A087991 -id:A087991 - cp's OEIS frontend

A062687 Numbers all of whose divisors are palindromic.

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 33, 44, 55, 66, 77, 88, 99, 101, 121, 131, 151, 181, 191, 202, 242, 262, 303, 313, 353, 363, 373, 383, 393, 404, 484, 505, 606, 626, 707, 727, 757, 787, 797, 808, 909, 919, 929, 939, 1111, 1331, 1441, 1661, 1991, 2222, 2662

Offset: 1

Erich Friedman, Jul 04 2001

			The divisors of 44 are 1, 2, 4, 11, 22 and 44, which are all palindromes, so 44 is in the sequence.
808 has divisors are 1, 2, 4, 8, 101, 202, 404, 808, so 808 is in the sequence.
818 is palindromic, but since it's 2 * 409, it's not in the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..221 from Indranil Ghosh)

Cf. A087991, A084325, A002385 (subset).

Subsequence of A002113.

isA062687 := proc(n)
    for d in numtheory[divisors](n) do
        if not isA002113(d) then
            return false;
        end if;
    end do;
    true ;
end proc: # R. J. Mathar, Sep 09 2015

palQ[n_] := Module[{idn = IntegerDigits[n]}, idn == Reverse[idn]]; Select[Range[2750], And@@palQ/@Divisors[#] &] (* Harvey P. Dale, Feb 27 2012 *)

isok(n) = {d = divisors(n); rd = vector(#d, i, subst(Polrev(digits(d[i])), x, 10)); (d == rd);} \\ Michel Marcus, Oct 10 2014

A087990 Number of palindromic divisors of n.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 4, 3, 3, 2, 5, 1, 3, 3, 4, 1, 5, 1, 4, 3, 4, 1, 6, 2, 2, 3, 4, 1, 5, 1, 4, 4, 2, 3, 6, 1, 2, 2, 5, 1, 5, 1, 6, 4, 2, 1, 6, 2, 3, 2, 3, 1, 5, 4, 5, 2, 2, 1, 6, 1, 2, 4, 4, 2, 8, 1, 3, 2, 4, 1, 7, 1, 2, 3, 3, 4, 4, 1, 5, 3, 2, 1, 6, 2, 2, 2, 8, 1, 6, 2, 3, 2, 2, 2, 6, 1, 3, 6, 4, 2, 4, 1, 4, 4

Offset: 1

Labos Elemer, Oct 08 2003

			n=132: divisors={1, 2, 3, 4, 6, 11, 12, 22, 33, 44, 66, 132}, revdivisors={1, 2, 3, 4, 6, 11, 21, 22, 33, 44, 66, 231}, a[132]=10; so 10 of 12 divisors of n are palindromic: {1, 2, 3, 4, 6, 11, 22, 33, 44, 66}.

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

Cf. A062687, A087991, A088000, A118031.

nd[x_, y_] := 10*x+y; tn[x_] := Fold[nd, 0, x]; rdi[x_] := tn[Reverse[IntegerDigits[x]]]; d0[x_] := DivisorSigma[0, x]; di[x_, i_] := Part[Divisors[x], i]; Table[Count[Divisors[s]-Table[rdi[di[s, w]], {w, 1, d0[s]}], 0], {s, 1, 256}]
palQ[n_] := Reverse[x = IntegerDigits[n]] == x; Table[Count[Divisors[n], ?(palQ[#] &)], {n, 105}] (* _Jayanta Basu, Aug 10 2013 *)
Table[Count[Divisors[n],?PalindromeQ],{n,110}] (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale, Jun 28 2017 *)

a(n) = sumdiv(n, d, my(dd=digits(d)); Vecrev(dd) == dd); \\ Michel Marcus, Apr 06 2020

def ispal(n):
    t = str(n)
    return t == t[::-1]
def A087990(n):
    s=0
    for i in range(1,n+1):
        if n%i==0 and ispal(i):
            s+=1
    return s # Indranil Ghosh, Feb 10 2017

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = A118031 = 3.370283... . - Amiram Eldar, Jan 01 2024

A356018 a(n) is the number of evil divisors (A001969) of n.

Original entry on oeis.org

0, 0, 1, 0, 1, 2, 0, 0, 2, 2, 0, 3, 0, 0, 3, 0, 1, 4, 0, 3, 1, 0, 1, 4, 1, 0, 3, 0, 1, 6, 0, 0, 2, 2, 1, 6, 0, 0, 2, 4, 0, 2, 1, 0, 5, 2, 0, 5, 0, 2, 3, 0, 1, 6, 1, 0, 2, 2, 0, 9, 0, 0, 3, 0, 2, 4, 0, 3, 2, 2, 1, 8, 0, 0, 4, 0, 1, 4, 0, 5, 3, 0, 1, 3, 3, 2

Offset: 1

Bernard Schott, Jul 23 2022

a(n) = 0 iff n is in A093696.

