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
Examples
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.
Links
- Michael S. Branicky, Table of n, a(n) for n = 2..8284 (terms 2..523 from Robert Israel)
Programs
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Maple
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
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PARI
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
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Python
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
Formula
A375201(a(n)) = n. - Robert Israel, Oct 15 2024
Extensions
Name clarified by Robert Israel, Oct 15 2024