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URL: https://oeis.org/A385452

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A385452
Numbers that are the concatenation of three (not necessarily distinct) primes whose sum is prime, and are also the product of three (not necessarily distinct) primes whose sum is prime.
1
775, 1975, 3115, 3157, 3175, 3311, 3535, 3553, 3731, 5117, 5135, 5335, 5537, 5593, 5735, 5797, 5957, 6775, 7511, 7675, 7733, 8957, 9737, 11297, 11315, 11473, 11713, 11753, 13115, 13135, 13433, 13615, 13715, 13717, 13783, 13895, 13937, 14935, 15175, 16337, 17297, 17347, 17437, 17537, 17719, 17759
OFFSET
1,1
COMMENTS
All terms are odd (see link for proof).
The two sums of primes are different mod 3 (see link for proof).
LINKS
EXAMPLE
a(3) = 3115 is a term because it is the concatenation of primes 3, 11 and 5 whose sum 19 is prime, and the product of primes 5, 7 and 89 whose sum 101 is prime.
The first term where the concatenation can be done in more than one way is a(30) = 13135 which is the concatenation of 13, 13 and 5 (with sum 31) and also of 131, 3 and 5 (with sum 139).
MAPLE
M:= 5: # for terms with up to M digits
PD[1]:= [2, 3, 5, 7]:
for d from 2 to M-2 do PD[d]:= select(isprime, [seq(i, i=10^(d-1)+1 .. 10^d-1, 2)]) od:
R:= NULL:
for d from 3 to M do
for d1 from 1 to d-2 do
for d2 from 1 to d-d1-1 do
for a in PD[d1] do
for b in PD[d2] do
for c in PD[d-d1-d2] do
if not isprime(a+b+c) then next fi;
t:= a + 10^d1*b + 10^(d1+d2)*c;
F:= ifactors(t)[2];
if add(s[2], s=F) = 3 and isprime(add(s[1]*s[2], s=F)) then
R:= R, t;
fi
od od od od od od:
sort(convert({R}, list));
CROSSREFS
Sequence in context: A250087 A252673 A077077 * A177845 A177843 A202893
KEYWORD
nonn,base
AUTHOR
Will Gosnell and Robert Israel, Jun 29 2025
STATUS
approved