A088712 -id:A088712

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A088784

Primes formed by concatenating a prime with the preceding prime.

+10
6

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

K. D. Bajpai, Table of n, a(n) for n = 1..10300

EXAMPLE

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

MATHEMATICA

concatpr[n_]:=FromDigits[Join[IntegerDigits[n], IntegerDigits[ NextPrime[ n, -1]]]]; Select[concatpr/@Prime[Range[400]], PrimeQ] (* Harvey P. Dale, May 12 2011 *)

PROG

(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

CROSSREFS

Cf. A088712.

KEYWORD

base,nonn

AUTHOR

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

EXTENSIONS

Terms a(30), a(31) and a(32) added by K. D. Bajpai, Aug 14 2014

STATUS

approved

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.

+10
1

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

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.

LINKS

Zak Seidov, Table of n, a(n) for n = 1..59542

EXAMPLE

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.

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 *)

PROG

(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

CROSSREFS

Cf. A088712, A088784.

KEYWORD

nonn,base

AUTHOR

Zak Seidov, Mar 07 2016

STATUS

approved

A178466

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

+10
0

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

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.

LINKS

Table of n, a(n) for n=1..44.

FORMULA

a(n) = A151799(A088712(n)).

EXAMPLE

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

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 *)

CROSSREFS

Cf. A030459, A030460, A088712.

KEYWORD

nonn,base

AUTHOR

Carmine Suriano, Jan 27 2011

STATUS

approved

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