A089695 Numbers m such that placing as many possible '+' signs anywhere in between the digits yields a prime in every case. Let abcd... be the digits of m; then abcd, a + bcd, ab + cd, abc + d, a + b + cd, a + bc + d, ab + c + d, a + b + c + d, ... are all prime.

2, 3, 5, 7, 11, 23, 29, 41, 43, 47, 61, 67, 83, 89, 101, 227, 229, 281, 401, 443, 449, 467, 601, 607, 647, 661, 683, 809, 821, 863, 881, 4001, 4463, 4643, 6007, 6067, 6803, 8009

Offset: 1

Author

Amarnath Murthy, Nov 10 2003

Keywords

• base
• nonn
• more

Comments

Though the first 27 terms match those of A089392, the next term of A089392 (2221) is not a member of this sequence. Conjecture: sequence is finite.

No more terms < 10^8. - David Wasserman, Oct 04 2005

Examples

			863 is a member 863, 8 + 63, 86 + 3, 8 + 6 + 3 are all prime.
		

Crossrefs

Cf. A089696.

Programs

Maple

with(combinat): ds:=proc(s) local j: RETURN(add(s[j]*10^(j-1),j=1..nops(s))):end: for d from 1 to 6 do sch:=[seq([1,op(i),d+1],i=choose([seq(j,j=2..d)]))]: for n from 10^(d-1) to 10^d-1 do sn:=convert(n,base,10): fl:=0: for s in sch do m:=add(j,j=[seq(ds(sn[s[i]..s[i+1]-1]),i=1..nops(s)-1)]): if not isprime(m) then fl:=1: break fi od: if fl=0 then printf("%d, ",n) fi od od: # C. Ronaldo
# second Maple program:
b:= proc(s) option remember; (n-> {s, seq(seq(seq(""||x||"+"||y,
        y=b(s[i+1..n])), x=b(s[1..i])), i=1..n-1)})(length(s))
    end:
q:= n-> andmap(isprime, map(parse, b(""||n))):
select(q, [$1..10000])[];  # Alois P. Heinz, Oct 29 2021

Mathematica

Select[Prime@Range@1010,And@@PrimeQ[n=#;Total/@(FromDigits/@#&/@Union[DeleteCases[SplitBy[#,#==-1&],{-1}]&/@(Insert[IntegerDigits@n,-1,#]&/@(List/@#&/@Rest@Subsets[Range@IntegerLength@n]))])]&] (* Giorgos Kalogeropoulos, Oct 29 2021 *)

Extensions

Corrected and extended by C. Ronaldo (aga_new_ac(AT)hotmail.com), Dec 25 2004