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.

A088712 Primes which when concatenated with the previous prime form a new prime.

Original entry on oeis.org

5, 53, 59, 67, 137, 179, 211, 223, 239, 263, 359, 541, 593, 613, 631, 653, 659, 757, 809, 977, 997, 1009, 1109, 1129, 1237, 1399, 1559, 1783, 1931, 1979, 1993, 2309, 2339, 2347, 2411, 2683, 2879, 3061, 3203, 3271, 3613, 3643, 3767, 4001
Offset: 1

Views

Author

Chuck Seggelin (barkeep(AT)plasteredDragon.com), Oct 15 2003

Keywords

Examples

			a(2) = 53 because 53 concatenated with the preceding prime (47) yields 5347 which is also prime.

Crossrefs

Cf. A088784.

Formula

If isprime(p(n) & p(n-1)) then sequence.add(p(n))

A133986 Consecutive prime pairs which after concatenation are prime (ascending or descending order).

Original entry on oeis.org

23, 53, 3137, 5347, 5953, 6761, 8389, 137131, 151157, 157163, 167173, 179173, 199211, 211199, 223211, 233239, 239233, 257263, 263257, 263269, 271277, 331337, 353359, 359353, 373379, 433439, 467479, 509521, 523541, 541523, 541547, 593587
Offset: 1

Views

Author

Enoch Haga, Sep 30 2007

Keywords

Comments

For the range above the prime concatenations occur after a mean of 3.1579 trials. Some primes are interlocking, as many as three in succession.

Examples

			5347 is a member: 53 and 47, prime; 4753 was also tested, but is composite.

Crossrefs

Cf. A088784.

Formula

Select consecutive prime pairs, concatenate and test for primality (ascending and descending order)

Extensions

Edited by Charles R Greathouse IV, Apr 26 2010

A242102 Semiprimes that are the concatenation of a prime and the previous prime.

Original entry on oeis.org

1713, 2319, 2923, 4341, 6159, 7167, 8983, 103101, 151149, 157151, 163157, 167163, 173167, 191181, 197193, 233229, 257251, 277271, 283281, 311307, 337331, 367359, 373367, 421419, 431421, 439433, 449443, 463461, 467463, 479467, 487479, 509503, 521509, 547541, 557547
Offset: 1

Views

Author

K. D. Bajpai, Aug 15 2014

Keywords

Examples

			13 and 17 are consecutive primes. Their reverse concatenation = 1713 = 3 * 571, which is semiprime.
19 and 23 are consecutive primes. Their reverse concatenation = 2319 = 3 * 773, which is semiprime.

Links

Crossrefs

Cf. A000040, A001358, A007796, A088784.

Programs

  • Maple
    select(k -> numtheory:-bigomega(k)=2, [seq(parse(cat(ithprime(n+1),ithprime(n))), n=1..200)]);
  • Mathematica
    A242102 = {}; Do[t = FromDigits[Flatten[IntegerDigits /@ {Prime[n], Prime[n - 1]}]]; If[PrimeOmega[t] == 2, AppendTo[A242102, t]], {n, 2, 200}]; A242102
  • PARI
    forprime(p=1,10^3,q=concat(Str(p),Str(precprime(p-1)));if(bigomega(eval(q))==2,print1(eval(q),", "))) \\ Derek Orr, Aug 15 2014

A269671 Integers n such that the concatenation of prime(n) and prime(n+1) and also concatenation of prime(n+1) and prime(n) are prime.

Original entry on oeis.org

46, 51, 55, 71, 99, 119, 164, 298, 345, 461, 509, 523, 588, 668, 779, 827, 844, 848, 999, 1100, 1151, 1215, 1306, 1321, 1408, 1553, 1568, 1616, 1779, 1900, 1931, 1953, 2102, 2150, 2221, 2444, 2653, 2677, 3116, 3405, 3527, 3731, 3776, 3890, 3898, 3989, 4070, 4188, 4257, 4546, 4556, 4574, 4681, 4694, 4846, 4947, 4948, 4974
Offset: 1

Views

Author

Zak Seidov, Mar 07 2016

Keywords

Comments

Difference between prime(n) and prime(n+1) is a multiple of 6, otherwise concatenation prime(n)//prime(n+1) is divisible by 3.

Examples

			prime(46)=199, prime(47)=211 and both 199211 and 211199 are prime,
prime(51)=233, prime(51)=239 and both 233239 and 239233 are prime,
prime(9999972)=179424263, prime(9999973)=179424269 and both 179424263179424269 and 179424269179424263 are prime.

Links

Crossrefs

Cf. A088712, A088784.

Programs

  • Mathematica
    PrimePi/@Select[Partition[Prime[Range[5000]],2,1],AllTrue[{FromDigits[ Join[ IntegerDigits[ #[[1]]],IntegerDigits[#[[2]]]]],FromDigits[ Join[ IntegerDigits[#[[2]]],IntegerDigits[#[[1]]]]]},PrimeQ]&][[All,1]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jun 14 2021 *)
  • PARI
    isok(n) = {my(sp = Str(prime(n))); my(sq = Str(prime(n+1))); isprime(eval(concat(sp, sq))) && isprime(eval(concat(sq, sp)));} \\ Michel Marcus, Mar 07 2016

A178466 Primes prime(k) such that the concatenation prime(k+1)//prime(k) is also prime.

Original entry on oeis.org

3, 47, 53, 61, 131, 173, 199, 211, 233, 257, 353, 523, 587, 607, 619, 647, 653, 751, 797, 971, 991, 997, 1103, 1123, 1231, 1381, 1553, 1777, 1913, 1973, 1987, 2297, 2333, 2341, 2399, 2677, 2861, 3049, 3191, 3259, 3607, 3637, 3761, 3989
Offset: 1

Views

Author

Carmine Suriano, Jan 27 2011

Keywords

Comments

53, 211, 653, 997, ... are also in A088712.
The role of the two primes is swapped in comparison to A030459.
The result of the concatenation is in A088784.

Examples

			The prime 53 is in the sequence because the next prime is 59 and 5953 is a prime.

Crossrefs

Cf. A030459, A030460, A088712.

Programs

  • Maple
    read("transforms") ;
for n from 1 to 600 do p := ithprime(n) ; q := nextprime(p) ; r := digcat2(q,p) ; if isprime(r) then printf("%d,",p) ; end if; end do: # R. J. Mathar, Jan 27 2011
  • Mathematica
    Transpose[Select[Partition[Prime[Range[600]],2,1],PrimeQ[FromDigits[ Flatten[ IntegerDigits/@Reverse[#]]]]&]][[1]]  (* Harvey P. Dale, Feb 02 2011 *)

Formula

a(n) = A151799(A088712(n)).
Showing 1-5 of 5 results.

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