A088000 -id:A088000 - cp's OEIS frontend

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

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

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

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

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

A183107 Sum of palindromic divisors of numbers k such that the sum of the palindromic divisors of k is palindromic.

Original entry on oeis.org

1, 3, 4, 7, 6, 8, 8, 1, 9, 1, 1, 11, 1, 6, 3, 1, 1, 3, 1, 3, 4, 1, 1, 3, 1, 8, 8, 4, 7, 1, 22, 4, 3, 1, 1, 3, 6, 1, 7, 4, 1, 33, 1, 3, 9, 7, 1, 3, 1, 6, 3, 4, 1, 8, 7, 4, 3, 6, 1, 1, 3, 1, 1, 22, 1, 6, 7, 3, 8, 3, 4, 7, 6, 1, 4, 8, 8, 3, 1, 1, 3, 33, 6, 3, 11, 7, 1

Offset: 1

Jaroslav Krizek, Dec 25 2010

Cf. A088000, A183106.

f[k_] := DivisorSum[k, # &, PalindromeQ[#] &]; Select[f /@ Range[150], PalindromeQ] (* Amiram Eldar, Aug 08 2024 *)

a(n) = A088000(A183106(n)).

More terms from Amiram Eldar, Aug 08 2024

A183108 Numbers m such that sum of divisors of m and sum of palindromic divisors of m are both palindromic.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 43, 130, 146, 166, 201, 205, 211, 221, 241, 244, 251, 271, 274, 281, 314, 325, 365, 388, 422, 433, 443, 463, 489, 519, 559, 633, 685, 793, 827, 857, 877, 887, 1841, 2021, 2111, 2221, 2284, 2305, 2441, 2551, 2561, 2666, 2751, 2881

Offset: 1

Jaroslav Krizek, Dec 25 2010

Numbers m such that A000203(m) and A088000(m) are both palindromic.

			a(8) = 130, divisors of 130: 1, 2, 5, 10, 13, 26, 65, 130; palindromic divisors of 130: 1, 2, 5; A000203(130) = 252, A088000(130) = 8; both numbers are palindromic.

Cf. A000203, A088000.

is_palindrome = lambda n, base=10: n.str(base) == n.str(base)[::-1]
A000203 = sigma
A088000 = lambda n: sum(d for d in divisors(n) if is_palindrome(d))
is_A183108 = lambda n: is_palindrome(A000203(n)) and is_palindrome(A088000(n)) # D. S. McNeil, Dec 28 2010

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

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A087990 Number of palindromic divisors of n.

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

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

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

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A183107 Sum of palindromic divisors of numbers k such that the sum of the palindromic divisors of k is palindromic.

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A183108 Numbers m such that sum of divisors of m and sum of palindromic divisors of m are both palindromic.

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