A380347 a(1) = 1, a(2) = 3; thereafter, a(n) is the smallest nonnegative integer not yet in the sequence such that both a(n-2) + a(n-1) + a(n) and their concatenation are primes.

1, 3, 7, 9, 27, 17, 23, 13, 31, 53, 73, 47, 29, 21, 33, 19, 37, 41, 11, 57, 63, 77, 51, 101, 59, 67, 151, 93, 109, 49, 39, 79, 81, 91, 181, 107, 161, 43, 163, 71, 83, 87, 69, 113, 89, 61, 131, 157, 233, 133, 137, 139, 203, 179, 117, 143, 171, 207, 241, 121, 159

Offset

1,2

Comments

Of course, more than 3 terms in a row can satisfy the requirement:

1 + 3 + 7 + 9 + 27 = 47 is prime like 137927;

3 + 7 + 9 + 27 + 17 + 23 + 13 + 31 + 53 + 73 + 47 + 29 + 21 = 353 is prime like 37927172313315373472921; etc.

Links

Table of n, a(n) for n=1..61.

Example

181 + 107 + 161 = 449 is prime like 181107161.

Maple

P:=proc(q) local a, n, ok, t, v; v:=[1, 3];
for n from 1 to q do for t from 5 by 2 do if numboccur(v, t)=0 then
a:=v[nops(v)]*10^length(t)+t; a:=v[nops(v)-1]*10^length(a)+a;
if isprime(v[nops(v)-1]+v[nops(v)]+t) and isprime(a) then v:=[op(v), t]; break; fi; fi; od;
od; op(v); end: P(59);

Mathematica

s={1, 3}; Do[i=1; Until[PrimeQ[i+s[[-1]]+s[[-2]]]&&PrimeQ[FromDigits[Join[IntegerDigits[s[[-2]]], IntegerDigits[s[[-1]]], IntegerDigits[i]]]&&!MemberQ[s, i]], i++]; AppendTo[s, i], {n, 59}]; s (* James C. McMahon, Jan 26 2025 *)

Prog

(Python)
from sympy import isprime
from itertools import count, islice
def agen(): # generator of terms
    yield 1
    an1, an, aset, m = 1, 3, {1}, 7
    for n in count(2):
        yield an
        aset.add(an)
        s2, str2 = an1 + an, str(an1) + str(an)
        k = next(j for j in count(m, 2) if j not in aset and j%10 != 5 and isprime(s2+j) and isprime(int(str2+str(j))))
        an1, an = an, k
        while m in aset or m%2 == 5: m += 2
print(list(islice(agen(), 61))) # Michael S. Branicky, Jan 22 2025
(PARI) F(n)={if (n==1, return(1)); my(s=[1, 3]); n--; while(n--, for(x=1, +oo, for(b=1, #s, if(x==s[b], next(2))); if(ispseudoprime(s[#s-1]+s[#s]+x), if(ispseudoprime(fromdigits(concat([Vec(digits(s[#s-1])), Vec(digits(s[#s])), Vec(digits(x))]))), s=concat(s, [x]); break())))); s} \\ Computes first n terms. - R. J. Cano, Jan 22 2025

Crossrefs

Cf. A000040.

Sequence in context: A118564 A098338 A033958 * A352016 A018827 A249664

Adjacent sequences: A380344 A380345 A380346 * A380348 A380349 A380350

Keyword

nonn,base,easy

Author

Paolo P. Lava, Jan 22 2025

Status

approved