A087990 -id:A087990 - cp's OEIS frontend

A093036 Number of palindromic divisors of a(n) sets a new record.

1, 2, 4, 6, 12, 24, 66, 132, 264, 792, 1848, 2772, 5544, 13332, 14652, 24024, 26664, 72072, 79992, 186648, 205128, 264264, 559944, 792792, 1333332, 2666664, 7279272, 7999992, 13333320, 14666652, 26690664, 29333304, 80071992, 134666532, 269333064, 807999192

Offset: 1

Jason Earls, May 08 2004

Beginning with 132, it appears that all entries are congruent mod 11*12; 11 to produce palindromic divisors and 12 for numerous divisors. - Robert G. Wilson v, May 14 2004
The number of palindromic divisors of a(n) are 1, 2, 3, 4, 5, 6, 8, 10, 12, 14, 15, 16, 19, 20, 21, 22, 24, 27, 28, 29, 30, 33, 37, 39, 43, 50, 52, 54, 57, 59, 61, 68, 72, 80, 90.
Every term is of the form Product_{i>=1} A226732(i)^e(i) for e(i) >= 0. - David A. Corneth, Jan 10 2021

Jason Earls, "Palindions," Mathematical Bliss, Pleroma Publications, 2009, pages 115-120. ASIN: B002ACVZ6O. [From Jason Earls, Nov 25 2009]

David A. Corneth, Conjectured next terms

Cf. A087990, A002093, A226732.

palindromicQ[n_, b_:10] := If[FromDigits[Reverse[IntegerDigits[n, b]], b] == n, True, False]; a = 0; Do[c = Count[palindromicQ[ # ] & /@ Divisors[n], True]; If[c > a, Print[n]; a = c], {n, 300000000}] (* Robert G. Wilson v, May 14 2004 with a small modification from Alonso del Arte to permit reuse in many other sequences' programs *)

Edited and extended by Robert G. Wilson v, May 14 2004

a(35)-a(36) from Chai Wah Wu, Jan 21 2021

A087997 a(n) is the least number with n palindromic divisors.

Original entry on oeis.org

1, 2, 4, 6, 12, 24, 72, 66, 484, 132, 616, 264, 2112, 792, 1848, 2772, 7326, 8712, 5544, 13332, 14652, 24024, 36036, 26664, 87912, 102564, 72072, 79992, 186648, 205128, 360360, 279972, 264264, 666666, 1213212, 879912, 559944, 888888, 792792

Offset: 1

Labos Elemer, Oct 13 2003

			n=24: a(24)=26664 has 32 divisors of which 24 are palindromic numbers: {1, 2, 3, 4, 6, 8, 11, 22, 33, 44, 66, 88, 101, 202, 303, 404, 606, 808, 1111, 2222, 3333, 4444, 6666, 8888}.
Some solutions are palindromic (like 2112), some are not (like 132).

Donovan Johnson, Table of n, a(n) for n = 1..150

Cf. A087990.

t = Table[Count[Divisors[k], ?(Reverse[x = IntegerDigits[#]] == x &)], {k, 2,15*10^5, 2}]; Join[{1}, Table[i = 1; While[t[[i]] != k, i++]; 2 i, {k, 2, 39}]] (* _Jayanta Basu, Aug 12 2013 *)
Module[{pds=Table[{n,Count[Divisors[n],?PalindromeQ]},{n,1214000}]},Table[ SelectFirst[pds,#[[2]]==k&],{k,39}]][[All,1]] (* _Harvey P. Dale, Dec 17 2021 *)

a(n)=Min{x; A087990[x]=n}

More terms from Ray Chandler, Oct 17 2003

A087991 Number of non-palindromic divisors of n.

Original entry on oeis.org

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

Offset: 1

Labos Elemer, Oct 08 2003

			For 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}, two of the 12 divisors of n are non-palindromic: {21,132}, so a(132) = 2.

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

Cf. A000005, A087990, A062687.

Cf. A001620, A118031.

palQ[n_] := Reverse[x = IntegerDigits[n]] == x; Table[Count[Divisors[n], ?(! palQ[#] &)], {n, 105}] (* _Jayanta Basu, Aug 10 2013 *)

def ispal(n):
    w=str(n)
    return w==w[::-1]
def A087991(n):
    s = 0
    for i in range(1, n+1):
        if n%i==0 and not ispal(i):
            s+=1
    return s
print([A087991(n) for n in range(1,60)]) # Indranil Ghosh, Feb 10 2017

a(n) = A000005(n) - A087990(n).

Sum_{k=1..n} a(k) ~ n * (log(n) + c), where c = 2*A001620 - 1 - A118031 = -3.2158519... . - Amiram Eldar, Apr 17 2025

A088000 a(n) is the sum of the palindromic divisors of n.

Original entry on oeis.org

1, 3, 4, 7, 6, 12, 8, 15, 13, 8, 12, 16, 1, 10, 9, 15, 1, 21, 1, 12, 11, 36, 1, 24, 6, 3, 13, 14, 1, 17, 1, 15, 48, 3, 13, 25, 1, 3, 4, 20, 1, 19, 1, 84, 18, 3, 1, 24, 8, 8, 4, 7, 1, 21, 72, 22, 4, 3, 1, 21, 1, 3, 20, 15, 6, 144, 1, 7, 4, 15, 1, 33, 1, 3, 9, 7, 96, 12, 1, 20, 13, 3, 1, 23, 6, 3

Offset: 1

Labos Elemer, Oct 14 2003

			n=14: a(14)=1+2+7=10;
n=101: a(101)=1+101=102;

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

Cf. A062687 (all divisors are palindromic), A087990 (number of palindromic divisors).

