A087999 -id:A087999 - cp's OEIS frontend

A334391 Numbers whose only palindromic divisor is 1.

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

Bernard Schott, Apr 26 2020

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

			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.

David A. Corneth, Table of n, a(n) for n = 1..10000

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

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

Select[Range[300], !AnyTrue[Rest @ Divisors[#], PalindromeQ] &] (* Amiram Eldar, Apr 26 2020 *)

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

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

A087990(a(n)) = 1.

A087999(a(n)) = 1.

A088001 a(n) is the sum of non-palindromic divisors of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 10, 0, 12, 13, 14, 15, 16, 17, 18, 19, 30, 21, 0, 23, 36, 25, 39, 27, 42, 29, 55, 31, 48, 0, 51, 35, 66, 37, 57, 52, 70, 41, 77, 43, 0, 60, 69, 47, 100, 49, 85, 68, 91, 53, 99, 0, 98, 76, 87, 59, 147, 61, 93, 84, 112, 78, 0, 67, 119, 92, 129, 71, 162, 73

Offset: 1

Labos Elemer, Oct 14 2003

Indranil Ghosh, Table of n, a(n) for n = 1..10000

Cf. A087999, A088000, A088002.

A088001 := proc(n)
        numtheory[sigma](n)-A088000(n) ;
end proc; # R. J. Mathar, Jul 28 2016

Table[Plus @@ Select[Divisors[k], Reverse[x = IntegerDigits[#]] != x &], {k, 73}] (* Jayanta Basu, Aug 12 2013 *)
Table[Total[Select[Divisors[n],!PalindromeQ[#]&]],{n,80}] (* Harvey P. Dale, May 15 2025 *)

def ispal(n):
    return n==int(str(n)[::-1])
def A088001(n):
    s=0
    for i in range(1, n+1):
        if n%i==0 and not ispal(i):
             s+=i
    return s # Indranil Ghosh, Feb 10 2017

a(n)=0 iff all divisors are palindromic. See A062687.

a(n)+A088000(n) = A000203(n). - R. J. Mathar, Sep 09 2015

A334139 Numbers that are equal to the LCM of their palindromic divisors.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 18, 20, 21, 22, 24, 28, 30, 33, 35, 36, 40, 42, 44, 45, 55, 56, 60, 63, 66, 70, 72, 77, 84, 88, 90, 99, 101, 105, 110, 111, 120, 121, 126, 131, 132, 140, 141, 151, 154, 161, 165, 168, 171, 180, 181, 191, 198, 202, 210

Offset: 1

Bernard Schott, Apr 15 2020

These terms are the fixed points of A087999.
All the palindromes are in the sequence.
Now, if m is non-palindromic, then m is a term iff m = q_1^r_1 *...* q_i^r_i *...* q_k^r_k, where q_1 <...=2, r_i >= 1 and every divisor q_i^r_i is a palindrome; these q_i^r_i are in A084092 (see examples).
The first 40 terms, from 1 to 99, are exactly the 40 smallest divisors of 27720, hence the first 40 terms of A178864, but this sequence, which is infinite, is not a duplicate. Also, 27720 is in this sequence.

			2, 5, 131 are terms as palindromic primes.
111 = 3 * 37 is a term because 111 is a palindrome, so LCM(1,3,37,111) = 111.
27720 = 2^3 * 3^2 * 5 * 7 * 11, every 2^3=8, 3^2=9, 5, 7, 11 is a palindrome so 27720 is another term, no palindromic.

Cf. A087999, A178864.

Subsequences: A002113, A002385, A062687, A084092.

Select[Range[200], LCM @@ Select[Divisors[#], PalindromeQ] == # &] (* Amiram Eldar, Apr 15 2020 *)

ispal(x) = my(d=digits(x)); d == Vecrev(d);
isok(n) = lcm(select(ispal,  divisors(n))) == n; \\ Michel Marcus, Apr 16 2020

A334321 Non-palindromic primes.

Original entry on oeis.org

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, 173, 179, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 317, 331, 337, 347, 349

Offset: 1

Bernard Schott, Apr 23 2020

Not the same as A052085, primes whose decimal digits are grouped together: 2, 3, 5, 7, 11 are not terms of this sequence, then the next difference occurs for prime 1013 that belongs to this sequence but not to A052085.

			97 is prime and is not a palindrome, hence 97 belongs to this sequence.

Equals A000040 \ A002385.

Intersection of A029742 and A000040.

Select[Range[350], PrimeQ[#] && !PalindromeQ[#] &] (* Amiram Eldar, Apr 23 2020 *)

isok(p) = if (isprime(p), my(d=digits(p)); d != Vecrev(d)); \\ Michel Marcus, Apr 23 2020

Formula : A087999(a(n)) = 1.

A334392 Numbers m such that the LCM of their palindromic divisors is neither 1 nor m.