			12 has 6 divisors: {1, 2, 3, 4, 6, 12} of which three {3, 6, 12} have an even number of 1's in their binary expansion with 11, 110 and 11100 respectively; hence a(12) = 3.

Cf. A000005, A001969, A093688, A093696 (location of 0s), A227872, A356019, A356020.

Similar sequences: A083230, A087990, A087991, A332268, A355302.

A356018 := proc(n)
    local a,d ;
    a := 0 ;
    for d in numtheory[divisors](n) do
        if isA001969(d) then
            a := a+1 ;
        end if;
    end do:
    a ;
end proc:
seq(A356018(n),n=1..200) ;  # R. J. Mathar, Aug 07 2022

a[n_] := DivisorSum[n, 1 &, EvenQ[DigitCount[#, 2, 1]] &]; Array[a, 100] (* Amiram Eldar, Jul 23 2022 *)

a(n) = my(v = valuation(n, 2)); n>>=v; d=divisors(n); sum(i=1, #d, bitand(hammingweight(d[i]), 1) == 0) * (v+1) \\ David A. Corneth, Jul 23 2022

from sympy import divisors
def c(n): return bin(n).count("1")&1 == 0
def a(n): return sum(1 for d in divisors(n, generator=True) if c(d))
print([a(n) for n in range(1, 101)]) # Michael S. Branicky, Jul 23 2022

a(n) = A000005(n) - A227872(n).

More terms from David A. Corneth, Jul 23 2022

A093037 Number of non-palindromic divisors of n sets a new record.

Original entry on oeis.org

1, 10, 20, 30, 48, 60, 120, 180, 240, 360, 420, 720, 840, 1260, 1440, 1680, 2520, 3360, 5040, 7560, 10080, 15120, 20160, 25200, 30240, 43680, 45360, 50400, 55440, 60480, 65520, 90720, 98280, 110880, 131040, 166320, 196560, 262080, 327600

Offset: 1

Jason Earls, May 08 2004

Cf. A087991.

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

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.

A335037 a(n) is the number of divisors of n that are themselves divisible by the product of their digits.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 4, 3, 3, 2, 6, 1, 3, 4, 4, 1, 5, 1, 4, 3, 3, 1, 8, 2, 2, 3, 4, 1, 6, 1, 4, 3, 2, 3, 8, 1, 2, 2, 5, 1, 5, 1, 4, 5, 2, 1, 8, 2, 3, 2, 3, 1, 5, 3, 5, 2, 2, 1, 8, 1, 2, 4, 4, 2, 5, 1, 3, 2, 4, 1, 10, 1, 2, 4, 3, 3, 4, 1, 5, 3, 2, 1, 7, 2, 2, 2, 5

Offset: 1

Bernard Schott, Jun 03 2020

Inspired by A332268.
A number that is divisible by the product of its digits is called Zuckerman number (A007602); e.g., 24 is a Zuckerman number because it is divisible by 2*4=8 (see links).
a(n) = 1 iff n = 1 or n is prime not repunit >= 13.
a(n) = 2 iff n is prime = 2, 3, 5, 7 or a prime repunit.
Numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 15, 24, 111111111111111111111 (repunit with 19 times 1's) have all divisors Zuckerman numbers. The sequence of numbers with all Zuckerman divisors is infinite iff there are infinitely many repunit primes (see A004023).

			For n = 4, the divisors are 1, 2, 4 and they are all Zuckerman numbers, so a(4) = 3.
For n = 14, the divisors are 1, 2, 7 and 14. Only 1, 2 and 7 are Zuckerman numbers, so a(14) = 3.

David A. Corneth, Table of n, a(n) for n = 1..10000
Giovanni Resta, Zuckerman numbers, Numbers Aplenty.
Gérard Villemin, Nombres de Zuckerman, Types de nombres.

Cf. A000005, A004023, A007602, A335038.

Similar with: A001227 (odd divisors), A087990 (palindromic divisors), A087991 (non-palindromic divisors), A242627 (divisors < 10), A332268 (Niven divisors).

zuckQ[n_] := (prodig = Times @@ IntegerDigits[n]) > 0&& Divisible[n, prodig]; a[n_] := Count[Divisors[n], ?(zuckQ[#] &)]; Array[a, 100] (* _Amiram Eldar, Jun 03 2020 *)

iszu(n) = my(p=vecprod(digits(n))); p && !(n % p);
a(n) = sumdiv(n, d, iszu(d)); \\ Michel Marcus, Jun 03 2020

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{n>=1} 1/A007602(n) = 3.26046... . - Amiram Eldar, Jan 01 2024

A355770 a(n) is the number of terms of A333369 that divide n.

