cp's OEIS Frontend

A385536 Integers k formed by the concatenation of 3 consecutive prime numbers such that k is divisible by at least one of those 3 primes.

%I A385536 #20 Jul 10 2025 22:52:24
%S A385536 235,357,71113,374143,616771,167173179,143914471451,
%T A385536 909071909089909091,104651162787110465116278991046511627907
%N A385536 Integers k formed by the concatenation of 3 consecutive prime numbers such that k is divisible by at least one of those 3 primes.
%C A385536 Subsequence of A132903.
%C A385536 If p is A255669, as it divides the concatenation of the next two primes, then p divides the concatenation of p with the next two primes. Thus, the first 4 terms of A255669 give rise to a(2), a(3), a(5) and a(6). In this sequence the number formed by concatenating 3 consecutive primes is allowed to be divisible by at least one of those 3, which generates more possibilities.
%C A385536 a(9) has p > 10^11 and thus >= 36 digits. - _Michael S. Branicky_, Jul 02 2025
%C A385536 a(9) has p <=  A258182(11) - 36 = 1046511627871.
%C A385536 a(10) has p <= A258182(22) - 226.
%e A385536 235 is a term, since it is the concatenation of the consecutive primes 2, 3, 5 and is divisible by 5.
%e A385536 2, 3 and 5: 235 = 5*47
%e A385536 3, 5, and 7: 357 = 3*119 = 7*51
%e A385536 7, 11 and 13: 71113 = 7*10159
%e A385536 37, 41 and 43: 374143 = 43*8701
%e A385536 61, 67 and 71: 616771 = 61*10111
%e A385536 167, 173 and 179: 167173179 = 167*1001037
%e A385536 1439, 1447 and 1451: 143914471451 = 1447*99457133
%e A385536 909071, 909089 and 909091: 909071909089909091 = 909091*999979000001
%p A385536 tcat:= proc(a,b,c)
%p A385536    c + (b + 10^(1+ilog10(b))*a)*10^(1+ilog10(c))
%p A385536 end proc:
%p A385536 R:= NULL: count:= 0:
%p A385536 q:= 2: r:= 3:
%p A385536 while count < 8 do
%p A385536   p:= q; q:= r; r:= nextprime(r);
%p A385536   x:= tcat(p,q,r);
%p A385536   if igcd(x,p*q*r)>1 then
%p A385536     R:= R,x; count:= count+1;
%p A385536   fi
%p A385536 od:
%p A385536 R; # _Robert Israel_, Jul 06 2025
%t A385536 cat[s_] := FromDigits[Flatten[IntegerDigits[s]]]; q[s_] := AnyTrue[s, Divisible[cat[s], #] &]; cat /@ Select[Partition[Prime[Range[72000]], 3, 1], q] (* _Amiram Eldar_, Jul 03 2025 *)
%Y A385536 Cf. A132903, A255669, A258182.
%K A385536 nonn,base,more
%O A385536 1,1
%A A385536 _Gonzalo Martínez_, Jul 02 2025
%E A385536 a(9) from _Michael S. Branicky_, Jul 03 2025