A334391 -id:A334391 - cp's OEIS frontend

A355695 a(n) is the smallest number that has exactly n nonpalindromic divisors (A029742).

1, 10, 20, 30, 48, 72, 60, 140, 144, 120, 210, 180, 300, 240, 560, 504, 360, 420, 780, 1764, 900, 960, 720, 1200, 840, 1560, 2640, 1260, 1440, 2400, 3900, 3024, 1680, 3120, 2880, 4800, 7056, 3600, 2520, 3780, 3360, 5460, 6480, 16848, 6300, 8820, 7200, 9240, 6720, 12480, 5040

Offset: 0

Bernard Schott, Jul 14 2022

			48 has 10 divisors: {1, 2, 3, 4, 6, 8, 12, 16, 24, 48}, only 12, 16, 24 and 48 are nonpalindromic; no positive integer smaller than 48 has four nonpalindromic divisors, hence a(4) = 48.

Michael S. Branicky, Table of n, a(n) for n = 0..710

Cf. A029742, A087991, A093037, A334391.

Similar sequences: A087997, A333456, A355303, A355594.

f[n_] := DivisorSum[n, 1 &, ! PalindromeQ[#] &]; seq[len_, nmax_] := Module[{s = Table[0, {len}], c = 0, n = 1, i}, While[c < len && n < nmax, i = f[n] + 1; If[i <= len && s[[i]] == 0, c++; s[[i]] = n]; n++]; s]; seq[50, 10^5] (* Amiram Eldar, Jul 14 2022 *)

isnp(n) = my(d=digits(n)); d!=Vecrev(d); \\ A029742
a(n) = my(k=1); while (sumdiv(k, d, isnp(d)) != n, k++); k; \\ Michel Marcus, Jul 14 2022

from sympy import divisors
from itertools import count, islice
def c(n): s = str(n); return s != s[::-1]
def f(n): return sum(1 for d in divisors(n, generator=True) if c(d))
def agen():
    n, adict = 0, dict()
    for k in count(1):
        fk = f(k)
        if fk not in adict: adict[fk] = k
        while n in adict: yield adict[n]; n += 1
print(list(islice(agen(), 51))) # Michael S. Branicky, Jul 27 2022

More terms from Michel Marcus, Jul 14 2022

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

Jaroslav Krizek, Dec 25 2010

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

Cf. A088000, A183107.

Subsequences: A334321, A334391.

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

q[k_] := PalindromeQ[DivisorSum[k, # &, PalindromeQ[#] &]]; Select[Range[120], q] (* Amiram Eldar, Aug 08 2024 *)

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

More terms from Amiram Eldar, Aug 08 2024

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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A355695 a(n) is the smallest number that has exactly n nonpalindromic divisors (A029742).

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

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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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