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 1-5 of 5 results.

A324989 Palindromes whose product of divisors is palindromic.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 11, 22, 101, 111, 121, 131, 151, 181, 191, 202, 313, 353, 373, 383, 727, 757, 787, 797, 919, 929, 1001, 1111, 10001, 10201, 10301, 10501, 10601, 11111, 11311, 11411, 12421, 12721, 12821, 13331, 13831, 13931, 14341, 14741, 15451, 15551, 16061
Offset: 1

Views

Author

Jaroslav Krizek, Mar 23 2019

Keywords

Comments

Numbers m such that m and A007955(m) = pod(m) are both in A002113.
Of 48025 terms < 10^11, all but 30 are prime. - Robert Israel, Apr 23 2019

Examples

			Product of divisors of palindrome number 22 with divisors 1, 2, 11 and 22 is 484 (palindrome number).

Links

Crossrefs

Cf. A007955, A002113, A069747.
Includes A002385.
Similar sequences for functions sigma(m) and tau(m): A028986, A324988.

Programs

  • Magma
    [n: n in [1..100000] | Intseq(n, 10) eq Reverse(Intseq(n, 10)) and Intseq(&*[d: d in Divisors(n)], 10) eq Reverse(Intseq(&*[d: d in Divisors(n)], 10))]
  • Maple
    revdigs:= proc(n)
local L, nL, i;
L:= convert(n, base, 10);
nL:= nops(L);
add(L[i]*10^(nL-i), i=1..nL);
end:
pals:= proc(d) local x, y;
  if d::even then [seq(x*10^(d/2)+revdigs(x), x=10^(d/2-1)..10^(d/2)-1)]
  else [seq(seq(x*10^((d+1)/2)+y*10^((d-1)/2)+revdigs(x), y=0..9), x=10^((d-1)/2-1)..10^((d-1)/2)-1)]
  fi
end proc:
pals(1):= [$1..9]:
filter:= proc(n) local v;
  v:= convert(numtheory:-divisors(n),`*`);
  revdigs(v)=v
end proc:
seq(op(select(filter, pals(d))),d=1..5); # Robert Israel, Apr 23 2019
  • Mathematica
    Select[Range[10^5], And[PalindromeQ@ #, PalindromeQ[Times @@ Divisors@ #]] &] (* Michael De Vlieger, Mar 24 2019 *)
Select[Range[17000],AllTrue[{#,Times@@Divisors[#]},PalindromeQ]&] (* Harvey P. Dale, Oct 13 2021 *)
  • PARI
    ispal(n) = my(d=digits(n)); Vecrev(d) == d;
isok(n) = ispal(n) && ispal(vecprod(divisors(n))); \\ Michel Marcus, Mar 23 2019
  • Python
    from math import isqrt
from itertools import chain, count, islice
from sympy import divisor_count
def A324989_gen(): # generator of terms
    return filter(lambda n:(s:=str(isqrt(n)**d if (d:=divisor_count(n)) & 1 else n**(d//2)))[:(t:=(len(s)+1)//2)]==s[:-t-1:-1],chain.from_iterable(chain((int((s:=str(d))+s[-2::-1]) for d in range(10**l,10**(l+1))), (int((s:=str(d))+s[::-1]) for d in range(10**l,10**(l+1)))) for l in count(0)))
A324989_list = list(islice(A324989_gen(),20)) # Chai Wah Wu, Jun 24 2022

A324988 Palindromes whose number of divisors is palindromic.

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, 141, 151, 161, 171, 181, 191, 202, 212, 222, 232, 242, 262, 282, 292, 303, 313, 323, 333, 343, 353, 363, 373, 383, 393, 404, 424, 434, 454, 474, 484, 494, 505, 515, 535, 545
Offset: 1

Views

Author

Jaroslav Krizek, Mar 23 2019

Keywords

Comments

Numbers m such that m and A000005(m) = tau(m) are both in A002113.

Examples

			Number of divisors of palindrome number 22 with divisors 1, 2, 11 and 22 is 4 (palindrome number).

Crossrefs

Cf. A000005, A002113, A069747.
Similar sequences for functions sigma(m) and pod(m): A028986, A324989.
Includes A002385, A046328 and A046329.

Programs

  • Magma
    [n: n in [1..1000] | Intseq(n, 10) eq Reverse(Intseq(n, 10)) and Intseq(NumberOfDivisors(n), 10) eq Reverse(Intseq(NumberOfDivisors(n), 10))]
  • Maple
    ispali:= proc(n) local L; L:= convert(n,base,10); L = ListTools:-Reverse(L) end proc:
select(t -> ispali(t) and ispali(numtheory:-tau(t)), [$1..10000]); # Robert Israel, Mar 26 2019
  • Mathematica
    Select[Range@ 600, And[PalindromeQ@ #, PalindromeQ@ DivisorSigma[0, #]] &] (* Michael De Vlieger, Mar 24 2019 *)
  • PARI
    ispal(n) = my(d=digits(n)); Vecrev(d) == d;
isok(n) = ispal(n) && ispal(numdiv(n)); \\ Michel Marcus, Mar 23 2019

A227947 Each term is a palindrome such that the sum of its proper divisors is a palindrome > 1.

