A345410 - OEIS

%I #25 May 27 2024 07:14:16

%S 44,1090,10450,5104,88888,10780,289982,299992,482174,478874,868868,

%T 499994,1073270,1087790,1071070,1069970,10904990,10794980,1091090,

%U 10892990,1100000,29955992,1101100,26688662,31022002,27599572,46400354,44688644,29821792,45289244,30122092,26988962

%N a(n) is the least number that is the sum of an emirp and its reversal in exactly n ways.

%C Interchanging an emirp and its reversal is not counted as a different way.

%C a(n) is the least number k such that there are exactly n unordered pairs of distinct primes (p,p') such that p' is the digit reversal of p and p+p' = k.

%C Are terms not divisible by 3? _Amiram Eldar_ finds proof they are; A056964(n) = n + reverse(n) is divisible by 3 if and only if n is divisible by 3. But emirps are primes (other than 3) so they are not divisible by 3. - _David A. Corneth_, Jun 19 2021

%H David A. Corneth, <a href="/A345410/b345410.txt">Table of n, a(n) for n = 1..423</a>

%H David A. Corneth, <a href="/A345410/a345410.gp.txt">A few examples</a>

%e a(3) = 10450 because 10450 = 1229+9221 = 1409+9041 = 3407+7043.

%p revdigs:= proc(n) local L,i; L:= convert(n,base,10); add(L[-i]*10^(i-1),i=1..nops(L)) end proc:

%p isemirp1:= proc(n) local r;

%p if not isprime(n) then return false fi;

%p r:= revdigs(n);

%p r > n and isprime(r)

%p end proc:

%p E:= select(isemirp1, [seq(seq(seq(i*10^d+j,j=1..10^d-1,2),i=[1,3,7,9]),d=1..5)]):

%p V:= sort(map(t -> t+revdigs(t),E)):

%p N:= nops(V):

%p W:= Vector(16):

%p i:= 1:

%p while i < N do

%p for j from 1 to N-i while V[i+j]=V[i] do od:

%p if j <= 16 and W[j] = 0 then W[j]:= V[i] fi;

%p   i:= i+j;

%p od:

%p convert(W,list);

%o (Python)

%o from itertools import product

%o from collections import Counter

%o from sympy import isprime, nextprime

%o def epgen(start=1, end=float('inf')): # generates unique emirp/prime pairs

%o     digits = 2

%o     while True:

%o       for first in "1379":

%o         for last in "1379":

%o           if last < first: continue

%o           for mid in product("0123456789", repeat=digits-2):

%o             strp = first + "".join(mid) + last

%o             revstrp = strp[::-1]

%o             if strp >= revstrp: continue

%o             p = int(strp)

%o             if p > end: return

%o             revp = int(strp[::-1])

%o             if isprime(p) and isprime(revp): yield (p, revp)

%o       digits += 1

%o def aupto(lim):

%o     alst = []

%o     c = Counter(sum(ep) for ep in epgen(1, lim) if sum(ep) <= lim)

%o     r = set(c.values())

%o     for i in range(1, max(r)+1):

%o         if i in r: alst.append(min(s for s in c if c[s] == i))

%o         else: break

%o     return alst

%o print(aupto(11*10**5)) # _Michael S. Branicky_, Jun 19 2021

%Y Cf. A006567, A345408, A345409.

%K nonn,base

%O 1,1

%A _J. M. Bergot_ and _Robert Israel_, Jun 18 2021

%E More terms from _David A. Corneth_, Jun 18 2021