cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A334391 Numbers whose only palindromic divisor is 1.

Original entry on oeis.org

1, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 103, 107, 109, 113, 127, 137, 139, 149, 157, 163, 167, 169, 173, 179, 193, 197, 199, 211, 221, 223, 227, 229, 233, 239, 241, 247, 251, 257, 263, 269, 271, 277, 281, 283, 289, 293, 299, 307
Offset: 1

Views

Author

Bernard Schott, Apr 26 2020

Keywords

Comments

Equivalent: Numbers such that the LCM of their palindromic divisors (A087999) is 1, or,
Numbers such that the number of palindromic divisors (A087990) is 1.
All terms are odd.
The 1st family consists of non-palindromic primes that form the subsequence A334321.
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).
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, ...}.
Also, equivalent: numbers all of whose divisors > 1 are nonpalindromic (A029742). - Bernard Schott, Jul 14 2022

Examples

			49 = 7^2, the divisor 7 is a palindrome so 49 is not a term.
169 = 13^2, divisors of 169 are {1, 13, 169} and 169 is a term.
391 = 17*23, divisors of 391 are {1,17,23,391} and 391 is a term.
307^2 = 94249 that is palindrome, so 94249 is not a term.

Links

Crossrefs

A334321 is a subsequence.
Cf. A008365, A087990, A087999, A334139.
Cf. A029742, A087991, A093037, A355695.

Programs

  • Maple
    notpali:= proc(n) local L;
  L:= convert(n,base,10);
  L <> ListTools:-Reverse(L)
end proc:
filter:= proc(n) option remember; andmap(notpali,numtheory:-divisors(n) minus {1}) end proc:
select(filter, [seq(i,i=1..400,2)]); # Robert Israel, Apr 28 2020
  • Mathematica
    Select[Range[300], !AnyTrue[Rest @ Divisors[#], PalindromeQ] &] (* Amiram Eldar, Apr 26 2020 *)
  • PARI
    ispal(n) = my(d=digits(n)); d == Vecrev(d);
isok(n) = fordiv(n, d, if (d>1 && ispal(d), return(0))); return(1); \\ Michel Marcus, Apr 26 2020
  • Python
    from sympy.ntheory import divisors, is_palindromic
def ok(n): return not any(is_palindromic(d) for d in divisors(n)[1:])
print(list(filter(ok, range(1, 308, 2)))) # Michael S. Branicky, May 08 2021

Formula

A087990(a(n)) = 1.
A087999(a(n)) = 1.

A183106 Numbers k such that sum of palindromic divisors of k (A088000(k)) is palindromic.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 10, 13, 15, 17, 19, 21, 23, 25, 26, 29, 31, 34, 37, 38, 39, 41, 43, 46, 47, 49, 50, 51, 52, 53, 56, 57, 58, 59, 61, 62, 65, 67, 68, 69, 71, 72, 73, 74, 75, 76, 79, 82, 83, 85, 86, 87, 89, 91, 92, 93, 94, 95, 97, 103, 106, 107, 109, 112, 113, 115, 116, 118, 119
Offset: 1

Views

Author

Jaroslav Krizek, Dec 25 2010

Keywords

Examples

			a(12) = 21; palindromic divisors of 21: 1, 3, 7; their sum is 11 (palindromic number).

Crossrefs

Cf. A088000, A183107.
Subsequences: A334321, A334391.

Programs

  • Maple
    isA183106 := proc(n)
    isA002113(A088000(n)) ;
end proc:
for n from 1 to 100 do
    if isA183106(n) then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, Sep 09 2015
  • Mathematica
    q[k_] := PalindromeQ[DivisorSum[k, # &, PalindromeQ[#] &]]; Select[Range[120], q] (* Amiram Eldar, Aug 08 2024 *)

Formula

A088000(a(n)) = A183107(n).

Extensions

More terms from Amiram Eldar, Aug 08 2024

A359510 Numbers that can't be written as a palindromic product, i.e., such that the concatenation of all digits of the factors yields a palindrome.

Original entry on oeis.org

23, 26, 29, 30, 34, 35, 37, 38, 43, 47, 53, 57, 59, 62, 65, 67, 70, 73, 74, 79, 82, 83, 85, 86, 87, 89, 92, 94, 95, 97, 103, 106, 107, 109, 123, 127, 130, 134, 137, 139, 140, 142, 145, 146, 148, 149, 152, 157, 158, 163, 167, 170, 173, 174, 178, 179, 182, 183, 185, 190, 193, 194, 197
Offset: 1

Views

Author

M. F. Hasler and Eric Angelini, Jan 03 2023

Keywords

Comments

Any number of factors 1 is allowed anywhere in the product.
The sequence contains all primes which are not palindromic when stripped of digits '1' on either side (for example 23, 29, 37, but not 13, 17, 19, 31 which can be written as 13*1, 17*1, 19*1, 1*31, etc., where the concatenation of all digits, "131", "171", ... is palindromic).

Examples

			Any palindrome is trivially a palindromic product and therefore not in the sequence. Also not in the sequence are 10 = 10*1, 12 = 12*1, ..., 20 = 2*5*2, 21 = 1*21. Therefore the first term is a(1) = 23.

Links

Crossrefs

Cf. A002113 (palindromes in base 10), A029742 (non-palindromes), A334321 (non-palindromic primes), A004176 (omit digits 1).
Showing 1-3 of 3 results.

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