A341715 a(n) = smallest prime of the form n||n+1||n+2||...||n+k, where || denotes decimal concatenation, or -1 if no such prime exists.

2, 3, 4567, 5, 67, 7, 89

Offset: 2

Author

N. J. A. Sloane, Feb 21 2021

Keywords

• nonn
• base
• more
• nice

Comments

a(1) is unknown, but is believed to exist (see A007908). The corresponding value of k, if it exists, is known to be at least 300000, so in any case this prime would be too large to include in an OEIS entry, which is why this sequence has offset 2.

a(9) = 9||10||...||187 (see Example section), but that is too large to show in the data field. a(A030457(n)) = A030457(n)||A030457(n)+1 and k = 1 for n > 1. If m is in A030470 but not in A030457, then a(m) = m||m+1||m+2||m+3 and k = 3. Of course a(p) = p and k = 0 for p prime. - Chai Wah Wu, Feb 22 2021

For the corresponding values of k and n+k, see A341716 and A341717.

See also A140793 = (23, 345...109, 4567, 567...17, ...), A341720, and A084559 for the variant with k >= 1, so that a(n) > n also for prime n. - M. F. Hasler, Feb 22 2021

Examples

			Starting at 12, 13, 14, 15, 17, 19, 20 we get the primes 1213, 13, 14151617, 1516171819, 17, 19, 20212223, which are all terms of this sequence.
Here is a(9) from _Chai Wah Wu_, Feb 22 2021, a 445-digit number:
910111213141516171819202122232425262728293031323334353637383940414243444546\
    47484950515253545556575859606162636465666768697071727374757677787980818\
    28384858687888990919293949596979899100101102103104105106107108109110111\
    11211311411511611711811912012112212312412512612712812913013113213313413\
    51361371381391401411421431441451461471481491501511521531541551561571581\
    59160161162163164165166167168169170171172173174175176177178179180181182\
    183184185186187
a(16) = 16||17||...||43 is prime. Also for a(10), I searched up to k <= 10000, so if it exists it will have tens of thousands of decimal digits. Some other big terms are: for n = 18, k = 3589; for n = 35, k = 568; for n = 66, k = 937; for n = 275, k = 814.  - _Chai Wah Wu_, Feb 22 2021
		

Links

• N. J. A. Sloane, Table of n, k, with a question-mark for unknown values
• N. J. A. Sloane, Exciting Number Sequences (video of talk), Mar 05 2021.

Crossrefs

Cf. A007908, A030457, A030470, A030471, A135605, A341716, A341717, A341718.

If k in the definition is allowed to be zero we get [the present sequence, A341716, A341717], but if we require k>0 we get [A140793, A341720, A084559].

See A075022 for the largest prime factor of 1||2||...||n.

Programs

Mathematica

Array[Block[{k = #, s = #}, While[! PrimeQ[s], k++; s = FromDigits[IntegerDigits[s]~Join~IntegerDigits[k]]]; s] &, 8, 2] (* Michael De Vlieger, Feb 22 2021 *)

PARI

A341715(n)=if(isprime(n),n,eval(concat([Str(k)|k<-[n..A084559(n)]]))) \\ M. F. Hasler, Feb 22 2021

Python

from sympy import isprime
def A341715(n):
    m, k = n, n
    while not isprime(m):
        k += 1
        m = int(str(m)+str(k))
    return m # Chai Wah Wu, Feb 22 2021

Formula

a(n) = concatenate(n, ..., A084559(n)) or a(n) = n if n is prime. - M. F. Hasler, Feb 22 2021