A087990 Number of palindromic divisors of n.

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

Author

Labos Elemer, Oct 08 2003

Keywords

• nonn
• base

Examples

			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}.
		

Links

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

Crossrefs

Cf. A062687, A087991, A088000, A118031.

Programs

Mathematica

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 *)

PARI

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

Python

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

Formula

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