A007632 -id:A007632 - cp's OEIS frontend

A249155 Palindromic in bases 6 and 15.

0, 1, 2, 3, 4, 5, 7, 14, 80, 160, 301, 602, 693, 994, 1295, 1627, 1777, 2365, 2666, 5296, 5776, 6256, 17360, 34720, 51301, 52201, 105092, 155493, 209284, 587846, 735644, 7904800, 11495701, 80005507, 80469907, 83165017, 89731777, 90196177

Offset: 1

Ray Chandler, Oct 27 2014

Intersection of A029953 and A029960.

			301 is a term since 301 = 1221 base 6 and 301 = 151 base 15.

Ray Chandler and Chai Wah Wu, Table of n, a(n) for n = 1..71 (terms < 6^28). First 65 terms from Ray Chandler.
Attila Bérczes and Volker Ziegler, On Simultaneous Palindromes, arXiv:1403.0787 [math.NT], 2014.

Cf. A007632, A060792, A249156, A249157, A249158.

palQ[n_Integer, base_Integer] := Block[{idn = IntegerDigits[n, base]}, idn == Reverse[idn]]; Select[Range[10^6] - 1, palQ[#, 6] && palQ[#, 15] &]

from gmpy2 import digits
def palQ(n, b): # check if n is a palindrome in base b
    s = digits(n, b)
    return s == s[::-1]
def palQgen(l, b): # generator of palindromes in base b of length <= 2*l
    if l > 0:
        yield 0
        for x in range(1, l+1):
            for y in range(b**(x-1), b**x):
                s = digits(y, b)
                yield int(s+s[-2::-1], b)
            for y in range(b**(x-1), b**x):
                s = digits(y, b)
                yield int(s+s[::-1], b)
A249155_list = [n for n in palQgen(8, 6) if palQ(n, 15)] # Chai Wah Wu, Nov 29 2014

A249157 Palindromic in bases 11 and 13.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 84, 366, 510, 732, 876, 1020, 1098, 1242, 1464, 10248, 30252, 31110, 62220, 103704, 146541, 3382050, 3698730, 4391268, 225622530, 272466250, 413186676, 713998530, 801837204, 848770222, 912265732

Offset: 1

Ray Chandler, Oct 27 2014

Intersection of A029956 and A029958.

			366 is a term since 366 = 303 base 11 and 366 = 222 base 13.

Ray Chandler, Table of n, a(n) for n = 1..69 (terms < 10^18)
Attila Bérczes and Volker Ziegler, On Simultaneous Palindromes, arXiv:1403.0787 [math.NT], 2014.

Cf. A007632, A060792, A249155, A249156, A249158.

palQ[n_Integer,base_Integer]:=Block[{idn=IntegerDigits[n,base]},idn==Reverse[idn]];Select[Range[10^6]-1,palQ[#,11]&&palQ[#,13]&]
Select[Range[0,44*10^5],AllTrue[IntegerDigits[#,{11,13}],PalindromeQ]&] (* The program generates the first 30 terms of the sequence. *) (* Harvey P. Dale, May 15 2025 *)

from gmpy2 import digits
def palQ(n, b): # check if n is a palindrome in base b
    s = digits(n, b)
    return s == s[::-1]
def palQgen(l, b): # unordered generator of palindromes in base b of length <= 2*l
    if l > 0:
        yield 0
        for x in range(1, b**l):
            s = digits(x, b)
            yield int(s+s[-2::-1], b)
            yield int(s+s[::-1], b)
A249157_list = sorted([n for n in palQgen(6,11) if palQ(n,13)]) # Chai Wah Wu, Nov 25 2014

A249158 Palindromic in bases 7 and 29.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 8, 16, 24, 150, 300, 5952, 7752, 7955, 9755, 9958, 11904, 13704, 13907, 14110, 15707, 15910, 392850, 751043, 4585544, 12737804, 12828748, 16380296, 19289406, 19380350, 20228253, 33115710, 395849700, 1339182534

Offset: 1

Ray Chandler, Oct 27 2014

			150 is a term since 150 = 303 base 7 and 150 = 55 base 27.

