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A353030
a(n) is the first emirp p such that there are exactly n unordered pairs (q,r) of emirps with p = q*r + q + r.
1
13, 1439, 100799, 3548879, 14061599, 38342303, 120355199, 12555446399
OFFSET
0,1
COMMENTS
a(n) is the first prime p such that the digit-reversal rev(p) of p is a prime and there are exactly n pairs (q,r) of primes such that q < r, rev(q) and rev(r) are primes, and p = q*r + q + r.
From David A. Corneth, Jan 14 2023: (Start)
a(8) <= 121347071999, a(9) <= 195271876799, a(10) <= 10175362797599, a(11) <= 17482966300799.
For n >= 2, n == 3 (mod 4) and (n + 1)/4 has at least 2*n divisors. (End)
EXAMPLE
a(3) = 3548879 because 3548879 = 17*197159 + 17 + 197159 = 359*9857 + 359 + 9857 = 953*3719 + 953 + 3719 and 3548879, 17, 197159, 359, 9857, 953, 3719 are emirps.
MAPLE
revdigs:= proc(n) local L, i; L:= convert(n, base, 10); add(L[-i]*10^(i-1), i=1..nops(L)) end proc:
isemirp:= proc(p) local r;
if not isprime(p) then return false fi;
r:= revdigs(p);
r <> p and isprime(r)
end proc:
g:= proc(n) local p, q, t, count;
count:= 0;
for t in select(`<`, numtheory:-divisors(n+1), floor(sqrt(n+1))) do
if isemirp(t-1) and isemirp((n+1)/t-1) then
count:= count+1;
fi
od;
count
end proc:
V:= Array(0..6): vcount:= 0:
p:= 2:
while vcount < 7 do
p:= nextprime(p);
d:= ilog10(p);
p1:= floor(p/10^d);
if p1=2 then p:= nextprime(3*10^d)
elif member(p1, {4, 5, 6}) then p:= nextprime(7*10^d)
elif p1=8 then p:= nextprime(9*10^d)
fi;
if isemirp(p) then
v:= g(p);
if V[v] = 0 then vcount:= vcount+1; V[v]:= p; fi;
fi
od:
convert(V, list);
CROSSREFS
Sequence in context: A203369 A350305 A197097 * A064962 A242562 A201357
KEYWORD
nonn,base,more
AUTHOR
J. M. Bergot and Robert Israel, Apr 18 2022
EXTENSIONS
a(7) from David A. Corneth, Jan 14 2023
STATUS
approved