A065531 -id:A065531 - cp's OEIS frontend

A106801 Records by number in A037183, by indices in A065531.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 13, 16, 17, 18, 20, 21, 25, 26, 30, 32, 35, 36, 38, 40, 43, 47, 49, 50, 51, 52, 54, 61, 66, 73, 76, 84, 85, 92, 97, 99, 101, 110, 113, 121, 122

Offset: 0

Michael Trott (mtrott(AT)wolfram.com) and Robert G. Wilson v, May 12 2005; extended Jun 06 2005

nonn

Michael Trott, The Mathematica Guide Book for Programming, Springer, 2004, page 218.

Cf. A037183, A065531, A107129.

palindromicBases[n_] := Module[{p}, Table[p = IntegerDigits[n, b]; If[ p == Reverse[p], {b, p}, Sequence @@ {}], {b, 2, n - 1}]]; lst = {0, 0}; Do[ If[ Length[ palindromicBases[n]] > lst[[ -1, 1]], AppendTo[lst, {c, n}]], {n, 200000}]; First[ Transpose[ lst]]

A037183 Smallest number that is palindromic (with at least 2 digits) in n bases.

Original entry on oeis.org

3, 5, 10, 21, 36, 60, 80, 120, 180, 264, 252, 360, 300, 960, 900, 720, 1080, 1440, 1800, 1680, 2160, 2880, 5616, 3780, 2520, 3600, 6120, 6720, 6300, 5040, 11340, 7560, 14112, 10800, 9240, 10080, 13860, 12600, 31200, 15120, 22680, 20160, 18480, 39312, 33264, 39600, 25200, 30240

Offset: 1

Erich Friedman, Dec 11 1999

Smallest number k that is palindromic in n bases b, 1 < b < k.
Only a(1), a(2), a(3), a(4) & a(7) are not congruent to 0 (mod 12). - Robert G. Wilson v, Oct 21 2014
First occurrence of k beginning with 0 in A135551. - Robert G. Wilson v, Jun 30 2017

			3 = 11 in base 2.
5 = 101 in base 2 and 11 in base 4.
10 is a palindrome in bases 3, 4 and 9: 101(3), 22(4) and 11(9). So a(3)=10.

Robert G. Wilson v, Table of n, a(n) for n = 1..342 (first 100 terms from Giovanni Resta)
Robert G. Wilson v, Smallest number which is palindromic in n bases or 0 if no such number is known

Cf. A065531, A107129, A135549.

f[n_] := Module[{idn, s = Floor@ Sqrt[n + 1] - 1}, lng = Table[ If[ Reverse[ idn = IntegerDigits[n, b]] == idn, {b}, Sequence @@ {}], {b, 2, s + 1}]; If[ IntegerQ@ Sqrt[n + 1], -1, 0] + Length@ lng + Count[ Mod[n, Range@ s], 0]]; f[n_] := 0 /; n < 3; t = Table[0, {700}]; k = 3; While[k < 1100000001, a = f[k]; If[ t[[a]] == 0, t[[a]] = k; Print[{a, k}]]; k++]; Take[t, 310] (* Robert G. Wilson v, Nov 02 2014 *)

More terms from David W. Wilson

A135549 Number of bases b, 1 < b < n-1, in which n is a palindrome.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 1, 1, 1, 2, 0, 1, 1, 1, 2, 2, 2, 2, 0, 2, 3, 1, 1, 3, 1, 3, 2, 3, 1, 2, 2, 2, 2, 2, 1, 4, 1, 2, 1, 4, 1, 3, 1, 2, 3, 3, 0, 4, 1, 3, 3, 4, 0, 3, 3, 3, 3, 1, 1, 5, 1, 2, 4, 3, 4, 3, 2, 3, 1, 3, 1, 5, 2, 2, 2, 2, 1, 5, 0, 6, 2, 3, 1, 5, 4, 2, 1, 4, 1, 4, 3, 4, 3, 1, 1, 5, 1, 4, 3, 6, 1, 3, 0, 5

Offset: 0

John P. Linderman, Feb 26 2008, Feb 28 2008

Every integer n is a palindrome when expressed in unary, or in base n-1 (where it will be 11). So here we assume 1 < b < n-1.

Records for a(n)>=1 are in A107129. - Dmitry Kamenetsky, Oct 22 2015

T. D. Noe, Table of n, a(n) for n = 0..10000
John P. Linderman, Description of A135549-A135551 and A016038
John P. Linderman, Perl program [Use the command: palin.pl]

Cf. A016038, A037183, A065531, A107129, A135550, A135551.

