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 1-6 of 6 results.

A028986 Palindromes whose sum of divisors is palindromic.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 333, 17571, 1757571, 1787871, 2249422, 4369634, 5136315, 412727214, 439838934, 518686815, 541626145, 17575757571, 52554845525, 4166253526614, 5136813186315, 5136868686315, 5806270726085, 7359770779537, 172757272757271, 513636363636315
Offset: 1

Views

Author

Patrick De Geest

Keywords

Comments

a(39) >= 10^18. - Hiroaki Yamanouchi, Sep 27 2014
Intersection of A002113 and of A028980. - Michel Marcus, Apr 06 2015

Links

Crossrefs

Cf. A002113 (palindromes), A028980 (sigma(n) is a palindrome).

Programs

  • Mathematica
    palQ[n_]:=Reverse[x=IntegerDigits[n]]==x; t={}; Do[If[palQ[n] && palQ[DivisorSigma[1,n]],AppendTo[t,n]],{n,5.2*10^6}]; t (* Jayanta Basu, May 17 2013 *)
Select[Range[52*10^6], AllTrue[{#, DivisorSigma[1, #]}, PalindromeQ] &] (* This naive program is not suitable for generating more than 13 terms of the sequence. *) (* Harvey P. Dale, Sep 21 2023 *)
  • PARI
    a(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]) \\ David A. Corneth in A002113, Jun 06 2014
pal(n)=d=digits(n);Vecrev(d)==d
for(n=2,10^5,if(pal(sigma(a(n))),print1(a(n),", "))) \\ Derek Orr, Apr 05 2015

Extensions

a(18)-a(24) from Donovan Johnson, Apr 19 2010
a(25)-a(26) from Donovan Johnson, Jun 16 2011

A244411 Nonprimes n such that the product of its divisors is a palindrome.

Original entry on oeis.org

1, 4, 22, 26, 49, 111, 121, 202, 1001, 1111, 2285, 10001, 10201, 11111, 100001, 1000001, 1001001, 1012101, 1100011, 1101011, 1109111, 1111111, 3069307, 10000001, 12028229, 12866669, 100000001, 101000101, 110000011, 110091011, 200010002, 10000000001, 10011111001
Offset: 1

Views

Author

Derek Orr, Jun 27 2014

Keywords

Comments

Primes trivially satisfy this property and are therefore not included in the sequence.
Numbers n such that A136522(A007955(n)) = 1.
A number is in the intersection of A002778 and A001358 iff it is in this sequence.
a(31) > 2*10^8.
a(32) > 4*10^8. - Chai Wah Wu, Aug 25 2015

Examples

			The divisors of 26 are 1,2,13,26. And 1*2*13*26 = 676 is a palindrome. Thus 26 is a member of this sequence.

Links

Crossrefs

Cf. A007955, A136522, A028980, A002778, A001358.

Programs

  • PARI
    rev(n)={r="";for(i=1,#digits(n),r=concat(Str(digits(n)[i]),r));return(eval(r))}
for(n=1,2*10^8,if(!isprime(n),d=divisors(n);ss=prod(j=1,#d,d[j]);if(ss==rev(ss),print1(n,", "))))
  • 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 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
A244411_list = [1]
for n in range(1,10**5):
    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]:
            A244411_list.append(n) # Chai Wah Wu, Aug 25 2015

Extensions

a(31) from Chai Wah Wu, Aug 25 2015
a(32)-a(33) from Giovanni Resta, Sep 20 2019

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

A074242 Numbers n such that sigma(n+1) = reverse(sigma(n)).

Original entry on oeis.org

5602, 42346, 184650339, 356930335, 453038125082
Offset: 1

Views

Author

Joseph L. Pe, Sep 19 2002

Keywords

Comments

a(6) > 10^13. - Giovanni Resta, Jun 26 2015

Examples

			sigma(5602 + 1) = 6048 = reverse(8406) = reverse(sigma(5602)), so 5602 is a term of the sequence.

Crossrefs

Cf. A028980 (sigma(n) = reverse(sigma(n))).