Ray Chandler, Table of n, a(n) for n = 1..80 (terms < 10^18)
Attila Bérczes and Volker Ziegler, On Simultaneous Palindromes, arXiv:1403.0787 [math.NT], 2014.

Cf. A007632, A060792, A249155, A249156, A249157.

palQ[n_Integer,base_Integer]:=Block[{idn=IntegerDigits[n,base]},idn==Reverse[idn]];Select[Range[10^6]-1,palQ[#,7]&&palQ[#,29]&]

from gmpy2 import digits
def palQ(n, b): # check if n is a palindrome in base b
    s = digits(n, b)
    return s == s[::-1]
def palQgen(l, b): # unordered generator of palindromes in base b of length <= 2*l
    if l > 0:
        yield 0
        for x in range(1, b**l):
            s = digits(x, b)
            yield int(s+s[-2::-1], b)
            yield int(s+s[::-1], b)
A249158_list = sorted([n for n in palQgen(8,7) if palQ(n,29)])
# Chai Wah Wu, Nov 25 2014

A118064 Decimal expansion of the sum of the reciprocals of the palindromic primes A002385 (Honaker's constant).

Original entry on oeis.org

1, 3, 2, 3, 9, 8, 2, 1, 4, 6, 8, 0, 6

Offset: 1

Martin Renner, May 11 2006

From Robert G. Wilson v, Nov 01 2010: (Start)
n \ sum to 10^n
1.267099567099567099567099567099567099567099567099567099567099567099567
1.320723244590290964212793334437872849720871258315369002493912638038324
1.323748402250648554164425746280035962754669829327727800040192015109270
1.323964105671202458016249150576217276147952428601889817773483085610332
1.323980718065525060936354534562000413901564393192688451911141729415146
1.323982026479475203850120990923294207966175748395470136325039323549015
1.323982136437462724794656629740867909978221153827990721566573347887836
1.323982145891606234777299440047139038371441916546100653011463101470839
1.323982146724859090645464845257681674740147563533254654075059843860490
1.323982146799188851138232927173756400348958236915409881890097448921521
1.323982146805857558347279363344557427339916178257233985191868031567947 (End)

			1.323982146806...

Carlos Rivera, Problems & Puzzles: Puzzle 056 - Honaker's Constant.
Eric Weisstein, Palindromic Prime.

Cf. A002385, A160910, A181442, A050251, A118031, A194097.

(* first obtain nextPalindrome from A007632 *) s = 1/11; c = 1; pp = 1; Do[ While[pp < 10^n, If[PrimeQ@ pp, c++; s = N[s + 1/pp, 64]]; pp = NextPalindrome@ pp]; If[ OddQ@ n, pp = 10^(n + 1); Print[{s, n, c}]], {n, 17}] (* Robert G. Wilson v, May 31 2009 *)
generate[n_] := Block[{id = IntegerDigits@n, insert = {{0}, {1}, {2}, {3}, {4}, {5}, {6}, {7}, {8}, {9}}}, FromDigits@ Join[id, #, Reverse@ id] & /@ insert]; sm = N[Plus @@ (1/{2, 3, 5, 7, 11}), 64]; k = 1; Do [While[k < 10^n, sm = N[sm + Plus @@ (1/Select[ generate@k, PrimeQ]), 128]; k++ ]; Print[{2 n + 1, sm}], {n, 9}] (* Robert G. Wilson v, Nov 01 2010 *)

Equals Sum_{p} 1/p, where p ranges over the palindromic primes.

Corrected by Eric W. Weisstein, May 14 2006
More terms from Robert G. Wilson v, Nov 01 2010
Entry revised by N. J. A. Sloane, May 05 2013

A248889 Palindromic in base 10 and 18.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 171, 323, 343, 505, 595, 686, 848, 1661, 2112, 3773, 23332, 46664, 69996, 262262, 583385, 782287, 859958, 981189, 1254521, 1403041, 1832381, 39388393, 54411445, 55499455, 88844888, 118919811, 191010191

Offset: 1

Mauro Fiorentini, Mar 05 2015

a(54) > 10^12.

