A066065 -id:A066065 - cp's OEIS frontend

A066064 a(n) = p.q in decimal notation where p = prime(n) and q is the smallest prime (A066065(n)) such that the concatenation p.q is a prime.

23, 37, 53, 73, 113, 137, 173, 193, 233, 293, 313, 373, 4111, 433, 4723, 5323, 593, 613, 673, 7129, 733, 797, 8311, 8923, 977, 1013, 1033, 10711, 1093, 11311, 1277, 13147, 1373, 13913, 1493, 15131, 15731, 1637, 16729, 1733, 17911, 18119, 1913

Offset: 1

Reinhard Zumkeller, Dec 01 2001

			A000040(2) = 3 and as 32, 33 and 35 are composite, the next prime 7 = A066065(2) yields a(2) = 37.

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A066065.

Table[Block[{q = 3, d = IntegerDigits[p], k}, While[! PrimeQ@ Set[k, FromDigits[Join[d, IntegerDigits[q]]]], q = NextPrime@ q]; k], {p, Prime@ Range@ 43}] (* Michael De Vlieger, Jun 19 2018 *)

a(n) = { my(p=prime(n)); forprime(q=3, oo, my(c=p*10^(logint(q,10)+1) + q); if(isprime(c), return(c))) } \\ Harry J. Smith, Nov 09 2009

A171154 Smallest prime whose decimal expansion begins with concatenation of first n primes in descending order.

Original entry on oeis.org

2, 3203, 5323, 75323, 11753221, 131175329, 171311753203, 19171311753229, 231917131175321, 292319171311753231, 3129231917131175327, 3731292319171311753239, 41373129231917131175321, 43413731292319171311753233, 4743413731292319171311753269, 534743413731292319171311753223

Offset: 1

Ulrich Krug (leuchtfeuer37(AT)gmx.de), Dec 04 2009

Sequence is conjectured to be infinite.
a(n) = "prime(n)...prime(1) R(n)".
R(n) for n>1: 03, 3, 3, 21, 9, 03, 29, 1, 31, 7, 39, 1, 33, 69, 23, 3, 59, 27, ...
It is conjectured that R(n)=1 for infinite many n.

			a(1) = 2 = prime(1) is the exceptional case, because no R(1).
a(2) = 3203 = prime(453) = "32 03", R(2)="03".
a(5) = 11753221 = prime(772902) = "prime(5)...prime(1) 21", R(5)=21.

Leonard E. Dickson, History of the Theory of numbers, vol. I, Dover Publications 2005.

Michael S. Branicky, Table of n, a(n) for n = 1..298

Cf. A000040, A066065, A019518, A089710, A053546.

from sympy import isprime, primerange, prime
def a(n):
    if n == 1: return 2
    c = int("".join(map(str, [p for p in primerange(2, prime(n)+1)][::-1])))
    pow10 = 10
    while True:
        c *= 10
        for b in range(1, pow10, 2):
            if b%5 == 0: continue
            if isprime(c+b):
                return c+b
        pow10 *= 10
print([a(n) for n in range(1, 17)]) # Michael S. Branicky, Mar 12 2022

a(14) and beyond from Michael S. Branicky, Mar 12 2022

cp's OEIS Frontend

A066064 a(n) = p.q in decimal notation where p = prime(n) and q is the smallest prime (A066065(n)) such that the concatenation p.q is a prime.

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A171154 Smallest prime whose decimal expansion begins with concatenation of first n primes in descending order.

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