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 2, 1, 3, 2, 1, 2, 2, 2, 4, 1, 2, 3, 2, 2, 3, 2, 1, 2, 2, 2, 3, 2, 1, 4, 2, 1, 2, 2, 4, 3, 2, 2, 4, 2, 1, 3, 1, 3, 5, 1, 1, 2, 2, 2, 4, 2, 2, 3, 2, 2, 4, 1, 2, 4, 1, 2, 4, 1, 3, 4, 1, 2, 2, 4, 2, 3, 2, 2, 5, 2, 2, 4, 2, 2, 3, 1, 1, 3, 3, 1, 2

Offset: 1

Bernard Schott, Jul 16 2022

Cf. A333369, A353735, A355771, A355772, A355773.

Similar sequences: A083230, A087990, A087991, A332268, A355302.

q[n_] := AllTrue[Tally @ IntegerDigits[n], EvenQ[Plus @@ #] &]; a[n_] := DivisorSum[n, 1 &, q[#] &]; Array[a, 100] (* Amiram Eldar, Jul 16 2022 *)

issimber(m) = my(d=digits(m), s=Set(d)); for (i=1, #s, if (#select(x->(x==s[i]), d) % 2 != (s[i] % 2), return (0))); return (1); \\ A333369
a(n) = sumdiv(n, d, issimber(d)); \\ Michel Marcus, Jul 18 2022

from sympy import divisors
def c(n): s = str(n); return all(s.count(d)%2 == int(d)%2 for d in set(s))
def a(n): return sum(1 for d in divisors(n, generator=True) if c(d))
print([a(n) for n in range(1, 88)]) # Michael S. Branicky, Jul 16 2022

More terms from Michael S. Branicky, Jul 16 2022

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

Original entry on oeis.org

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

A179937 a(n) is the product of the non-palindromic divisors of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 10, 1, 12, 13, 14, 15, 16, 17, 18, 19, 200, 21, 1, 23, 288, 25, 338, 27, 392, 29, 4500, 31, 512, 1, 578, 35, 7776, 37, 722, 507, 8000, 41, 12348, 43, 1, 675, 1058, 47, 221184, 49, 12500, 867, 17576, 53, 26244, 1, 21952, 1083, 1682, 59

Offset: 1

Jaroslav Krizek, Jan 12 2011

			For n = 20, set of non-palindromic divisors is {10, 20}; a(12) = 10*20 = 200.

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

Cf. A087990, A087991, A088000, A088001, A184392.

Table[Times@@Select[Divisors[n],!PalindromeQ[#]&],{n,60}] (* Harvey P. Dale, May 15 2023 *)

def ispal(n):
    return n==int(str(n)[::-1])
def A179937(n):
    s=1
    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) = A007955(n) / A184392(n).

More terms from Indranil Ghosh, Feb 10 2017

A184392 a(n) is the product of palindromic divisors of n.

Original entry on oeis.org

1, 2, 3, 8, 5, 36, 7, 64, 27, 10, 11, 144, 1, 14, 15, 64, 1, 324, 1, 40, 21, 484, 1, 1152, 5, 2, 27, 56, 1, 180, 1, 64, 1089, 2, 35, 1296, 1, 2, 3, 320, 1, 252, 1, 85184, 135, 2, 1, 1152, 7, 10, 3, 8, 1, 324, 3025, 448, 3, 2, 1, 720, 1, 2, 189, 64, 5, 18974736, 1, 8, 3, 70, 1, 10368, 1, 2, 15, 8, 5929, 36, 1, 320, 27, 2, 1, 1008, 5, 2, 3, 59969536, 1, 1620, 7, 8, 3, 2, 5, 1152, 1, 14, 970299, 40

Offset: 1

Jaroslav Krizek, Jan 12 2011

			For n = 20, set of palindromic divisors is {1, 2, 4, 5}; a(12) = 1*2*4*5 = 40.

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

Cf. A007955, A087990, A087991, A088000, A088001, A179937.

palQ[n_]:=Module[{idn=IntegerDigits[n]}, idn==Reverse[idn]]; f[n_]:=Times@@Select[Divisors[n],palQ]; Table[f[n],{n,100}]  (* Harvey P. Dale, Jan 21 2011 *)

def ispal(n):
    return n==int(str(n)[::-1])
def A184392(n):
    s=1
    for i in range(1, n+1):
        if n%i==0 and ispal(i):
             s*=i
    return s # Indranil Ghosh, Feb 10 2017

a(n) = A007955(n) / A179937(n).

More terms from Harvey P. Dale, Jan 21 2011

cp's OEIS Frontend

A062687 Numbers all of whose divisors are palindromic.

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A087990 Number of palindromic divisors of n.

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A356018 a(n) is the number of evil divisors (A001969) of n.

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A093037 Number of non-palindromic divisors of n sets a new record.

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A334391 Numbers whose only palindromic divisor is 1.

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A335037 a(n) is the number of divisors of n that are themselves divisible by the product of their digits.

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A355770 a(n) is the number of terms of A333369 that divide n.

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