			848 in decimal is 2B2 in base 18, so 848 is in the sequence.
1661 in decimal is 525 in base 18, so 1661 is in the sequence.
1771 in decimal is 587 in base 18, which is not a palindrome, so 1771 is not in the sequence.

M. Fiorentini and Chai Wah Wu, Table of n, a(n) for n = 1..68 a(n) for n = 1..53 from M. Fiorentini.

Cf. A007632, A007633, A029961, A029962, A029963, A029964, A029965, A029966, A029967, A029968, A029969, A029970, A028731, A097855, A248899

[n: n in [0..2*10^7] | Intseq(n) eq Reverse(Intseq(n)) and Intseq(n,18) eq Reverse(Intseq(n,18))]; // Vincenzo Librandi, Mar 21 2015

IsPalindromic := proc(n, Base)
    local Conv, i;
    Conv := convert(n, base, Base);
    for i from 1 to nops(Conv) / 2 do
        if Conv [i] <> Conv [nops(Conv) + 1 - i] then
            return false;
        fi:
    od:
    true;
end proc:
Base := 18;
A := [];
for i from 1 to 10^6 do:
   S := convert(i, base, 10);
   V := 0;
   if i mod 10 = 0 then
      next;
   fi;
   for j from 1 to nops(S) do:
      V := V * 10 + S [j];
   od:
   for j from 0 to 10 do:
      V1 := V * 10^(nops(S) + j) + i;
      if IsPalindromic(V1, Base) then
         A := [op(A), V1];
      fi;
   od:
   V1 := (V - (V mod 10)) * 10^(nops(S) - 1) + i;
   if IsPalindromic(V1, Base) then
      A := [op(A), V1];
   fi;
od:
sort(A);

palindromicQ[n_, b_:10] := TrueQ[IntegerDigits[n, b] == Reverse[IntegerDigits[n, b]]]; Select[Range[0, 499], palindromicQ[#] && palindromicQ[#, 18] &] (* Alonso del Arte, Mar 21 2015 *)

isok(n) = (n==0) || ((d = digits(n, 10)) && (Vecrev(d) == d) && (d = digits(n, 18)) && (Vecrev(d) == d)); \\ Michel Marcus, Mar 14 2015

def palgen10(l): # generator of palindromes of length <= 2*l
    if l > 0:
        yield 0
        for x in range(1,l+1):
            n = 10**(x-1)
            n2 = n*10
            for y in range(n,n2):
                s = str(y)
                yield int(s+s[-2::-1])
            for y in range(n,n2):
                s = str(y)
                yield int(s+s[::-1])
def palcheck(n, b): # check if n is a palindrome in base b
    s = digits(n, b)
    return s == s[::-1]
A248889_list = [n for n in palgen10(9) if palcheck(n, 18)]
# Chai Wah Wu, Mar 23 2015

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

A069024 Numbers that are palindromic in base 2 as well as in base 10 (initial zeros may be prepended).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 20, 30, 33, 40, 60, 66, 80, 90, 99, 252, 272, 292, 313, 330, 585, 626, 660, 717, 990, 2112, 2720, 2772, 2920, 4224, 5850, 6336, 7447, 7470, 8448, 8580, 9009, 15351, 21120, 22122, 25752, 32223, 39993, 40904, 42240, 44244, 48384

Offset: 1

Amarnath Murthy, Apr 02 2002

			66 in base 2 is 1000010, which is palindromic if rewritten as 01000010.

Robert Israel, Table of n, a(n) for n = 1..243

Cf. A007632.

