A384726 a(n) is the least number that is both the product of n distinct primes and the concatenation of n distinct primes.

2, 35, 273, 11235, 237615, 11237835, 1123317195, 111371237835, 11132343837615, 1113172923477615, 111317233377372295, 11131723677292413195, 1113172377671953734135, 111317192375336174123715

Offset: 1

Author

Robert Israel, Jun 08 2025

Keywords

• nonn
• base
• more

Comments

a(n) is odd for n >= 2, because a number whose last digit is 2 and second-last is odd is divisible by 4.

Examples

			a(4) = 11235 is a term because 11235 is the product of four distinct primes 3, 5, 7, 107 and the concatenation of four distinct primes 11, 2, 3, 5, and no smaller number works.
		

Crossrefs

Cf. A083427, A374665.

Programs

Maple

cdp:= proc(x, n, S)
  local i,y;
  if n = 1 then return (not(member(x,S)) and isprime(x)) fi;
  for i from 1 to ilog10(x)+2-n do
    y:= x mod 10^i;
    if member(y,S) or not isprime(y) then next fi;
    if procname((x-y)/10^i, n-1, S union {y}) then return true fi;
  od;
  false
end proc:
f:= proc(n) uses priqueue; local pq, t, p, x, i, L, v, Lp;
  initialize(pq);
  L:= [seq(ithprime(i), i=2..n+1)];
  v:= convert(L, `*`);
  insert([-v, L], pq);
  do
    t:= extract(pq);
    x:= -t[1];
    if cdp(x,n,{}) then return x fi;
    L:= t[2];
    p:= nextprime(L[-1]);
    for i from n to 1 by -1 do
      if i < n and L[i] <> prevprime(L[i+1]) then break fi;
      Lp:= [op(L[1..i-1]), op(L[i+1..n]), p];
      insert([-convert(Lp, `*`), Lp], pq)
  od od;
end proc:
f(1):= 2:
map(f, [$1..9]);

Extensions

a(11)-a(14) from Jinyuan Wang, Jun 12 2025