A344936 -id:A344936 - cp's OEIS frontend

A344937 a(n) is the largest k such that when strings of zeros of lengths t = 1..k are inserted between every pair of adjacent digits of prime(n), the resulting numbers are all primes.

1, 1, 1, 3, 0, 0, 0, 1, 2, 0, 0, 2, 2, 1, 2, 4, 0, 1, 0, 2, 4, 0, 0, 1, 1, 2, 0, 3, 0, 0, 0, 1, 0, 0, 2, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0

Offset: 5

Felix Fröhlich, Jun 03 2021

Initially, except for n = 1..4, similar to A290174, but the two sequences differ from n = 28 onwards.

			For n = 8: prime(8) = 19 and the numbers 109, 1009 and 10009 are all prime, while 100009 is not. Thus it is possible to insert strings of zeros of lengths 1, 2 and 3 between all adjacent digits of 19 such that the resulting number is prime. Since 3 is the largest length of such a string in case of 19, a(8) = 3.

Cf. A290174, A344637, A344936.

Table[k=0;While[PrimeQ@FromDigits@Flatten@Riffle[IntegerDigits@Prime@n,{Table[0,k]}],k++];k-1,{n,5,100}] (* Giorgos Kalogeropoulos, Jun 03 2021 *)

eva(n) = subst(Pol(n), x, 10)
insert_zeros(num, len) = my(d=digits(num), v=[]); for(k=1, #d-1, v=concat(v, concat([d[k]], vector(len)))); v=concat(v, d[#d]); eva(v)
a(n) = my(p=prime(n), ip=p); for(k=1, oo, ip=insert_zeros(p, k); if(!ispseudoprime(ip), return(k-1)))

from sympy import isprime, prime
def insert_zeros(n, k): return int(("0"*k).join(list(str(n))))
def a(n):
  pn, k = prime(n), 1
  while isprime(insert_zeros(pn, k)): k += 1
  return k - 1
print([a(n) for n in range(5, 92)]) # Michael S. Branicky, Jun 03 2021

A341899 a(n) is the smallest prime p > 10 such that when strings of n zeros are inserted between every pair of adjacent digits the result is also a prime.

Original entry on oeis.org

11, 19, 17, 13, 13, 23, 17, 17, 31, 13, 23, 41, 127, 61, 23, 13, 13, 67, 53, 89, 19, 227, 17, 29, 61, 151, 31, 37, 107, 53, 1741, 263, 167, 23, 31, 89, 61, 13, 43, 241, 53, 347, 1319, 19, 79, 419, 521, 19, 809, 677, 97, 97, 1223, 89, 13, 79, 67, 257, 17, 499

Offset: 1

Felix Fröhlich, Jun 04 2021

First differs from A306920 at n = 13.

a(n) = A306920(n) if A306920(n) is < 100, i.e., is a two-digit number.

			For n = 13: Inserting 13 zeros between all adjacent digits of 127 gives 10000000000000200000000000007, which is prime. Since 127 is the smallest prime where inserting exactly 13 zeros between all adjacent digits results in a number that is also prime, a(13) = 127.

Cf. A306920, A344637, A344936, A344937.

eva(n) = subst(Pol(n), x, 10)
insert_zeros(num, len) = my(d=digits(num), v=[]); for(k=1, #d-1, v=concat(v, concat([d[k]], vector(len)))); v=concat(v, d[#d]); eva(v)
a(n) = forprime(p=10, , if(ispseudoprime(insert_zeros(p, n)), return(p)))

cp's OEIS Frontend

A344937 a(n) is the largest k such that when strings of zeros of lengths t = 1..k are inserted between every pair of adjacent digits of prime(n), the resulting numbers are all primes.

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