Intersection of A061917 and A057890.

nextpal:= proc(p,d,V,b)
  local i,i2,pp,m,m2;
  pp:=p;
  V[1]:= V[1]+1;
  m2:= floor(d/2);
  i2:= ceil(d/2);
  if d::odd then pp:= pp + b^m2 else pp:= pp + b^m2 + b^(m2-1) fi;
  for i from 1 while V[i] = b do
    V[i]:= 0:
    if i = i2 then
      if d::even then
        ArrayTools:-Extend(V,[1],inplace);
        return b^d+1, d+1, V
      else
        V[i2]:= 1;
        return b^d+1, d+1, V;
      fi;
    fi;
    V[i+1]:= V[i+1]+1;
    if (d::odd and i=1) then pp:= pp + b^(i2-i-1) else
      pp:= pp + b^(i2-i-1) - b^(i2-i+1) fi;
  od;
  return pp, d, V
end proc:
count:= 1:
S:= 0:
p2[0]:=1: V2[0]:= <1>: d2[0]:= 1:m2:= 0:
p10[0]:= 1: V10[0]:= <1>: d10[0]:= 1: m10:= 0:
while count < 100 do
  i2:= min[index]([seq(p2[i],i=0..m2)])-1; p2o:= p2[i2];
  i10:= min[index]([seq(p10[i],i=0..m10)])-1; p10o:= p10[i10];
  if p2o = p10o then
    S:= S, p2o; count:= count+1;
  fi;
  if p2o <= p10o then x, d2[i2], V2[i2]:= nextpal(p2o/2^i2, d2[i2], V2[i2],2); p2[i2]:= 2^i2 *x;
    if i2 = m2 then m2:= m2+1; p2[m2]:= 2^m2; V2[m2]:= <1>; d2[m2]:= 1;
 fi;
  else
    x, d10[i10], V10[i10]:= nextpal(p10o/10^i10, d10[i10], V10[i10],10);
    p10[i10]:= 10^i10 * x;
    if i10 = m10 then m10:= m10+1; p10[m10]:= 10^m10; V10[m10]:= <1>; d10[m10]:= 1
fi fi od:
S; # Robert Israel, Apr 01 2024

pal[n_, b_] := (z=IntegerDigits[n, b]) == Reverse[z]; extpal[n_, b_] := If[Mod[n, b]>0, pal[n, b], extpal[n/b, b]]; Select[Range[50000], extpal[ #, 10]&&extpal[ #, 2]&]

Edited by Dean Hickerson, Apr 06 2002

0 inserted by Sean A. Irvine, Mar 29 2024

A069025 Smallest power of 2 with digital sum (A007953) n, or 0 if no such number exists.

Original entry on oeis.org

1, 2, 0, 4, 32, 0, 16, 8, 0, 64, 128, 0, 256, 2048, 0, 0, 0, 0, 4096, 8192, 0, 16384, 0, 0, 65536, 32768, 0, 0, 524288, 0, 1048576, 0, 0, 0, 134217728, 0, 16777216, 0, 0, 67108864, 8388608, 0, 268435456, 0, 0, 4398046511104, 2147483648, 0, 0

Offset: 1

Amarnath Murthy, Apr 02 2002

a(3k)=0. In general about half the entries are nonzero.

			Both 2^4=16 and 2^10=1024 have a digital sum of 7 but 2^4 is the smaller so it is the one presented.

Cf. A007632.

a = Table[0, {50}]; Do[b = Plus @@ IntegerDigits[2^n]; If[b < 51 && a[[b]] == 0, a[[b]] = 2^n], {n, 0, 10^4}]; a

Edited by Robert G. Wilson v, Nov 05 2002

A124334 Nonpalindromes in base 10 that are palindromes in base 2.

Original entry on oeis.org

15, 17, 21, 27, 31, 45, 51, 63, 65, 73, 85, 93, 107, 119, 127, 129, 153, 165, 189, 195, 219, 231, 255, 257, 273, 297, 325, 341, 365, 381, 387, 403, 427, 443, 455, 471, 495, 511, 513, 561, 633, 645, 693, 765, 771, 819, 843, 891, 903, 951, 975

Offset: 1

Tanya Khovanova, Dec 26 2006

			17(10) = 10001(2), a palindrome.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A007632 = numbers that are palindromic in bases 2 and 10.

