A062687 -id:A062687 - cp's OEIS frontend

A084325 Palindromes such that at least one divisor is non-palindromic; palindromes not in A062687.

111, 141, 161, 171, 212, 222, 232, 252, 272, 282, 292, 323, 333, 343, 414, 424, 434, 444, 454, 464, 474, 494, 515, 525, 535, 545, 555, 565, 575, 585, 595, 616, 636, 646, 656, 666, 676, 686, 696, 717, 737, 747, 767, 777, 818, 828, 838, 848, 858, 868, 878

Offset: 1

Jason Earls, Jun 21 2003

			141 is in the sequence because the divisors of 141 are 1, 3, 47 and 141, from which 47 is the only non-palindromic number. - _Indranil Ghosh_, Feb 10 2017

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

palQ[n_] := Reverse[x = IntegerDigits[n]] == x; Select[Range[880], palQ[#] && And @@ palQ /@ Divisors[#] == False &] (* Jayanta Basu, Aug 10 2013 *)

A084982 Erroneous version of A062687.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 33, 44, 55, 66, 77, 88, 99, 101, 111, 121, 131, 151, 181

Offset: 1

dead

A087992 Duplicate of A062687.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 33, 44, 55, 66, 77, 88, 99, 101, 121, 131, 151, 181, 191

Offset: 1

dead

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

A329419 Numbers all of whose divisors are binary palindromes.

Original entry on oeis.org

1, 3, 5, 7, 9, 15, 17, 21, 27, 31, 45, 51, 63, 73, 85, 93, 107, 119, 127, 153, 189, 219, 255, 257, 313, 365, 381, 443, 511, 765, 771, 1193, 1241, 1285, 1453, 1533, 1571, 1619, 1787, 1799, 1831, 1879, 2313, 3579, 3855, 4369, 4889, 5113, 5189, 5397, 5557, 5869

Offset: 1

Amiram Eldar, Nov 29 2019

Subsequence of A163410, and differs from it from n = 65.

			15 is in the sequence since the binary representations of its divisors, 1, 3, 5, and 15, are all palindromes: 1, 11, 101, and 1111.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A006995, A062687, A163410.

Supersequence of A016041.

binPalQ[n_] := PalindromeQ @ IntegerDigits[n, 2]; seqQ[n_] := binPalQ[n] && AllTrue[Most @ Divisors[n], binPalQ]; Select[Range[10^4], seqQ]

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

A093696 Numbers n such that all divisors of n have an odd number of 1's in their binary expansions.

Original entry on oeis.org

1, 2, 4, 7, 8, 11, 13, 14, 16, 19, 22, 26, 28, 31, 32, 37, 38, 41, 44, 47, 49, 52, 56, 59, 61, 62, 64, 67, 73, 74, 76, 79, 82, 88, 91, 94, 97, 98, 103, 104, 107, 109, 112, 118, 121, 122, 124, 127, 128, 131, 133, 134, 137, 143, 146, 148, 151, 152, 157, 158, 164, 167, 173

Offset: 1

Jason Earls, May 16 2004

Subsequence of A000069. - Michel Marcus, Feb 09 2014

Numbers all of whose divisors are odious. - Bernard Schott, Jul 22 2022

			14 is in the sequence because its divisors are [1, 2, 7, 14] and in binary: 1, 10, 111 and 1110, all have an odd number of 1's.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000069, A001969, A227872, A330289, A355968, A355969.
Similar sequences: A062687, A190217, A337741, A337941, A355596.
A000079 is a subsequence.

isA001969 := proc(n)
    if wt(n) mod 2 = 0 then
        true;
    else
        false;
    end if;
end proc:
isA093696 := proc(n)
    for d in numtheory[divisors](n) do
        if isA001969(d) then
            return false;
        end if;
    end do;
    true;
end proc:
for n from 1 to 200 do
    if isA093696(n) then
        printf("%d,",n);
    end if;
end do: # R. J. Mathar, Feb 13 2014

odiousQ[n_] := OddQ @ DigitCount[n, 2][[1]]; Select[Range[200], AllTrue[ Divisors[#], odiousQ ] &] (* Amiram Eldar, Dec 09 2019 *)

is(n)=fordiv(n,d,if(hammingweight(d)%2==0, return(0))); 1 \\ Charles R Greathouse IV, Mar 29 2013

from sympy import divisors, isprime
def c(n): return bin(n).count("1")&1
def ok(n): return n > 0 and all(c(d) for d in divisors(n, generator=True))
print([k for k in range(174) if ok(k)]) # Michael S. Branicky, Jul 24 2022

{n: A356018(n) =0 }. - R. J. Mathar, Aug 07 2022

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

A308851 Numbers >= 2 all of whose divisors > 1 are Brazilian.