Original entry on oeis.org

4, 6, 8, 9, 333, 646, 656, 979, 1001, 3553, 10801, 11111, 18581, 31713, 34943, 48484, 57375, 95259, 99099, 158851, 262262, 569965, 1173711, 1216121, 1399931, 1439341, 1502051, 1925291, 3203023, 3436343, 3659563, 3662663, 3803083, 3888883, 5185815, 5352535, 5893985, 5990995, 6902096, 9341439, 9452549
Offset: 1

Views

Author

Derek Orr, Oct 03 2013

Keywords

Comments

All terms are composite numbers. - Chai Wah Wu, Dec 23 2015

Examples

			4 has proper divisors 1 and 2. 1 + 2 = 3 is also a palindrome. So 4 is a member of this sequence.

Links

Crossrefs

Cf. A001065, A032350, A028986.

Programs

  • Mathematica
    palQ[n_] := Block[{d = IntegerDigits@ n}, d == Reverse@ d]; fQ[n_] := Block[{s = DivisorSigma[1, n] - n}, palQ@ s && s > 1]; Select[
Select[Range@ 1000000, palQ], fQ] (* Michael De Vlieger, Apr 06 2015 *)
spdQ[n_]:=Module[{spd=DivisorSigma[1,n]-n},n==IntegerReverse[n] && spd>1 && spd==IntegerReverse[spd]]; Select[Range[10^7],spdQ] (* The program uses the IntegerReverse function from Mathematica version 10 *) (* Harvey P. Dale, Jan 03 2016 *)
  • PARI
    pal(n)=d=digits(n);Vecrev(d)==d
for(n=1,10^6,s=sigma(n)-n;if(pal(n)&&pal(s)&&s>1,print1(n,", "))) \\ Derek Orr, Apr 05 2015
  • Python
    from sympy import divisors
def pal(n):
  r = ''
  for i in str(n):
    r = i + r
  return r == str(n)
{print(n,end=', ') for n in range(1,10**7) if pal(n) and pal(sum(divisors(n))-n) and len(divisors(n)) > 2}
## Simplified by Derek Orr, Apr 05 2015

Extensions

Initial terms 0 and 1 removed and more terms added by Derek Orr, Apr 05 2015
Definition edited by Derek Orr, Apr 05 2015
Definition edited by Harvey P. Dale, Jan 03 2016

A240466 Palindromes for which both the numerator (A017665) and the denominator (A017666) of sigma(n)/n are palindromes, where sigma is the sum of divisors (A000203).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 333, 17571, 40004, 93939, 569965, 1787871, 2316132, 541626145, 17575757571, 5806270726085, 7359770779537, 520524424425025, 17275787578757271, 17878787578787871
Offset: 1

Views

Author

Michel Marcus, Apr 06 2014

Keywords

Comments

Compare with A028986 (Palindromes whose sum of divisors is palindromic).
These terms of A028986 also belong here: 1, 2, 3, 4, 5, 7, 333, 17571, 1787871, 541626145, 17575757571, 5806270726085, 7359770779537.
a(22) > 10^18. - Hiroaki Yamanouchi, Sep 27 2014

Crossrefs

Cf. A000203, A017665, A017666, A028986.

Programs

  • PARI
    reverse(expr)=my(v=Vec(Str(expr)),n=length(v));eval(concat(vector(n,i,v[n-i+1])));
isok(n) = (rn = reverse(n)) && (rn == n) && (ab = sigma(n)/n) && (abr = sigma(rn)/rn) && (numerator(abr) == reverse(numerator(ab))) && (denominator(abr) == reverse(denominator(ab)));

Extensions

a(16)-a(21) from Hiroaki Yamanouchi, Sep 27 2014

A327324 Palindromes whose number and sum of divisors are both also palindromic.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 333, 17571, 1757571, 1787871, 5136315, 518686815, 541626145, 17575757571, 5136813186315, 5136868686315, 5806270726085, 172757272757271, 513636363636315, 17275787578757271, 17578787578787571, 17878787578787871, 51363636363636315
Offset: 1

Views

Author

Jaroslav Krizek, Aug 30 2019

Keywords

Comments

Numbers m such that m, A000005(m) = tau(m) and A000203(m) = sigma(m) are all in A002113.
Corresponding values of tau(a(n)): 1, 2, 2, 3, 2, 2, 6, 4, 4, 4, 8, 8, 8, 4, 8, 8, 8, 4, 8, ...
Corresponding values of sigma(a(n)): 1, 3, 4, 7, 6, 8, 494, 23432, 2343432, 2383832, ...
Intersection of A028986 and A324988.

Examples

			tau(333) = A000005(333) = 6; sigma(333) = A000203(333) = 494.

Crossrefs

Cf. A000005, A000203, A002113, A028986, A324989, A324988.

Programs

  • Magma
    [m: m in [1..1000000] | Intseq(m, 10) eq Reverse(Intseq(m, 10)) and Intseq(NumberOfDivisors(m), 10) eq Reverse(Intseq(NumberOfDivisors(m), 10)) and Intseq(&+[d: d in Divisors(m)], 10) eq Reverse(Intseq(&+[d: d in Divisors(m)], 10))];
  • Mathematica
    Select[Range[2*10^6], PalindromeQ[#] && PalindromeQ[DivisorSigma[0, #]] && PalindromeQ[DivisorSigma[1, #]] &] (* Amiram Eldar, Aug 31 2019 *)
  • PARI
    ispal(n) = my(d=digits(n)); d == Vecrev(d);
isok(n) = ispal(n) && ispal(numdiv(n)) && ispal(sigma(n)); \\ Michel Marcus, Sep 02 2019

Extensions

a(20)-a(23) with the help of Daniel Suteu
Showing 1-5 of 5 results.

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