Cf. A016038 (non-palindromic numbers in any base 1 < b < n-1)

a = {0, 0, 0}; For[n = 4, n < 100, n++, c = 0; For[b = 2, b < n - 1, b++, If[IntegerDigits[n, b] == Reverse[IntegerDigits[n, b]], c++ ]]; AppendTo[a, c]]; a (* Stefan Steinerberger, Feb 27 2008 *)
Table[cnt=0; Do[d=IntegerDigits[n,b]; If[d==Reverse[d], cnt++ ], {b,2,n-2}]; cnt, {n,0,100}] (* T. D. Noe, Feb 28 2008 *)
Table[Total[Boole[Table[PalindromeQ[IntegerDigits[n,b]],{b,2,n-2}]]],{n,0,120}] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Apr 14 2020 *)

a(n) = A065531(n)-1 = A126071(n)-2 for n>2. - T. D. Noe, Feb 28 2008

A107129 Numbers n which are palindromic in more bases b, 1

Original entry on oeis.org

1, 3, 5, 10, 21, 36, 60, 80, 120, 180, 252, 300, 720, 1080, 1440, 1680, 2160, 2520, 3600, 5040, 7560, 9240, 10080, 12600, 15120, 18480, 25200, 27720, 36960, 41580, 45360, 50400, 55440, 83160, 110880, 131040, 166320, 221760, 277200, 332640, 360360

Offset: 0

Michael Trott (mtrott(AT)wolfram.com) and Robert G. Wilson v, May 12 2005

Records by number in A037183, by indices in A065531.

Except for 3, 5 and 21 they are all even and except for the first seven, they are all multiples of twelve.

			1 has no palindromic representation in bases 2 to n.
3 = 11_2.
5 = 101_2, 11_4.
10 = 101_3, 22_4, 11_9.
21 = 10101_2, 111_4, 33_6, 11_20.
36960 = 5775_19, 3(90)3_97, (176)(176)_209, (168)(168)_219,
(165)(165)_223, (160)(160)_230, (154)(154)_239, (140)(140)_263, (132)(132)_279,
(120)(120)_307, (112)(112)_329, (110)(110)_335, (105)(105)_351, (96)(96)_384,
(88)(88)_419, (84)(84)_439, (80)(80)_461, (77)(77)_479, (70)(70)_527,
(66)(66)_559, (60)(60)_615, (56)(56)_659, (55)(55)_671, (48)(48)_769,
(44)(44)_839, (42)(42)_879, (40)(40)_923, (35)(35)_1055, (33)(33)_1119,
(32)(32)_1154, (30)(30)_1231, (28)(28)_1319, (24)(24)_1539, (22)(22)_1679,
(21)(21)_1759, (20)(20)_1847, (16)(16)_2309, (15)(15)_2463, (14)(14)_2639,
(12)(12)_3079, (11)(11)_3359, (10)(10)_3695, 88_4619, 77_5279, 66_6159, 55_7391,
44_9239, 33_12319, 22_18479, 11_36959.

Michael Trott, The Mathematica GuideBook for Programming, Springer, 2004, page 218.

Robert G. Wilson v, Table of n, a(n) for n = 0..91

Cf. A037183, A065531, A106801, A135549.

f[n_] := Block[{s = Floor@ Sqrt[n + 1] - 1, b = 2, c = If[IntegerQ@ Sqrt[n + 1], -2, -1]}, While[b < s + 2, idn = IntegerDigits[n, b]; If[ idn == Reverse@ idn, c++]; b++]; c + Count[ Mod[n, Range@ s], 0]]; f[n_] := 0 /; n < 3;
k = 0; mx = -1; lst = {}; While[ k < 360000001, c = f@ k; If[ c > mx, AppendTo[lst, k]; mx = c]; k++]; lst

A135551 Number of bases b, 1 < b < n, in which n is a palindrome.

