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 1-3 of 3 results.

A088784 Primes formed by concatenating a prime with the preceding prime.

Original entry on oeis.org

53, 5347, 5953, 6761, 137131, 179173, 211199, 223211, 239233, 263257, 359353, 541523, 593587, 613607, 631619, 653647, 659653, 757751, 809797, 977971, 997991, 1009997, 11091103, 11291123, 12371231, 13991381, 15591553, 17831777, 19311913, 19791973, 19931987, 23092297
Offset: 1

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Author

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

Keywords

Examples

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

Links

Crossrefs

Cf. A088712.

Programs

  • Mathematica
    concatpr[n_]:=FromDigits[Join[IntegerDigits[n],IntegerDigits[ NextPrime[ n,-1]]]]; Select[concatpr/@Prime[Range[400]],PrimeQ] (* Harvey P. Dale, May 12 2011 *)
  • PARI
    for(n=1,10^3,p=prime(n); q=concat(Str(p),Str(precprime(p-1))); if(isprime(eval(q)), print1(q,", "))) \\ Derek Orr, Aug 14 2014
  • Python
    from itertools import islice
from sympy import isprime, nextprime
def agen(): # generator of terms
    p, pstr = 2, "2"
    while True:
        q = nextprime(p)
        qstr = str(q)
        t = int(qstr + pstr)
        if isprime(t):
            yield t
        p, pstr = q, qstr
print(list(islice(agen(), 32))) # Michael S. Branicky, Jan 05 2022

Extensions

Terms a(30), a(31) and a(32) added by K. D. Bajpai, Aug 14 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 1-3 of 3 results.

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