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A089695
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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.
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3
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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
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OFFSET
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1,1
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COMMENTS
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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.
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LINKS
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EXAMPLE
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863 is a member 863, 8 + 63, 86 + 3, 8 + 6 + 3 are all prime.
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MAPLE
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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))):
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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base,nonn,more
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AUTHOR
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EXTENSIONS
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Corrected and extended by C. Ronaldo (aga_new_ac(AT)hotmail.com), Dec 25 2004
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STATUS
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approved
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