N:= 10000: # to get the first N entries
count:= 0:
for d from 1 while count < N do
   d1:= ceil(d/2); d2:= d - d1;
   for x from 2^(d1-1) to 2^d1 - 1 while count < N do
      if d::even then x1:= x else x1 := floor(x/2) fi;
      L:= convert(x1,base,2);
      y:= 2^(d2)*x + add(L[j]*2^(d2-j),j=1..d2);
      L10:= convert(y,base,10);
      if ListTools[Reverse](L10) = L10 then next fi;
      count:= count+1;
      A[count]:= y;
   od
od:
seq(A[n],n=1..N);
# Robert Israel, Apr 20 2014

Select[Range[1000], Reverse[IntegerDigits[ # ]] != IntegerDigits[ # ] && Reverse[IntegerDigits[ #, 2]] == IntegerDigits[ #, 2] &]
pal2[n_]:=With[{c=IntegerDigits[n,2]},c==Reverse[c]]; Select[Range[1000],!PalindromeQ[#]&&pal2[#]&] (* Harvey P. Dale, Nov 10 2024 *)

A248899 Numbers that are palindromic in bases 10 and 19.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 666, 838, 1771, 432234, 864468, 1551551, 1897981, 2211122, 155292551, 330050033, 453848354, 467535764, 650767056, 666909666, 857383758, 863828368, 47069796074, 62558085526, 67269596276, 87161116178, 96060106069, 121791197121, 127673376721, 139103301931, 234595595432, 246025520642

Offset: 1

Mauro Fiorentini, Mar 06 2015

Next term > 10^12.

			838 = 262 in base 19.

Cf. A007632, A007633, A029961, A029962, A029963, A029964, A029965, A029966, A029967, A029968, A029969, A029970, A029731, A097855, A248889.

[n: n in [0..2*10^7] | Intseq(n) eq Reverse(Intseq(n))and Intseq(n, 19) eq Reverse(Intseq(n, 19))]; // Vincenzo Librandi, Mar 08 2015

IsPalindromic := proc(n, Base)   local Conv, i;
   Conv := convert(n, base, Base);
for i from 1 to nops(Conv) / 2 do:
    if Conv [i] <> Conv [nops(Conv) + 1 - i] then
       return false:
    fi:
od:
return true;
end proc;
Base := 19;
A := [];
for i from 1 to 10^6 do:
   S := convert(i, base, 10);
   V := 0;
   if i mod 10 = 0 then
      next;
   fi;
   for j from 1 to nops(S) do:
      V := V * 10 + S [j];
   od:
   for j from 0 to 10 do:
      V1 := V * 10^(nops(S) + j) + i;
      if IsPalindromic(V1, Base) then
         A := [op(A), V1];
      fi;
   od:
   V1 := (V - (V mod 10)) * 10^(nops(S) - 1) + i;
   if IsPalindromic(V1, Base) then
      A := [op(A), V1];
   fi;
od:
sort(A);

palQ[n_, b_] := Block[{d = IntegerDigits[n, b]}, If[d == Reverse@ d, True, False]]; Select[Range[0, 10^6], And[palQ[#, 10], palQ[#, 19]] &] (* Michael De Vlieger, Mar 07 2015 *)
b1=10; b2=19; lst={}; Do[d1=IntegerDigits[n, b1]; d2=IntegerDigits[n, b2]; If[d1==Reverse[d1]&&d2==Reverse[d2], AppendTo[lst, n]], {n, 0, 10^7}]; lst (* Vincenzo Librandi, Mar 08 2015 *)

isok(n) = (n==0) || ((d = digits(n, 10)) && (Vecrev(d) == d) && (d = digits(n, 19)) && (Vecrev(d) == d)); \\ Michel Marcus, Mar 07 2015

cp's OEIS Frontend

A249155 Palindromic in bases 6 and 15.

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A249157 Palindromic in bases 11 and 13.

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A249158 Palindromic in bases 7 and 29.

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A118064 Decimal expansion of the sum of the reciprocals of the palindromic primes A002385 (Honaker's constant).

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A248889 Palindromic in base 10 and 18.

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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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A069024 Numbers that are palindromic in base 2 as well as in base 10 (initial zeros may be prepended).

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