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 1, 2, 2, 2, 3, 1, 2, 2, 2, 3, 3, 3, 3, 1, 3, 4, 2, 2, 4, 2, 4, 3, 4, 2, 3, 3, 3, 3, 3, 2, 5, 2, 3, 2, 5, 2, 4, 2, 3, 4, 4, 1, 5, 2, 4, 4, 5, 1, 4, 4, 4, 4, 2, 2, 6, 2, 3, 5, 4, 5, 4, 3, 4, 2, 4, 2, 6, 3, 3, 3, 3, 2, 6, 1, 7, 3, 4, 2, 6, 5, 3, 2, 5, 2, 5, 4, 5, 4, 2, 2, 6, 2, 5, 4, 7, 2, 4, 1, 6

Offset: 0

John P. Linderman, Feb 26 2008, Feb 28 2008

Every integer n is a palindrome when expressed in unary, or in base n-1 (where it will be 11).
First occurrence in A037183.
a(n) is always less than A001221(n) except for 2 and 6; a(n) is always less than A001222(n) except for even powers of twos and 6, 12, 81, 243, 625, 729, 2187, 19683, 59049, ..., . - Robert G. Wilson v, Jul 17 2016

Robert G. Wilson v, Table of n, a(n) for n = 0..100000
John P. Linderman, Description of A135549-A135551 and A016038
John P. Linderman, Perl program [Use the command: BASEDELTA=0 palin.pl]

Cf. A037183, A135549, A135550, A016038.

Essentially the same as A065531.

palindromicBases[n_] := Module[{p}, Table[p = IntegerDigits[n, b]; If[p == Reverse[p], {b, p}, Sequence @@ {}], {b, 2, n - 1}]]; Array[ Length@ palindromicBases@# &, 105, 0] (* Robert G. Wilson v, Oct 15 2014 *)
palQ[n_Integer, base_Integer] := Module[{idn = IntegerDigits[n, base]}, idn == Reverse[idn]];
f[n_] := Block[{s = Ceiling@ Sqrt@ n, b = 2, c = If[ IntegerQ@ Sqrt[4n + 1], -1, 0]}, While[b < s, If[ palQ[n, b], c++]; b++]; c + Count[ Mod[n, Range[s - 1]], 0]]; f[0] = 0; Array[f, 105, 0] (* much faster for large Ns *) (* Robert G. Wilson v, Oct 20 2014 *)

a(n) = A135549(n) + 1 for n>2; otherwise a(n) = A135549(n) = 0. - Michel Marcus, Oct 15 2014

a(n) = A126071(n) - 1. - Michel Marcus, Mar 07 2015

A087911 Smallest prime p that is a palindrome in n different bases < p.

Original entry on oeis.org

2, 3, 5, 17, 191, 257, 1009, 4561, 4591, 21601, 57601, 54121, 86677, 176401, 415801, 291721, 950041, 1259701, 3049201, 1670761, 6098401, 3880801, 5654881, 13759201, 18618601, 14414401, 18960481, 15135121, 31600801, 45405361, 35814241

Offset: 1

Randy L. Ekl, Oct 17 2003

a(n) = A000040(A137779^(-1)(n)). - Attila Olah (jolafix(AT)gmail.com), May 06 2008, corrected May 08 2008

The sequence is not monotonic: a(10) > a(11) = 54121. - Attila Olah (jolafix(AT)gmail.com), May 06 2008, corrected May 08 2008

			a(4) = 191 because 191 base 6 = 515, 191 base 9 = 232, 191 base 10 = 191 and 191 base 190 = 11, all palindromes. No numbers less than 191 can be represented in 4 such ways.
a(12) = 54121 because 54121 is a palindrome in 12 different bases, including base 1 and base 54120.

Karl Hovekamp, Jan 01 2007, Table of n, a(n) for n = 1..55
K. Hovekamp, Palindromic numbers [Broken link?]

Cf. A137779, A037183, A107129, A065531, A135549.

q=1; forprime(m=3,20000,count=0; for(b=2,m-1, w=b+1; k=0; i=m; while(i>0,k=k*w+i%b; i=floor(i/b)); l=0; j=k; while(j>0,l=l*w+j%w; j=floor(j/w)); if(l==k,count=count+1,); if(count>q,print1(m,", "); q=count,)))

More terms from David Wasserman, Jun 20 2005
Terms a(17)-a(22) computed by Karl Hovekamp, sent by David Wasserman, Dec 19 2006
More terms from Karl Hovekamp, Jan 01 2007

A375350 a(n) is the smallest number k such that the sum of the bases b, 1 < b < k-1, for which k is palindromic, equals n . If no such number exists, a(n) = -1.