Original entry on oeis.org

16, 25, 26, 27, 32, 34, 38, 39, 46, 48, 49, 50, 51, 52, 54, 57, 58, 62, 64, 65, 68, 69, 74, 75, 76, 78, 80, 81, 82, 85, 86, 87, 91, 92, 93, 94, 95, 96, 98, 100, 102, 104, 106, 108, 112, 114, 115, 116, 117, 118, 119, 122, 123, 124, 125, 128, 129, 130, 133, 134

Offset: 1

Bernard Schott, May 04 2020

A334139, A334391 and this sequence form a partition of the set of positive integers N* (A000027).

The integers {2^k, k >= 4, 2^k non-palindrome} form a subsequence whose first few terms are : 16, 32, 64, 128, ...

			50 has 3 palindromic divisors {1, 2, 5} then A087999(50) = 10 and 50 is a term.

Wikipedia, Partition of a set

Cf. A087990, A087999.

Cf. A334391 [LCM(palindromic divisors of m) = 1], A334139 [LCM(palindromic divisors of m) = m], this sequence [LCM(palindromic divisors of m) != 1 and != m].

Select[Range[125], !MemberQ[{1, #}, LCM @@ Select[Divisors[#], PalindromeQ]] &] (* Amiram Eldar, May 05 2020 *)

ispal(x) = my(d=digits(x)); d == Vecrev(d);
isok(m) = my(d=divisors(m), lcmpd = lcm(select(x->ispal(x), d))); (lcmpd != 1) && (lcmpd != m); \\ Michel Marcus, May 05 2020

A339624 Perfect powers p^k, k >= 2 of palindromic primes p when p^k is not a palindrome.

Original entry on oeis.org

16, 25, 27, 32, 49, 64, 81, 125, 128, 243, 256, 512, 625, 729, 1024, 2048, 2187, 2401, 3125, 4096, 6561, 8192, 15625, 16384, 16807, 17161, 19683, 22801, 32761, 32768, 36481, 59049, 65536, 78125, 97969, 117649, 124609, 131072, 139129, 146689, 161051, 177147, 262144

Offset: 1

Bernard Schott, Dec 10 2020

Equivalently: numbers m with only one prime factor such that the LCM of their palindromic divisors is neither 1 nor m: subsequence of A334392.
G. J. Simmons conjectured there are no palindromes of form n^k for k >= 5 (and n > 1) (see Simmons p. 98). According to this conjecture, these perfect powers are terms: {2^k, k>=4}, {3^k, k>=3}, {5^k, k>=2}, {7^k, k=2 and k>=4}, {11^k, k>=5}, {101^k, k>= 5}, {131^k, k>=2}, ...
From a(1) = 16 to a(17) = 2187, the data is the same as A056781(10) until A056781(26), then a(18) = 2401 and A056781(27) = 4096.

			5^2 = 25, 2^6 = 64, 3^4 = 81 are terms.
7^2 = 49 is a term, 7^3 = 343 is not a term, and 7^4 = 2401 is a term.
101^2 = 10201 and 11^4 = 14641 are not terms.

Murray S. Klamkin, Problems in applied mathematics: selections from SIAM review, (1990), p. 520.

Gustavus J. Simmons, Palindromic Powers, J. Rec. Math., 3 (No. 2, 1970), 93-98 [Annotated scanned copy].
Wikipedia, Palindromic number, Perfect Powers.

Intersection of A025475 and A334392.
Cf. A087990, A087999, A334139, A334391.
Subsequences: A000079 \ {1,2,4,8}, A000244 \ {1,3,9}, A000351 \ {1,5}, A000420 \ {1,7,343}, A001020 \ {1,11,121,1331,14641}, A096884 \ {1,101, 10201, 1030301, 104060401}.

q[n_] := Module[{f = FactorInteger[n]}, Length[f] == 1 && f[[1, 2]] > 1 && PalindromeQ[f[[1, 1]]]]; Select[Range[10^5], !PalindromeQ[#] && q[#] &] (* Amiram Eldar, Dec 10 2020 *)

ispal(n) = my(d=digits(n)); Vecrev(d)==d;
isok(k) = my(p); isprimepower(k, &p) && isprime(p) && ispal(p) &&!ispal(k); \\ Michel Marcus, Dec 10 2020

More terms from Amiram Eldar, Dec 10 2020

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A334391 Numbers whose only palindromic divisor is 1.

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A088001 a(n) is the sum of non-palindromic divisors of n.

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A334139 Numbers that are equal to the LCM of their palindromic divisors.

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A334321 Non-palindromic primes.

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A334392 Numbers m such that the LCM of their palindromic divisors is neither 1 nor m.

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A339624 Perfect powers p^k, k >= 2 of palindromic primes p when p^k is not a palindrome.

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