A361444 - OEIS

%I #30 Mar 19 2023 13:35:18

%S 1,2,3,4,7,6,5,8,9,22,141,88,111,202,55,222,11,212,99,232,121,252,101,

%T 66,131,242,191,272,77,282,151,292,171,262,181,606,313,414,343,444,

%U 353,44,303,424,33,434,323,404,383,474,535,484,373,454,333,464,363

%N Lexicographically earliest sequence of distinct positive base-10 palindromes such that a(n) + a(n+1) is prime.

%H Michael De Vlieger, <a href="/A361444/b361444.txt">Table of n, a(n) for n = 1..10000</a>

%H Michael De Vlieger, <a href="/A361444/a361444.png">Log log scatterplot of a(n)</a>, n = 1..10^4, with a color function showing number of decimal digits of a(n) where red = 1, orange = 2, ..., magenta = 8.

%H Michael De Vlieger, <a href="/A361444/a361444_1.png">Log log scatterplot of a(n)</a>, n = 1..10^4, with a color function showing a(n) mod 10, where red = 1, orange = 2, ..., magenta = 9.

%e a(10) = 22, the smallest unused positive palindrome which can be added to a(9) = 9 to get a prime; 9 + 22 = 31.

%t nn = 60; kk = 5*10^4; c[_] = False; a[1] = j = 1; c[1] = True; u = 2;

%t MapIndexed[Set[s[First[#2]], #1] &, Select[Range[kk], PalindromeQ]];

%t Do[k = u; While[Or[c[k], CompositeQ[s[k] + j]], k++];

%t  Set[{a[n], c[k], j}, {k, True, s[k]}];

%t  If[k == u, While[c[u], u++]], {n, 2, nn}];

%t Array[s @* a, nn] (* _Michael De Vlieger_, Mar 18 2023 *)

%o (Python)

%o from sympy import isprime

%o from itertools import count, islice, product

%o def pals(): # generator of palindromes

%o     digits = "0123456789"

%o     for d in count(1):

%o         for p in product(digits, repeat=d//2):

%o             if d > 1 and p[0] == "0": continue

%o             left = "".join(p); right = left[::-1]

%o             for mid in [[""], digits][d%2]:

%o                 yield int(left + mid + right)

%o def agen(): # generator of terms of sequence

%o     pg, passed = pals(), []

%o     an = next(p for p in pg if p > 0) # start at 1

%o     while True:

%o         yield an

%o         for p in passed:

%o             if isprime(an+p):

%o                 passed.remove(p)

%o                 break

%o         else:

%o              while not isprime(an + (p:=next(pg))):

%o                  passed.append(p)

%o         an = p

%o print(list(islice(agen(), 57))) # _Michael S. Branicky_, Mar 12 2023

%o (PARI) nextpal(k) = my(d=digits(k)); while (d!=Vecrev(d), k++; d = digits(k)); k;

%o lista(nn) = my(va = vector(nn)); va[1] = 1; for (n=2, nn, my(k=1); while(!isprime(va[n-1]+k) || #select(x->(x==k), va), k=nextpal(k+1)); va[n] = k;); va; \\ _Michel Marcus_, Mar 19 2023

%Y Cf. A002113, A082979.

%K base,nonn

%O 1,2

%A _Jodi Spitz_, Mar 12 2023

%E a(25) and beyond from _Michael S. Branicky_, Mar 12 2023