cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A244423 Nonprime palindromes n such that the product of divisors of n is also a palindrome.

Original entry on oeis.org

1, 4, 22, 111, 121, 202, 1001, 1111, 10001, 10201, 11111, 100001, 1000001, 1001001, 1012101, 1100011, 1101011, 1111111, 10000001, 100000001, 101000101, 110000011, 200010002, 10000000001, 10011111001, 11000100011, 11001010011, 11100100111, 11101010111, 20000100002
Offset: 1

Views

Author

Derek Orr, Jun 27 2014

Keywords

Comments

Primes trivially satisfy this property and are therefore not included in the sequence.
These are the palindromes in A244411.

Examples

			The divisors of 22 are 1, 2, 11 and 22. 1*2*11*22 = 484 is a palindrome. Since 22 is also a palindrome, 22 is a member of this sequence.

Links

Crossrefs

Cf. A244411, A002113.

Programs

  • Mathematica
    palQ[n_] := Block[{d = IntegerDigits@ n}, Reverse@ d == d]; lim = 15000000; Select[Complement[Range@ lim, Prime@ Range@ PrimePi@ lim], And[palQ@ #, palQ[Times @@ Divisors@ #]] &] (* Michael De Vlieger, Aug 25 2015 *)
Select[Range[200002*10^5],!PrimeQ[#]&&AllTrue[{#,Times@@Divisors[#]},PalindromeQ]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jul 28 2020 *)
  • PARI
    rev(n)={r="";dig=digits(n);for(i=1,#dig,r=concat(Str(dig[i]),r));return(eval(r))}
for(n=1,10^8,if(rev(n)==n&&(!isprime(n)), d=divisors(n);ss=prod(j=1,#d,d[j]);if(ss==rev(ss),print1(n,", "))))
  • PARI
    /* david(n) returns the n-th palindrome from David A. Corneth, Jun 06 2014 */
david(n)={my(d, i, r); r=vector(#digits(n-10^(#digits(n\11)))+#digits(n\11)); n=n-10^(#digits(n\11)); d=digits(n); for(i=1, #d, r[i]=d[i]; r[#r+1-i]=d[i]); sum(i=1, #r, 10^(#r-i)*r[i])}
rev(n)={r="";dig=digits(n);for(i=1,#dig,r=concat(Str(dig[i]),r));return(eval(r))}
for(n=2,10^6,pal=david(n);if(!isprime(pal),d=divisors(pal);ss=prod(j=1,#d,d[j]);if(ss==rev(ss),print1(pal,", "))))
  • Python
    import sympy
from sympy import isprime
from sympy import divisors
def rev(n):
  r = ""
  for i in str(n):
    r = i + r
  return int(r)
def a():
  for n in range(1,10**8):
    if rev(n) == n and not isprime(n):
      p = 1
      for i in divisors(n):
        p*=i
      if rev(p)==p:
        print(n,end=', ')
a()
  • Python
    from sympy import divisor_count, sqrt
def palgen(l,b=10): # generator of palindromes in base b of length <= 2*l
    if l > 0:
        yield 0
        for x in range(1,l+1):
            n = b**(x-1)
            n2 = n*b
            for y in range(n,n2):
                k, m = y//b, 0
                while k >= b:
                    k, r = divmod(k,b)
                    m = b*m + r
                yield y*n + b*m + k
            for y in range(n,n2):
                k, m = y, 0
                while k >= b:
                    k, r = divmod(k,b)
                    m = b*m + r
                yield y*n2 + b*m + k
A244423_list = [1]
for n in palgen(6):
    d = divisor_count(n)
    if d > 2:
        q, r = divmod(d,2)
        s = str(n**q*(sqrt(n) if r else 1))
        if s == s[::-1]:
            A244423_list.append(n) # Chai Wah Wu, Aug 25 2015

Extensions

Edited name by Chai Wah Wu, Aug 25 2015

A327325 Integers with palindromic product of divisors.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 11, 22, 26, 49, 101, 111, 121, 131, 151, 181, 191, 202, 313, 353, 373, 383, 727, 757, 787, 797, 919, 929, 1001, 1111, 2285, 10001, 10201, 10301, 10501, 10601, 11111, 11311, 11411, 12421, 12721, 12821, 13331, 13831, 13931, 14341, 14741, 15451
Offset: 1

Views

Author

Jaroslav Krizek, Aug 30 2019

Keywords

Comments

Numbers m such that A007955(m) = pod(m) are in A002113.
Corresponding values of pod(a(n)): 1, 2, 3, 8, 5, 7, 11, 484, 676, 343, 101, 12321, 1331, 131, 151, ...

Examples

			A007955(49) = 343.

Links

Crossrefs

Cf. A002113, A007955, A028980, A180925, A244411.

Programs

  • Magma
    [m: m in [1..10^5] | Intseq(&*[d: d in Divisors(m)], 10) eq Reverse(Intseq(&*[d: d in Divisors(m)], 10))];
  • Mathematica
    Select[Range[16000], PalindromeQ[#^(DivisorSigma[0, #]/2)] &] (* Amiram Eldar, Aug 31 2019 *)
  • PARI
    isok(n) = my(d=digits(vecprod(divisors(n)))); Vecrev(d) == d; \\ Michel Marcus, Sep 02 2019

A261534 Nonprime palindromes n with only the digits 0, 1, 2 such that the product of divisors of n is also a palindrome.

Original entry on oeis.org

1, 22, 111, 121, 202, 1001, 1111, 10001, 10201, 11111, 100001, 1000001, 1001001, 1012101, 1100011, 1101011, 1111111, 10000001, 100000001, 101000101, 110000011, 200010002, 10000000001, 10011111001, 11000100011, 11001010011, 11100100111, 11101010111, 20000100002
Offset: 1

Views

Author

Chai Wah Wu, Aug 31 2015

Keywords

Comments

A subsequence of A244423.

Links

Crossrefs

Cf. A244411, A244423.

Programs

  • Mathematica
    lim = 1000000; palQ[n_] := Block[{d = IntegerDigits@ n}, d == Reverse@ d]; c = Complement[Range@ lim, Prime@ Range@ PrimePi@ lim]; t = Select[c, Total@ Take[RotateRight@ DigitCount@ #, -7] == 0 &]; Select[t, palQ[Times @@ Divisors@ #] &] (* Michael De Vlieger, Sep 02 2015 *)
Rest[Select[FromDigits/@Tuples[{0,1,2},11],!PrimeQ[#]&&AllTrue[{#,Times@@ Divisors[ #]},PalindromeQ]&]] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Feb 02 2020 *)
  • Python
    from _future_ import division
from sympy import divisor_count
from gmpy2 import isqrt, t_divmod, digits
def palgen(l,b=10): # generator of palindromes in base b of length <= 2*l
    if l > 0:
        yield 0
        for x in range(1,l+1):
            n = b**(x-1)
            n2 = n*b
            for y in range(n,n2):
                k, m = y//b, 0
                while k >= b:
                    k, r = t_divmod(k,b)
                    m = b*m + r
                yield y*n + b*m + k
            for y in range(n,n2):
                k, m = y, 0
                while k >= b:
                    k, r = t_divmod(k,b)
                    m = b*m + r
                yield y*n2 + b*m + k
A261534_list = [1]
for m in palgen(17,3):
    n = int(digits(m,3))
    d = int(divisor_count(n))
    if d > 2:
        q, r = t_divmod(d,2)
        s = digits(n**q*(isqrt(n) if r else 1))
        if s == s[::-1]:
            A261534_list.append(n)
Showing 1-3 of 3 results.

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