Original entry on oeis.org

5, 8, 25, 12, 14, 10, 89, 107, 16, 67, 20, 18, 109, 331, 187, 227, 95, 157, 26, 409, 28, 24, 45, 191, 65, 241, 58, 85, 57, 44, 161, 299, 63, 62, 401, 42, 40, 337, 50, 36, 74, 56, 99, 52, 94, 1129, 86, 145, 129, 54, 68, 64, 1613, 76, 48, 1073, 175, 533, 559, 341

Offset: 2

Jean-Marc Rebert, Aug 14 2024

			a(7) = 10, because 10 is palindromic in bases 3 (as 101) and 4 (as 22), which are both less than 9. The sum of these bases (3 + 4) is 7, and no smaller number has this property.
Table begins:
  a(2) = 5 = 101_2,
  a(3) = 8 = 22_3,
  a(4) = 25 = 121_4,
  a(5) = 12 = 22_5,
  a(6) = 14 =  22_6,
  a(7) = 10 = 101_3 = 22_4,
  a(8) = 89 = 131_8,
  a(9) = 107 = 1101011_2 = 212_7,
  a(10) = 16 = 121_3 = 22_7.

Michael S. Branicky, Table of n, a(n) for n = 2..8284 (terms 2..523 from Robert Israel)

Cf. A002113,A007632, A007953, A065531, A107129, A135549, A253594, A375201.

ispali:= proc(x,b) local F; F:= convert(x,base,b);
  andmap(t -> F[t] = F[-t], [$1.. nops(F)/2])
end proc:
f:= proc(k) convert(select(b -> ispali(k,b),[$2..k-2]),`+`) end proc:
N:= 100: # for a(2) .. a(N)
V:= Vector(N): count:= 0:
for x from 5 while count < N-1 do
   v:= f(x);
   if v >= 2 and v <=N and V[v] = 0 then V[v]:= x; count:= count+1;  fi
od:
convert(V[2..N],list); # Robert Israel, Oct 14 2024

isok(k, n) = my(s=0); for(b=2, k-2, my(d=digits(k, b)); if (d == Vecrev(d), s += b)); s == n;
a(n) = my(k=1); while (!isok(k, n), k++); k; \\ Michel Marcus, Aug 14 2024

from itertools import count, islice
from sympy.ntheory import is_palindromic
def f(n): return sum(b for b in range(2, n-2) if is_palindromic(n, b))
def agen(): # generator of terms
    adict, n = dict(), 2
    for k in count(4):
        v = f(k)
        if v not in adict:
            adict[v] = k
            while n in adict: yield adict[n]; n += 1
print(list(islice(agen(), 60))) # Michael S. Branicky, Oct 15 2024

A375201(a(n)) = n. - Robert Israel, Oct 15 2024

Name clarified by Robert Israel, Oct 15 2024

A065809 a(n) is the smallest number m > n such that m is palindromic in base n and is not palindromic in bases b with 2 <= b < n.

Original entry on oeis.org

3, 4, 25, 6, 14, 32, 54, 30, 11, 84, 39, 140, 75, 176, 102, 198, 19, 220, 147, 110, 69, 384, 175, 416, 486, 420, 58, 570, 279, 544, 429, 306, 245, 684, 296, 380, 663, 880, 615, 1134, 258, 1012, 1035, 1104, 47, 1392, 539, 1500, 1071, 1508, 53, 2106

Offset: 2

Naohiro Nomoto, Dec 06 2001

Index at which first occurrence of n occurs in A016026 when the palindrome is multidigit. Only the first two terms of A016026 are single-digit palindromes. - Robert G. Wilson v, Dec 22 2021

			From _Robert G. Wilson v_, Dec 22 2021: (Start)
a(2) = 3 since A016026(3) = 2;
a(3) = 4 since A016026(4) = 3;
a(4) = 25 since A016026(25) = 4; etc. (End)

Robert G. Wilson v, Table of n, a(n) for n = 2..10000 (first 441 terms from Robert Israel).

Cf. A016026, A065531, A249634.

N:= 100:
f:= proc(m) local k,L;
   for k from 2 to m do
     L:= convert(m,base,k);
     if L = ListTools:-Reverse(L) then return k fi
   od;
   FAIL
end proc:
V:= Array(2..N):
count:= 0:
for m from 3 while count < N-1 do
  v:= f(m);
  if v = FAIL or v > N then next fi;
  if V[v] = 0 then count:= count+1; V[v]:= m;
  fi
od:
convert(V,list); # Robert Israel, Apr 08 2019

fpb = Compile[{{n, Integer}}, Module[{b = 2, idn}, While[ Reverse[idn = IntegerDigits[n, b]] != idn, b++]; b]]; k = 3; t[] := 0; While[k < 2200, a = fpb@ k; If[ t[a] == 0, t[a] = k]; k++]; t@# & /@ Range[2, 53] (* Robert G. Wilson v, Dec 22 2021 *)

Definition edited by N. J. A. Sloane, Apr 08 2019

A375387 a(n) is the least number k whose sum of digits in base 10 is n and that is palindromic in base n, or -1 if no such number exists.

