A344637 -id:A344637 - cp's OEIS frontend

A344936 a(n) is the smallest prime p such that a string s of n zeros can be inserted between all adjacent digits of p simultaneously such that the resulting number is also prime and is also prime for each s of length k with 0 < k < n.

11, 19, 19, 71, 98689, 130049597, 78736136153

Offset: 1

Felix Fröhlich, Jun 03 2021

a(n) is the smallest prime p such that A344937(i) >= n, where i is the index of p in A000040.

			For n = 5: 98689, 908060809, 9008006008009, 90008000600080009, 900008000060000800009 and 9000008000006000008000009 are all prime. Since 98689 is the smallest prime where strings of zeros of successive lengths up to 5 can be inserted between all adjacent digits such that each resulting number is also prime, a(5) = 98689.

Cf. A000040, A344637, A344937.

Table[m=1;While[!And@@Table[PrimeQ@FromDigits@Flatten@Riffle[IntegerDigits@Prime@m,{Table[0,k]}],{k,n}],m++];Prime@m,{n,5}] (* 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) = forprime(p=10, , for(k=1, n, if(!ispseudoprime(eva(insert_zeros(p, k))), break, if(k==n, return(p)))))

from sympy import isprime, nextprime
def insert_zeros(n, k): return int(("0"*k).join(list(str(n))))
def ok(p, n): return all(isprime(insert_zeros(p, k)) for k in range(1, n+1))
def a(n, startat=11):
  p = startat
  while True:
    if ok(p, n): return p
    p = nextprime(p)
print([a(n) for n in range(1, 6)]) # Michael S. Branicky, Jun 03 2021

a(7) from Michael S. Branicky, Jun 11 2021

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.

Original entry on oeis.org

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)))

A358318 For n >= 5, a(n) is the number of zeros that need to be inserted to the left of the ones digit of the n-th prime so that the result is composite.

Original entry on oeis.org

2, 2, 2, 4, 1, 1, 1, 2, 3, 1, 1, 3, 3, 2, 3, 5, 1, 2, 1, 3, 5, 1, 1, 1, 3, 3, 1, 3, 3, 1, 4, 1, 1, 1, 3, 1, 2, 2, 3, 1, 2, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 1, 2, 1, 2, 5, 2, 2, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 6, 1, 2, 2, 1, 2, 2, 1, 1, 2, 1, 1, 1, 1, 2, 3, 1, 2, 1, 1, 1, 1, 3

Offset: 5

Rida Hamadani, Nov 09 2022

Conjecture: the sequence is bounded.

			For n = 8, prime(8) is 19. 109, 1009, and 10009 are all primes, while 100009 is not, thus a(8) = 4.
For n = 30, prime(30) is 113. 1103 and 11003 are prime, while 110003 is not, thus a(30) = 3.

Cf. A000040, A344637.

a[n_] := Module[{p = Prime[n], c = 1, q, r}, r = Mod[p, 10]; q = 10*(p - r); While[PrimeQ[q + r], q *= 10; c++]; c]; Array[a, 100, 5] (* Amiram Eldar, Nov 27 2022 *)

a(n) = n=prime(n); for(i=1,oo, isprime(n=10*n-n%10*9) || return(i)); \\ Kevin Ryde, Dec 08 2022

from sympy import isprime, prime
def a(n):
    s, c = str(prime(n)), 1
    while isprime(int(s[:-1] + '0'*c + s[-1])): c += 1
    return c
print([a(n) for n in range(5, 92)]) # Michael S. Branicky, Nov 09 2022

If a(n) > 1 then a(pi(k)) = a(n) - 1 where k = 100*floor(p/10) + p mod 10 and p = prime(n) (i.e., k is the result when a single 0 is inserted to the left of the ones digit of p).

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A344936 a(n) is the smallest prime p such that a string s of n zeros can be inserted between all adjacent digits of p simultaneously such that the resulting number is also prime and is also prime for each s of length k with 0 < k < n.

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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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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.

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A358318 For n >= 5, a(n) is the number of zeros that need to be inserted to the left of the ones digit of the n-th prime so that the result is composite.

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