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%I #52 Jul 14 2022 15:03:02
%S 1,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97,103,
%T 107,109,113,127,137,139,149,157,163,167,169,173,179,193,197,199,211,
%U 221,223,227,229,233,239,241,247,251,257,263,269,271,277,281,283,289,293,299,307
%N Numbers whose only palindromic divisor is 1.
%C Equivalent: Numbers such that the LCM of their palindromic divisors (A087999) is 1, or,
%C Numbers such that the number of palindromic divisors (A087990) is 1.
%C All terms are odd.
%C The 1st family consists of non-palindromic primes that form the subsequence A334321.
%C The 2nd family consists of {p^k, p prime, k >= 2} such that p^j for 1 <= j <= k is not a palindrome {169 = 13^2, 289 = 17^2, 361 = 19^2, ..., 2197 = 13^3, ...} (see examples).
%C The 3rd family consists of products p_1^q_1 * ... * p_k^q_k with k >= 2, all of whose divisors are nonpalindromic {221 = 13 * 27, 247 = 13 * 19, 299 = 13 * 23, 377 = 13 * 29, 391 = 17 * 23, 403 = 13 * 31, 481 = 13 * 37, ...}.
%C Also, equivalent: numbers all of whose divisors > 1 are nonpalindromic (A029742). - _Bernard Schott_, Jul 14 2022
%H David A. Corneth, <a href="/A334391/b334391.txt">Table of n, a(n) for n = 1..10000</a>
%F A087990(a(n)) = 1.
%F A087999(a(n)) = 1.
%e 49 = 7^2, the divisor 7 is a palindrome so 49 is not a term.
%e 169 = 13^2, divisors of 169 are {1, 13, 169} and 169 is a term.
%e 391 = 17*23, divisors of 391 are {1,17,23,391} and 391 is a term.
%e 307^2 = 94249 that is palindrome, so 94249 is not a term.
%p notpali:= proc(n) local L;
%p L:= convert(n,base,10);
%p L <> ListTools:-Reverse(L)
%p end proc:
%p filter:= proc(n) option remember; andmap(notpali,numtheory:-divisors(n) minus {1}) end proc:
%p select(filter, [seq(i,i=1..400,2)]); # _Robert Israel_, Apr 28 2020
%t Select[Range[300], !AnyTrue[Rest @ Divisors[#], PalindromeQ] &] (* _Amiram Eldar_, Apr 26 2020 *)
%o (PARI) ispal(n) = my(d=digits(n)); d == Vecrev(d);
%o isok(n) = fordiv(n, d, if (d>1 && ispal(d), return(0))); return(1); \\ _Michel Marcus_, Apr 26 2020
%o (Python)
%o from sympy.ntheory import divisors, is_palindromic
%o def ok(n): return not any(is_palindromic(d) for d in divisors(n)[1:])
%o print(list(filter(ok, range(1, 308, 2)))) # _Michael S. Branicky_, May 08 2021
%Y A334321 is a subsequence.
%Y Cf. A008365, A087990, A087999, A334139.
%Y Cf. A029742, A087991, A093037, A355695.
%K nonn,base
%O 1,2
%A _Bernard Schott_, Apr 26 2020
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