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A379138 a(n) is the first number that is the sum of two palindromic primes in exactly n ways.

%I A379138 #7 Dec 16 2024 02:15:38
%S A379138 0,4,10,504,25242,1110,28782,46764,46254,86058,50094,47874,107880,
%T A379138 108180,110100,108990,107070,109800,2726262,2830272,2698962,3029292,
%U A379138 2900982,2799972,2979792,3100002,2998992,4498944,4409034,4709064,4510044,4916184,4790874,4787874,4869684,4959594,4896984,4891884
%N A379138 a(n) is the first number that is the sum of two palindromic primes in exactly n ways.
%C A379138 a(n) is the least k such that there are exactly n numbers j <= k/2 where both j and k - j are in A002385.
%e A379138 a(5) = 1110 because 1110 = 181 + 929 = 191 + 919 = 313 + 797 = 353 + 757 = 383 + 727 is the sum of two palindromic primes in exactly 5 ways, and no smaller even number works.
%p A379138 digrev:= proc(n) local L,i;
%p A379138   L:= convert(n,base,10);
%p A379138   add(L[-i]*10^(i-1),i=1..nops(L))
%p A379138 end proc:
%p A379138 F:= proc(d) # d-digit odd palindromic primes, d >= 3
%p A379138   local R,x,rx,i;
%p A379138     select(isprime,map(t -> seq(10^((d+1)/2)*t + i*10^((d-1)/2) + digrev(t),i=0..9), [$(10^((d-3)/2)) .. 10^((d-1)/2)-1]))
%p A379138 end proc:
%p A379138 PP:= [3,5,7,11,op(F(3)),op(F(5)),op(F(7))]: nPP:= nops(PP):
%p A379138 V:= Vector(2*PP[-1],datatype=integer[1]):
%p A379138 for i from 1 to nPP do for j from 1 to i do
%p A379138    x:= PP[i]+PP[j];
%p A379138    V[x]:= V[x]+1
%p A379138 od od:
%p A379138 M:= max(V):
%p A379138 W:= Array(0..M,-1):
%p A379138 W[0]:= 0: W[1]:= 4:
%p A379138 for x from 1 to 2*PP[-1] do
%p A379138   if W[V[x]] = -1 then W[V[x]]:= x fi
%p A379138 od:
%p A379138 convert(W,list); # entries of -1 indicate values > 10^8
%Y A379138 Cf. A002385, A377848.
%K A379138 nonn
%O A379138 0,2
%A A379138 _Robert Israel_, Dec 15 2024