A088000 := proc(n)
    a := 0 ;
    for d in numtheory[divisors](n) do
        if isA002113(d) then
            a := a+d ;
        end if;
    end do;
    a ;
end proc:
seq(A088000(n),n=1..100) ; # R. J. Mathar, Sep 09 2015

Table[Plus @@ Select[Divisors[k], Reverse[x = IntegerDigits[#]] == x &], {k, 86}] (* Jayanta Basu, Aug 12 2013 *)

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

def ispal(n):
    return n==int(str(n)[::-1])
def A088000(n):
    s=0
    for i in range(1, n+1):
        if n%i==0 and ispal(i):
             s+=i
    return s
print([A088000(n) for n in range(1,30)]) # Indranil Ghosh, Feb 10 2017

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

A087999 a(n) is the LCM of palindromic divisors of n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 1, 14, 15, 8, 1, 18, 1, 20, 21, 22, 1, 24, 5, 2, 9, 28, 1, 30, 1, 8, 33, 2, 35, 36, 1, 2, 3, 40, 1, 42, 1, 44, 45, 2, 1, 24, 7, 10, 3, 4, 1, 18, 55, 56, 3, 2, 1, 60, 1, 2, 63, 8, 5, 66, 1, 4, 3, 70, 1, 72, 1, 2, 15, 4, 77, 6, 1, 40, 9, 2, 1, 84, 5, 2, 3, 88

Offset: 1

Labos Elemer, Oct 14 2003

Sequence is not multiplicative. For example, a(141) = 141 != a(3)*a(47) = 3 * 1. - Franklin T. Adams-Watters, Oct 27 2006

			n=252: a(252)=252=n,since palindromic divisors = {1,2,3,4,6,7,9,252};
n=255: a(255)=15<n, palind.div ={1,3,5}.

Indranil Ghosh, Table of n, a(n) for n = 1..50000

Cf. A087990.

Table[LCM @@ Select[Divisors[k], Reverse[x = IntegerDigits[#]] == x &], {k, 88}] (* Jayanta Basu, Aug 12 2013 *)

ispal(x) = my(d=digits(x)); d == Vecrev(d);
a(n) = lcm(select(x->ispal(x), divisors(n))); \\ Michel Marcus, Mar 27 2020

a(n)=1 for non-palindromic primes like 13.

A355699 a(n) is the smallest number that has exactly n repdigit divisors.

Original entry on oeis.org

1, 2, 4, 6, 12, 24, 72, 66, 666, 132, 1332, 264, 2664, 792, 13320, 3960, 14652, 26664, 48840, 29304, 79992, 341880, 146520, 399960, 1333332, 1025640, 2799720, 8879112, 2666664, 18666648, 7999992, 44395560, 13333320, 93333240, 39999960, 279999720, 269333064

Offset: 1

Bernard Schott, Jul 14 2022

			72 has 12 divisors: {1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72}, only {1, 2, 3, 4, 6, 8, 9} are repdigits; no positive integer smaller than 72 has seven repdigit divisors, hence a(7) = 72.

David A. Corneth, Table of n, a(n) for n = 1..135
David A. Corneth, PARI program

Cf. A010785, A087990, A190217, A340548, A355698.

Similar sequences: A087997, A333456, A355303, A355695.

f[n_] := DivisorSum[n, 1 &, Length[Union[IntegerDigits[#]]] == 1 &]; seq[len_, nmax_] := Module[{s = Table[0, {len}], c = 0, n = 1, i}, While[c < len && n < nmax, i = f[n]; If[i <= len && s[[i]] == 0, c++; s[[i]] = n]; n++]; s]; seq[24, 10^6] (* Amiram Eldar, Jul 15 2022 *)

isrep(n) = 1==#Set(digits(n)); \\ A010785
a(n) = my(k=1); while (sumdiv(k, d, isrep(d)) != n, k++); k; \\ Michel Marcus, Jul 15 2022

\\ See PARI link. - David A. Corneth, Jul 26 2022

from sympy import divisors
from itertools import count, islice
def c(n): return len(set(str(n))) == 1
def f(n): return sum(1 for d in divisors(n, generator=True) if c(d))
def agen():
    n, adict = 1, 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(), 21))) # Michael S. Branicky, Jul 26 2022

a(9)-a(35) from Michael S. Branicky, Jul 14 2022

a(36)-a(37) from Michael S. Branicky, Jul 15 2022

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

cp's OEIS Frontend

A093036 Number of palindromic divisors of a(n) sets a new record.

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A087997 a(n) is the least number with n palindromic divisors.

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A087991 Number of non-palindromic divisors of n.

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A088000 a(n) is the sum of the palindromic divisors of n.

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

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A087999 a(n) is the LCM of palindromic divisors of n.

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A355699 a(n) is the smallest number that has exactly n repdigit divisors.

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