Original entry on oeis.org

-1, 130, 41, 123, 16, 170, -1, 55, 155, 39, 274, 239, 96, 187, 494, 2925, 685, 1784, 1389, 859, 599, 1779, 1978, 989, 6597, 5887, 6968, 8499, 5989, 17969, 29859, 17899, 28898, 435897, 38989, 2089469, 1788960, 498847, 2886278, 487878, 919996, 4098689, 898794, 1896967

Offset: 3

Jean-Marc Rebert, Aug 13 2024

A positive integer that is a multiple of 3 ends with 0 in base 3, so it cannot be a palindrome in base 3.
A positive integer that is a multiple of 9 ends with 0 in base 9, so it cannot be a palindrome in base 9.
From Michael S. Branicky, Aug 15 2024: (Start)
Regarding a(2): To be a palindrome in base 2, it must end with 1, hence odd. To be odd and have digit sum 2 in base 10, it must be of the form t_d = 10^(d-1) + 1, d > 1 (a d-digit base-10 number).  t_d is not divisible by 3, and base-2 palindromes with even length (i.e., number of binary digits) are divisible by 3, so, if a(2) exists, it must be a base-2 palindrome with odd length.
Computer search shows no such terms with d <= 10^6, so a(2), if it exists, has > 10^6 decimal digits. (End)

			a(5) = 41, because 4 + 1 = 5 and 41 = 131_5, and no lesser number has this property.
First terms are:
  130 = 2002_4
  41  = 131_5
  123 = 3323_6
  16  = 22_7
  170 = 252_8

Michael S. Branicky, Table of n, a(n) for n = 3..149
Michael S. Branicky, Python programs for OEIS A375387
Jean-Marc Rebert, a375387_1e10

Cf. A002113, A007953, A051885, A065531, A107129, A228915, A253594.

isok(k, n) = if (sumdigits(k)==n, my(d=digits(k, n)); d==Vecrev(d));
a(n) = if ((n==3) || (n==9), return((-1))); my(k=1); while (!isok(k,n), k++); k; \\ Michel Marcus, Aug 13 2024

# see Links for faster variants
from itertools import count
from sympy.ntheory import is_palindromic
def a(n):
    if n in {3, 9}: return -1
    return next(k for k in count(10**(n//9)-1) if sum(map(int, str(k)))==n and is_palindromic(k, n))
print([a(n) for n in range(3, 47)]) # Michael S. Branicky, Aug 13 2024

A065708 a(n) is the position of A037183(n) in a sorted list of the terms of A037183.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 10, 13, 12, 16, 15, 14, 17, 18, 20, 19, 21, 25, 22, 26, 24, 30, 23, 27, 29, 28, 32, 35, 36, 34, 31, 38, 37, 33, 40, 43, 42, 41, 47, 49, 48, 39, 45, 50, 44, 46, 51, 52, 54, 61, 57, 56, 58, 55, 66, 60, 62, 64, 73, 59, 65, 76, 63

Offset: 1

Naohiro Nomoto, Dec 04 2001

			a(11)=10 because A037183(11)=252 is the 10th largest term in A037181. - _Sean A. Irvine_, Sep 09 2023

Cf. A037183, A065531.

Name clarified and more terms from Sean A. Irvine, Sep 09 2023

cp's OEIS Frontend

A106801 Records by number in A037183, by indices in A065531.

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A037183 Smallest number that is palindromic (with at least 2 digits) in n bases.

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A135549 Number of bases b, 1 < b < n-1, in which n is a palindrome.

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A107129 Numbers n which are palindromic in more bases b, 1

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A135551 Number of bases b, 1 < b < n, in which n is a palindrome.

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A087911 Smallest prime p that is a palindrome in n different bases < p.

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A375350 a(n) is the smallest number k such that the sum of the bases b, 1 < b < k-1, for which k is palindromic, equals n . If no such number exists, a(n) = -1.

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A065809 a(n) is the smallest number m > n such that m is palindromic in base n and is not palindromic in bases b with 2 <= b < n.

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