)

Links

Crossrefs

Cf. A001359, A069830.

Programs

  • Maple
    for i from 1 to 200 do a := msolve(ithprime(i)*n=1,ithprime(i+1)); b[i] := rhs(convert(a,list)[1])*ithprime(i); od:seq(b[k],k=1..200);
# alternative
f:= (p,q)->(1/p mod q)*p:
seq(f(ithprime(i),ithprime(i+1)),i=1..100); # Robert Israel, Apr 21 2021
  • Mathematica
    p = 2; Table[q = NextPrime[p]; i = 2; While[Mod[y = i*p, q] != 1, i++]; p = q; y, {n, 43}] (* Jayanta Basu, Jul 02 2013 *)

Formula

a(n) = prime(n)*A069830(n). - Robert Israel, Apr 21 2021

Extensions

Corrected and extended by Sascha Kurz, Aug 10 2002

A194367 Smallest k such that prime(n) divides k*prime(n+1)+1.

Original entry on oeis.org

1, 1, 2, 5, 5, 3, 8, 14, 19, 14, 5, 9, 20, 32, 39, 44, 29, 10, 50, 35, 12, 59, 69, 11, 24, 50, 77, 53, 27, 8, 95, 109, 68, 125, 74, 25, 26, 122, 139, 144, 89, 18, 95, 48, 98, 116, 123, 167, 113, 57
Offset: 1

Views

Author

Juri-Stepan Gerasimov, Aug 23 2011

Keywords

Examples

			a(4) = 5 as prime(4)=7 divides 5*11+1, where 11=prime(5).
a(7) = 8 = (17*9-1)/19. - _Bob Selcoe_, Aug 21 2016

Links

Crossrefs

Cf. A077005.
Cf. A000040 (prime numbers), A069830.

Programs

  • Maple
    seq(-ithprime(i+1)^(-1) mod ithprime(i),i=1..100); # Robert Israel, Aug 25 2016
  • Mathematica
    Table[k = 1; While[! Divisible[k Prime[n + 1] + 1, Prime@ n], k++]; k, {n, 50}] (* Michael De Vlieger, Aug 22 2016 *)
  • PARI
    a(n)=my(p=prime(n),q=nextprime(p+1));lift(Mod(-1,p)/q) \\ Charles R Greathouse IV, Sep 03 2011

Formula

a(n) = (prime(n)*A069830(n) - 1)/prime(n+1). - Bob Selcoe, Aug 21 2016

A309578 Multiplicative inverse of Fibonacci(prime(n)) modulo Fibonacci(prime(n+1)).

Original entry on oeis.org

1, 3, 8, 48, 144, 329, 2584, 15456, 104005, 832040, 1866294, 34111385, 267914296, 1602508992, 10783446409, 193501094490, 1548008755920, 3472236254411, 166151337293088, 498454011879264, 1118049290473933, 53500214605455696, 360008399296352015, 2460986135945634432
Offset: 1

Views

Author

Dragos Ristache, Aug 08 2019

Keywords

Comments

Since Fibonacci numbers have the property that gcd(x,y) = gcd(Fibonacci(x), Fibonacci(y)), the modular inverse will always exist for this sequence.

Crossrefs

Cf. A000045, A030426, A069830.

Programs

  • Maple
    a:= n-> (f-> (1/f(n) mod f(n+1)))(j->combinat[fibonacci](ithprime(j))):
seq(a(n), n=1..25);  # Alois P. Heinz, Aug 12 2019
  • Mathematica
    Table[ModularInverse[Fibonacci[Prime[n]], Fibonacci[Prime[n+1]]], {n, 20}]
  • PARI
    a(n)={lift(1/(Mod(fibonacci(prime(n)), fibonacci(prime(n+1)))))} \\ Andrew Howroyd, Aug 12 2019

Formula

a(n) = invmod(Fibonacci(prime(n)), Fibonacci(prime(n+1))).
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Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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