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 1-10 of 11 results. Next

A263618 Number of palindromic squares with exactly n digits.

Original entry on oeis.org

4, 0, 3, 0, 7, 1, 5, 0, 11, 0, 5, 1, 19, 0, 13, 1, 25, 0, 18, 0, 48, 1, 31, 0, 70, 1, 44, 2, 105, 0, 70, 1, 153, 1, 98, 3, 209, 0, 132, 0, 291, 1, 181, 1, 384, 0, 234, 2, 496, 1, 301, 1, 636, 0, 383, 0, 798, 1, 474, 1, 981, 0, 578, 0, 1199, 2, 701
Offset: 1

Views

Author

N. J. A. Sloane, Oct 23 2015

Keywords

Comments

Number of terms in A002779 with exactly n digits.
a(24) = a(30) = a(38) = a(40) = 0. - Robert Price, Apr 26 2019
a(2*k+1) > 0 since (10^k+1)^2 is a palindrome of 2*k+1 digits. - Chai Wah Wu, Jun 14 2024

Links

  • G. J. Simmons, Palindromic powers, J. Rec. Math., 3 (No. 2, 1970), 93-98. [Annotated scanned copy] See page 95.

Crossrefs

Cf. A002778, A002779, A263617, A263617, A263619, A263620.
Cf. A034822 (positions of zeros).

Programs

  • Mathematica
    Table[Length[Select[Range[If[n == 1, 0, Ceiling[Sqrt[10^(n - 1)]]],Floor[Sqrt[10^n]]], #^2 == IntegerReverse[#^2] &]], {n, 15}] (* Robert Price, Apr 26 2019 *)

Extensions

a(13)-a(19) from Chai Wah Wu, Oct 24 2015
a(20) from Robert Price, Apr 26 2019
a(21)-a(44) (using A002778) from Chai Wah Wu, Sep 16 2021
a(45)-a(67) from A002778 added by Max Alekseyev, Apr 08 2025

A059868 There exist no palindromic pentagonal numbers of length a(n).

Original entry on oeis.org

3, 9, 11, 12, 24, 30, 32, 33
Offset: 1

Views

Author

Patrick De Geest, Feb 15 2000

Keywords

Links

Crossrefs

Cf. A034822, A059869, A059870, A034307, A000326, A028386, A002069.

Programs

  • Mathematica
    A002069 = {0, 1, 5, 22, 1001, 2882, 15251, 720027, 7081807, 7451547, 26811862, 54177145, 1050660501, 1085885801, 1528888251, 2911771192, 2376574756732, 5792526252975, 5875432345785, 10810300301801, 264571020175462, 5292834004382925, 10808388588380801, 15017579397571051, 76318361016381367, 150621384483126051, 735960334433069537, 1003806742476083001, 1087959810189597801, 2716280733370826172};
A059868[n_] := Length[Select[A002069, IntegerLength[#] == n  || (n == 1 && # == 0) &]];
Select[Range[18], A059868[#] == 0 &] (* Robert Price, Apr 26 2019 *)
  • Python
    def ispal(n): s = str(n); return s == s[::-1]
def penpals(limit):
  for k in range(limit+1):
    if ispal(k*(3*k-1)//2): yield k*(3*k-1)//2
def aupto(limit):
  lengths = set(range(1, limit+1))
  for p in penpals(10**limit):
    lp, minlen = len(str(p)), min(lengths)
    for li in list(lengths):
      if li < lp: print(li, "in A059868"); lengths.discard(li)
    if lp in lengths: lengths.discard(lp); print("... discarding", lp)
    if len(lengths) == 0: return
aupto(15) # Michael S. Branicky, Mar 09 2021

Extensions

Name clarified by David A. Corneth, Apr 26 2019
a(6)-a(8) from Bert Dobbelaere, Apr 15 2025

A059869 Numbers k such that there exist no palindromic heptagonals of length k.

Original entry on oeis.org

8, 9, 14, 16, 32
Offset: 1

Views

Author

Patrick De Geest, Feb 15 2000

Keywords

Links

Crossrefs

Cf. A034822, A034307, A059868, A059870, A000566, A054971, A054910.