Programs

  • Mathematica
    Do[ If[FromDigits[Reverse[IntegerDigits[DivisorSigma[1, n]]]] == DivisorSigma[1, n + 1], Print[n]], {n, 1, 10^7}]
  • PARI
    isok(n) = digits(sigma(n+1)) == Vecrev(digits(sigma(n))); \\ Michel Marcus, Jun 26 2015

Extensions

a(3)-a(4) from Donovan Johnson, Feb 01 2009
a(5) from Giovanni Resta, Jun 26 2015

A274028 Numbers whose sum of divisors and sum of anti-divisors are both palindromes.

Original entry on oeis.org

1, 2, 3, 4, 5, 208, 211, 489, 39765, 41689, 43545, 45772, 1226372, 2028209, 3131006, 5639781, 45224913, 402664481, 509561899, 534611505, 30392347941, 37824872279, 42100531202, 67332408085, 208185050013, 363340615629, 1316050604902, 1792459658755, 2465601425469
Offset: 1

Views

Author

Paolo P. Lava, Jun 07 2016

Keywords

Comments

Intersection of A028980 and A274049. - Michel Marcus, Jun 08 2016

Examples

			Anti-divisors of 208 are 3, 5, 32, 83, 139 and their sum is 262 ; sigma(208) = 434.

Crossrefs

Cf. A000203, A002113, A028980, A066417, A073956, A274049.

Programs

  • Maple
    with(numtheory): T:=proc(w) local x, y, z; x:=w; y:=0;for z from 1 to ilog10(x)+1 do y:=10*y+(x mod 10); x:=trunc(x/10); od; y; end:
P:=proc(q) local a,j,k,n,t; print(1); for n from 1 to q do k:=0; j:=n;
while j mod 2 <> 1 do k:=k+1; j:=j/2; od;
a:=sigma(2*n+1)+sigma(2*n-1)+sigma(n/2^k)*2^(k+1)-6*n-2;
if T(a)=a and sigma(n)=T(sigma(n)) then print(n); fi; od; end: P(10^6);

Extensions

Missing a(1)-a(2) and a(17)-a(20) from Giovanni Resta, Jun 19 2016
a(21)-a(29) from Max Alekseyev, Jan 28 2024

A259541 Numbers n such that antisigma(n) is palindromic.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 13, 23, 30, 31, 36, 109, 119, 158, 351, 1645, 1653, 2003, 3476, 3520, 3934, 4913, 8037, 9379, 35324, 36516, 91951, 128955, 200003, 390066, 402603, 1068869, 2000003, 2144992, 2467458, 2867828, 3392245, 3607663
Offset: 1

Views

Author

Paolo P. Lava, Jun 30 2015

Keywords

Comments

Primes of the form 2*10^k+3 belong the sequence (see A177134 and A081677).

Examples

			antisigma(1) = 1*2/2 - sigma(1) = 1 - 1 = 0;
antisigma(13) = 13*14/2 - sigma(13) = 91 - 14 = 77;
antisigma(109) = 109*110/2 - sigma(109) = 5995 - 110 = 5885.

Crossrefs

Cf. A024816, A028980, A081677.

Programs

  • Maple
    with(numtheory): T:=proc(w) local x, y, z; x:=w; y:=0;
for z from 1 to ilog10(x)+1 do y:=10*y+(x mod 10); x:=trunc(x/10);
od; y; end: P:=proc(q) local a,n;
for n from 1 to q do a:=n*(n+1)/2-sigma(n); if a=T(a) then print(n);
fi; od; end: P(10^9);
  • Mathematica
    palQ[n_] := Block[{d = IntegerDigits@ n}, d == Reverse@ d]; Select[Range@ 4000000, palQ[# (# + 1)/2 - DivisorSigma[1, #]] &] (* Michael De Vlieger, Jul 01 2015 *)
  • PARI
    isok(n) = my(d = digits(n*(n+1)/2 - sigma(n))); Vecrev(d)==d; \\ Michel Marcus, Jul 01 2015
Showing 1-6 of 6 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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