A065112 -id:A065112 - cp's OEIS frontend

A030670 Smallest prime formed by appending a number to the n-th prime.

23, 31, 53, 71, 113, 131, 173, 191, 233, 293, 311, 373, 419, 431, 479, 5323, 593, 613, 673, 719, 733, 797, 839, 8923, 971, 1013, 1031, 10711, 1091, 11311, 1277, 1319, 1373, 1399, 1493, 1511, 1571, 1637, 16729, 1733, 17911, 1811, 1913, 1931

Offset: 1

Patrick De Geest

Previous name: Smallest prime whose decimal expansion begins (nontrivially) with the n-th prime.
Add digits to p (starting with a nonzero digit) until another prime is reached.
This differs from A064792 in that there the appended digits may start with a 0. The first difference occurs at a(16) = 5323, while A064792(16) = 5303. - M. F. Hasler, Jan 15 2025

			a(16) = 5323 because 53 is the 16th prime, and 23 is the smallest number that can be appended to 53 to give another prime. 5303 is not allowed because 03 starts with zero. - _David Radcliffe_, Jan 08 2025

Robert Israel, Table of n, a(n) for n = 1..10000

See A064792 for another version. Note that A064792 <= a(n). Cf. A065112, A178220.

f:= proc(p) local d,x;
  for d from 1 do
    x:= nextprime(10^d*p+10^(d-1)-1);
    if x < 10^d*(p+1) then return x fi
  od
end proc:
map(f @ ithprime, [$1..100]); # Robert Israel, Aug 12 2018

f[n_] := Block[{k = 1, p = Prime@ n}, While[a = 10^Floor[1 + Log10@ k] p + k; !PrimeQ@ a, k += 2]; a]; Array[f, 44]

apply( {A030670(n)=n=prime(n);for(L=1,oo, n*=10; forstep(s=bitor(10^(L-1),1),10^L-1,2, isprime(n+s)&& return(n+s)))}, [1..44]) \\ M. F. Hasler, Jan 15 2025

from sympy import prime, isprime
from itertools import count
def a030670(n):
  p = str(prime(n))
  return next(x for k in count(1) if isprime(x:=int(p+str(k)))) # David Radcliffe, Jan 08 2025

Title changed by David Radcliffe, Jan 08 2025

A088606 Smallest number k such that concatenation of k and prime(n) is a prime, or 0 if no other number exists. a(1) = a(3) = 0.

Original entry on oeis.org

0, 1, 0, 1, 2, 1, 3, 4, 2, 2, 1, 1, 2, 4, 3, 3, 3, 4, 1, 2, 1, 1, 2, 3, 1, 5, 1, 5, 1, 2, 4, 2, 2, 4, 11, 1, 4, 1, 3, 6, 2, 1, 3, 1, 5, 6, 4, 1, 5, 1, 5, 2, 4, 2, 3, 6, 2, 3, 1, 2, 1, 2, 1, 2, 3, 6, 3, 4, 2, 4, 6, 3, 1, 1, 6, 2, 2, 4, 12, 1, 5, 4, 5, 1, 1, 5, 3, 3, 3, 3, 2, 5, 1, 3, 1, 2, 17, 2, 1, 3, 3, 2, 5, 5

Offset: 1

Amarnath Murthy, Oct 15 2003

Subsidiary sequences: (set(1)) Index of the start of the first occurrence of a string of n consecutive 1's or 2's or 3's etc. (set (2)): a(n) = smallest prime such that concatenation of 1 with n successive primes starting from a(n) gives primes in each case. (n primes are obtained.) Similarly for 2, 3, etc. Conjecture: The subsidiary sequences are infinite.

A065112(n) = a(n) concatenated with prime(n). - Bill McEachen, May 27 2021

Cf. A096915, A065112.

a(n) = if ((n==1) || (n==3), 0, my(k=1); while (!isprime(eval(Str(k, prime(n)))), k++); k); \\ Michel Marcus, Jul 11 2021

More terms from Ray Chandler, Oct 18 2003

A064792 Append more digits to the n-th prime (leading zeros are permitted) until another prime is reached.

Original entry on oeis.org

23, 31, 53, 71, 113, 131, 173, 191, 233, 293, 311, 373, 419, 431, 479, 5303, 593, 613, 673, 719, 733, 797, 839, 8923, 971, 1013, 1031, 10709, 1091, 11311, 1277, 1319, 1373, 1399, 1493, 1511, 1571, 1637, 16703, 1733, 17903, 1811, 1913, 1931, 1973

Offset: 1

Reinhard Zumkeller, Oct 17 2001

Differs from A030670 in that here the appended digits may start with 0 which is not possible there (although zeros are allowed there, too, if they are not the first appended digit): The first difference appears at a(16) = 5303 where A030670(16) = 5323. - M. F. Hasler, Jan 15 2025

Daniel Mondot, Table of n, a(n) for n = 1..10000

Cf. A030665.

See A030670 for another version. Note that A030670 >= a(n). Cf. A065112.

apply( {A064792(n)=n=prime(n);for(L=1,oo, n*=10; forstep(s=1,10^L-1,2, isprime(n+s)&& return(n+s)))}, [1..44]) \\ M. F. Hasler, Jan 15 2025

a(n) = A030665(prime(n)). - Michel Marcus, Jan 17 2025

cp's OEIS Frontend

A030670 Smallest prime formed by appending a number to the n-th prime.

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A088606 Smallest number k such that concatenation of k and prime(n) is a prime, or 0 if no other number exists. a(1) = a(3) = 0.

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A064792 Append more digits to the n-th prime (leading zeros are permitted) until another prime is reached.

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