A253594 -id:A253594 - cp's OEIS frontend

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.

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

A087166 Primes which are palindromes in 3 or more bases.

Original entry on oeis.org

17, 31, 67, 73, 107, 109, 127, 151, 157, 173, 181, 191, 197, 211, 227, 241, 257, 271, 277, 307, 313, 337, 353, 373, 379, 401, 409, 419, 421, 433, 443, 457, 461, 463, 487, 521, 523, 541, 577, 587, 601, 617, 619, 631, 647, 661, 673, 683, 701, 719, 727, 743, 757, 761, 773, 787, 797, 809, 857, 859

Offset: 1

Randy L. Ekl, Oct 18 2003

For the purposes of this sequence, single digits are not counted as palindromes (otherwise every number n is a palindrome in all bases > n). - Robert Israel, May 01 2020

			31 is in the list, as 31 base 2 = 11111, 31 base 5 = 111 and 31 base 30 = 11, i.e. three different ways.

Robert Israel, Table of n, a(n) for n = 1..5867

Primes in A253594.

N:= 1000: # for all terms <= N
digrev:= proc(n,b)
  local L,i;
  L:= convert(n,base,b);
  add(L[-i]*b^(i-1),i=1..nops(L))
end proc:
bpalis:= proc(b, N)
  local Res,dmax,d,m;
  dmax:= floor(log[b](N))+1;
  if dmax < 2 then return [] fi;
  Res:= seq(i*(b+1),i=1..b-1);
  for d from 3 to dmax do
    if d::even then
      m:= d/2;
      Res:= Res, seq(n*b^m + digrev(n,b),n=b^(m-1)..b^m-1);
    else
      m:= (d-1)/2;
      Res:= Res, seq(seq(n*b^(m+1)+y*b^m+digrev(n,b), y=0..b-1), n=b^(m-1)..b^m-1);
    fi
  od;
  select(`<=`,[Res], N)
end proc:
V:= Vector(N):
for b from 2 to N-1 do
  bp:= bpalis(b,N);
  V[bp]:= V[bp] +~ 1
od:
select(p -> isprime(p) and V[p] >= 3, [seq(i,i=3..N,2)]); # Robert Israel, May 01 2020

Corrected by Robert Israel, May 01 2020

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

A374561 Integers which are palindromes when expressed in more than one base 2 to 10.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 15, 16, 17, 18, 20, 21, 24, 26, 27, 28, 31, 33, 36, 40, 45, 46, 50, 51, 52, 55, 57, 63, 65, 67, 73, 78, 80, 82, 85, 88, 91, 92, 93, 98, 99, 100, 104, 105, 107, 109, 111, 114, 119, 121, 127, 129, 130, 135, 141, 142, 150, 151, 154, 160, 164, 170, 171, 173, 178

Offset: 1

Paul Duckett, Jul 11 2024

Sequence is infinite because all integers of the form 4^n-1 are palindromic in bases 2 and 4.

			5 is a term since it's palindromic in more than one base: base 2 (101) and base 4 (11).
121 is a term since it's palindromic in base 3 (11111) and base 7 (232), and also in fact in bases 8 and 10.

Cf. A050812, A253594.

q[n_] := Count[Range[2, 10], ?(PalindromeQ[IntegerDigits[n, #]] &)] > 1; Select[Range[180], q] (* _Amiram Eldar, Jul 20 2024 *)

isok(k) = sum(b=2, 10, my(v=digits(k, b)); v==Vecrev(v)) > 1; \\ Michel Marcus, Aug 03 2024

from sympy.ntheory import is_palindromic
def ok(n):
    c = 0
    for b in range(2, 11):
        c += int(is_palindromic(n, b))
        if c > 1: return True
    return False
print([k for k in range(1, 180) if ok(k)]) # Michael S. Branicky, Aug 02 2024

A050812(a(n)) >= 2. - Michael S. Branicky, Aug 02 2024

cp's OEIS Frontend

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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A087166 Primes which are palindromes in 3 or more bases.

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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.

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PARI

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A374561 Integers which are palindromes when expressed in more than one base 2 to 10.

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Mathematica

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