Programs

  • Mathematica
    A054910 = {0, 1, 7, 55, 616, 3553, 4774, 60606, 848848, 4615164, 5400045, 6050506, 7165445617, 62786368726, 65331413356, 73665056637, 91120102119, 345546645543, 365139931563, 947927729749, 3646334336463, 7111015101117, 717685292586717, 19480809790808491, 615857222222758516, 1465393008003935641, 8282802468642082828, 15599378333387399551, 20316023422432061302};
A059869[n_] := Length[Select[A054910, IntegerLength[#] == n || (n == 1 && # == 0) &]];
Select[Range[19], A059869[#] == 0 &] (* Robert Price, Apr 28 2019 *)

A059870 Numbers n such that there exist no palindromic octagonal numbers of length n.

Original entry on oeis.org

2, 3, 5, 6, 7, 8, 10, 11, 15, 17, 18, 20, 22, 23, 26, 31, 34
Offset: 1

Views

Author

Patrick De Geest, Feb 15 2000

Keywords

Links

Crossrefs

Cf. A034822, A034307, A059868, A059869, A000567, A057106, A057107.

Extensions

a(13)-a(17) from World!Of Numbers link entered by Michel Marcus, Mar 04 2014

A082721 There exist no palindromic hexagonals of length n.

Original entry on oeis.org

3, 8, 9, 12, 22, 24, 27, 30, 36, 38, 40
Offset: 1

Views

Author

Patrick De Geest, Apr 13 2003

Keywords

Links

Crossrefs

Cf. A034822, A054969, A059868, A059869, A059870, A082722, A034307.

Programs

  • Mathematica
    A054969 = {0, 1, 6, 66, 3003, 5995, 15051, 66066, 617716, 828828, 1269621, 1680861, 5073705, 5676765, 1264114621, 5289009825, 6172882716, 13953435931, 1313207023131, 5250178710525, 6874200024786, 61728399382716, 602224464422206, 636188414881636, 1250444114440521, 16588189498188561, 58183932923938185, 66056806460865066, 67898244444289876, 514816979979618415, 3075488771778845703, 6364000440440004636, 15199896744769899151};
A082721[n_] := Length[Select[A054969, IntegerLength[#] == n || (n == 1 && # == 0) &]];
Select[Range[19], A082721[#] == 0 &] (* Robert Price, Apr 27 2019 *)
  • Python
    def ispal(n): s = str(n); return s == s[::-1]
def hexpals(limit):
  yield from (k*(2*k-1) for k in range(limit+1) if ispal(k*(2*k-1)))
def aupto(limit):
  lengths = set(range(1, limit+1))
  for h in hexpals(10**limit):
    if len(lengths) == 0: return
    lh, minlen = len(str(h)), min(lengths)
    if lh > minlen: print(minlen, "in A082721"); lengths.discard(minlen)
    if lh in lengths: lengths.discard(lh); print("... discarding", lh)
aupto(14) # Michael S. Branicky, Mar 08 2021

A082722 Numbers k for which there exist no palindromic 9-gonals (also known as nonagonals or enneagonals) of length k.

Original entry on oeis.org

2, 6, 13, 14, 15, 16, 20, 25, 27, 28, 29, 30, 31, 32
Offset: 1

Views

Author

Patrick De Geest, Apr 13 2003

Keywords

Comments

Previous name was: There exist no palindromic nonagonals (enneagonals) of length n.

Links

Crossrefs

Cf. A034822, A059868, A082721, A059869, A059870, A034307.

Programs

  • Mathematica
    A082723 = {0, 1, 9, 111, 474, 969, 6666, 18981, 67276, 4411144, 6964696, 15444451, 57966975, 448707844, 460595064, 579696975, 931929139, 994040499, 1227667221, 9698998969, 61556965516, 664248842466, 699030030996, 99451743334715499, 428987160061789824, 950178723327871059, 1757445628265447571, 4404972454542794044, 9433971680861793349, 499583536595635385994, 1637992008558002997361, 19874891310701319847891};
A082722[n_] := Length[Select[A082723, IntegerLength[#] == n || (n == 1 && # == 0) &]];
Select[Range[22], A082722[#] == 0 &] (* Robert Price, Apr 29 2019 *)

Extensions

Definition edited by Jon E. Schoenfield, Sep 15 2013

A307717 Number of palindromic squares, k^2, of length n such that k is also palindromic.