Original entry on oeis.org

7, 13, 31, 43, 73, 91, 127, 157, 211, 217, 241, 301, 307, 403, 421, 463, 511, 559, 601, 757, 889, 949, 1093, 1099, 1123, 1333, 1477, 1483, 1651, 1687, 1723, 2041, 2149, 2263, 2551, 2743, 2801, 2821, 2947, 2971, 3133, 3139, 3241, 3307, 3541, 3907, 3913, 3937

Offset: 1

Bernard Schott, Jun 28 2019

nonn

The terms of this sequence are the Brazilian primes and the products of two or more distinct Brazilian primes.

There are no even numbers because 2 is not Brazilian.

			91 is a term because all divisors of 91 that are > 1: {7, 13, 91} are Brazilian numbers with 7 = 111_2, 13 = 111_3 and 91 = 77_12.

Amiram Eldar, Table of n, a(n) for n = 1..1000
Index entries for sequences related to Brazilian numbers

Cf. A085104 (subsequence), A125134.

Similar with even numbers: A000079, with odd numbers: A005408, with palindromes: A062687, with repdigits: A190217.

brazQ[n_] := Block[{k, b, ok}, If[FindInstance[k (1 + b) == n && 1 < b < n - 1 && 0 < k < b, {k, b}, Integers] != {}, True, b = 2; ok = False; While[1 + b + b^2 <= n && ! ok, ok = Length@ Union@ IntegerDigits[n, b++] == 1]; ok]]; Select[ Range[3, 4000, 2], AllTrue[ Rest@ Divisors@ #, brazQ] &] (* Giovanni Resta, Jun 29 2019 *)
max = 5000; fQ[n_] := Module[{b = 2, found = False}, While[b < n - 1 && Length[Union[IntegerDigits[n, b]]] > 1, b++]; b < n - 1]; A125134 = Select[Range[4, max], fQ]; Select[Range[2, max], Intersection[A125134, Rest[Divisors[#]]] == Rest[Divisors[#]] &] (* Vaclav Kotesovec, Jun 29 2019, using a subroutine from T. D. Noe *)

isb(n) = for(b=2, n-2, d=digits(n, b); if(vecmin(d)==vecmax(d), return(1)));
isok(n) = {fordiv(n, d, if ((d>1) && ! isb(d), return (0));); return (1);} \\ Michel Marcus, Jun 29 2019

A337741 Numbers all of whose divisors are Niven numbers (A005349).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 18, 20, 21, 24, 27, 36, 40, 54, 63, 72, 81, 108, 162, 216, 243, 324, 486, 648, 972, 1944

Offset: 1

Amiram Eldar, Sep 17 2020

Since the only prime Niven numbers are the single-digit primes 2, 3, 5 and 7, all the terms are 7-smooth numbers (A002473).

If k is a term, all the divisors of k are also terms. Since all the terms are 7-smooth, every term is of the form p * k, where p is in {2, 3, 5, 7} and k is a smaller term. Thus it is easy to verify that there are only 31 terms in this sequence, and 1944 being the last term.

			6 is a term since all the divisors of 6, i.e., 1, 2, 3 and 6, are Niven numbers.

Subsequence of A002473 and A005349.
Cf. A332268, A335708, A335709.
Similar sequences: A062687, A190217, A329419.

nivenQ[n_] := Divisible[n, Plus @@ IntegerDigits[n]]; allQ[n_] := AllTrue[Divisors[n], nivenQ]; p = {1, 2, 3, 5, 7}; s = {1}; n = 0; While[Length[s] != n, n = Length[s]; s = Select[Union @ Flatten @ Outer[Times, s, p], allQ]]; s

cp's OEIS Frontend

A084325 Palindromes such that at least one divisor is non-palindromic; palindromes not in A062687.

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A084982 Erroneous version of A062687.

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A087992 Duplicate of A062687.

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A087990 Number of palindromic divisors of n.

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A329419 Numbers all of whose divisors are binary palindromes.

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A087991 Number of non-palindromic divisors of n.

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A093696 Numbers n such that all divisors of n have an odd number of 1's in their binary expansions.

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

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A308851 Numbers >= 2 all of whose divisors > 1 are Brazilian.

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