A333577 a(2) = 0, and otherwise a(n) is the least multiple of prime(n+1) whose decimal representation ends with that of prime(n).

12, 0, 35, 77, 611, 1513, 817, 1219, 2523, 1829, 2331, 2337, 3741, 3243, 5247, 3953, 1159, 5561, 5467, 1971, 6873, 1079, 4183, 3589, 9797, 48101, 3103, 46107, 33109, 15113, 120127, 77131, 67137, 76139, 45149, 38151, 104157, 165163, 100167, 87173, 101179, 170181

Offset

1,1

Comments

Inspired by the 134th problem of Project Euler (see link).

a(n) > 1 iff n != 2.

Some particular terms:

a(3) = 35 is the concatenation of prime(2) and prime(3),

a(4) = 77 is the palindrome prime(4) and prime(4),

a(13) = 3741 is the concatenation of prime(12) and prime(13),

a(25) = 9797 is the concatenation of prime(25) and prime(25).

Links

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Project Euler, Problem 134: Prime pair connection.

Example

For prime(2) = 3 and prime(3) = 5, there does not exist any integer that ends with 3 and is divisible by 5, hence a(2)=0 and it is the only term equal to 0.
For prime(5) = 11 and prime(6) = 13, 611 ends with 11 and 611=13*47 is divisible by 13, and no integer < 611 satisfies these two conditions, so a(5)= 611.

Mathematica

a[2] = 0; a[n_] := Module[{p = Prime[n], q, r}, q = NextPrime[p]; r = 10^Ceiling[Log10[p]]; While[!Divisible[p, q], p += r]; p]; Array[a, 100] (* Amiram Eldar, Mar 27 2020 *)

Prog

(PARI) a(n) = {if (n==2, return(0)); my(p = prime(n), q = prime(n+1), x = p, k = 0); until ((x % q) == 0, k++; x = eval(concat(Str(k), Str(p))); ); x; } \\ Michel Marcus, Mar 28 2020
(PARI) a(n) = { if (n==2, return (0), my (p=prime(n), q=nextprime(p+1)); lift(chinese(Mod(p, 10^#digits(p)), Mod(0, q)))) } \\ Rémy Sigrist, Mar 29 2020
(Python)
from sympy import prime, nextprime, mod_inverse
def A333577(n):
    if n == 2:
        return 0
    p = prime(n)
    q, r = nextprime(p), 10**len(str(p))
    return p*q*mod_inverse(q, r) % (q*r) # Chai Wah Wu, Mar 31 2020

Crossrefs

Cf. A000040, A333845 (variant).

Sequence in context: A307170 A332053 A225951 * A278711 A331911 A307841

Adjacent sequences: A333574 A333575 A333576 * A333578 A333579 A333580

Keyword

nonn,base

Author

Bernard Schott, Mar 27 2020

Extensions

More terms from Amiram Eldar, Mar 27 2020

Name improved by Rémy Sigrist, Mar 29 2020

Status

approved