Original entry on oeis.org

4, 0, 2, 0, 5, 0, 3, 0, 8, 0, 5, 0, 13, 0, 9, 0, 22, 0, 16, 0, 37, 0, 27, 0, 60, 0, 43, 0, 93, 0, 65, 0, 138, 0, 94, 0, 197, 0, 131, 0, 272, 0, 177, 0, 365, 0, 233, 0, 478, 0, 300, 0, 613, 0, 379, 0, 772, 0, 471, 0, 957, 0, 577, 0, 1170, 0, 698, 0, 1413, 0
Offset: 1

Views

Author

Robert Price, Apr 23 2019

Keywords

Examples

			There are only two palindromic squares of length 3 whose root is also palindromic. 11^2=121 and 22^2=484. Thus, a(3)=2.

Links

Crossrefs

Cf. A002778, A002779, A034307, A034822, A218035.

Programs

  • Mathematica
    Table[Length[Select[Range[If[n == 1, 0, Ceiling[Sqrt[10^(n - 1)]]], Floor[Sqrt[10^n]]], # == IntegerReverse[#] && #^2 == IntegerReverse[#^2] &]], {n, 15}]

Formula

From Christoph Koutschan, Feb 19 2022: (Start)
a(2n-1) = A218035(n).
a(n) is given by a quasi-polynomial (for a proof, see A218035):
a(1) = 4;
a(2n) = 0;
a(4n+1) = (n^3-3*n^2+11*n+6)/3 (n > 0);
a(4n+3) = (n^3+5*n+12)/6 (n >= 0). (End)

Extensions

a(16)-a(20) from Robert Price, Apr 25 2019
a(21)-a(70) from Giovanni Resta, Apr 28 2019

A065378 Primes p such that p^2 is a palindromic square.

Original entry on oeis.org

2, 3, 11, 101, 307, 30001253, 100111001, 110111011, 111010111, 111091111, 1011099011101, 1100011100011, 1100101010011, 1101010101011, 100110101011001, 100110990111001, 101000010000101, 101011000110101, 101110000011101
Offset: 1

Views

Author

Patrick De Geest, Nov 03 2001

Keywords

Comments

Record prime base number of a sporadic palindromic square is 13661181333262459.

Examples

			E.g. a(6) = 900075181570009 = p^2 with p = 30001253 and prime.

Links

Crossrefs

Cf. A065379, A002778, A002779, A034822, A028816, A016106, A059744, A016113, A007573.

Programs

  • Mathematica
    t={}; Do[p=Prime[n]; If[FromDigits[Reverse[IntegerDigits[p^2]]]==p^2,AppendTo[t,p]],{n,10^7}]; t (* Jayanta Basu, May 11 2013 *)

Extensions

Corrected by Jayanta Basu, May 11 2013

A307348 Numbers k such that there are no palindromic triangular numbers of length k.

Original entry on oeis.org

12, 30, 40
Offset: 1

Views

Author

Robert Price, May 01 2019

Keywords

Links

Crossrefs

Cf. A000217, A003098, A008509, A034822.

Programs

  • Mathematica
    A003098 = Select[PolygonalNumber[3, Range[0, 10^6]], PalindromeQ];
(* Set Range to level of desired running time. *)
nopal[n_] := Length[Select[A003098, IntegerLength[#] == n || (n == 1 && # == 0) &]]; Select[Range[12], nopal[#] == 0 &]
(* Set Range to encompass length of last value in A003098. *)

A065379 Palindromic squares with a prime root.

Original entry on oeis.org

4, 9, 121, 10201, 94249, 900075181570009, 12124434743442121, 12323244744232321, 12341234943214321, 1022321210249420121232201, 1210024420147410244200121, 1210222232227222322220121
Offset: 1

Views

Author

Patrick De Geest, Nov 03 2001

Keywords

Comments

Record sporadic palindromic square with a prime root is 186627875420278656872024578726681.

Examples

			a(6) = 900075181570009 = p^2 with p = 30001253, a prime.

Links

Crossrefs

Cf. A065378, A002778, A002779, A034822, A028816, A016106, A059744, A016113, A007573.
Showing 1-10 of 11 results. Next

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

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