A393176 The smallest positive number k such that the digit reversal of k, when it is written in all bases 2 to n, shares a factor with k.

3, 6, 6, 6, 10, 10, 12, 12, 12, 12, 24, 24, 24, 24, 24, 24, 42, 42, 42, 42, 42, 42, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 84, 84, 84, 84, 84, 84, 84, 84, 84, 84, 84, 84, 108, 108, 108, 108, 108, 108, 120, 120, 120, 120, 120, 120

Offset

2,1

Comments

It is unknown if the sequence is infinite.

Links

Scott R. Shannon, Table of n, a(n) for n = 2..1427

Formula

a(n+1) >= a(n). - Michael S. Branicky, Feb 06 2026

Example

a(2) = 3 as 3 = 11_2 and the reverse of 11_2 = R(11_2) = 3 and 3 shares a factor with 3.
a(3) = 6 as 6 = 110_2 and R(110_2) = 11_2 = 3 and 3 shares a factor with 6, and 6 = 20_3 and R(20_3) = 2_3 = 2 and 2 shares a factor with 6.
a(6) = 10 as 10 = 1010_2 and R(1010_2) = 101_2 = 5, 10 = 101_3 and R(101_3) = 101_3 = 10, 10 = 22_4 and R(22_4) = 22_4 = 10, 10 = 20_5 and R(20_5) = 2_5 = 2, 10 = 14_6 and R(14_6) = 41_6 = 25, and 5, 10, 10, 2, 25 all share a factor with 10.

Prog

(Python)
from itertools import count
from math import gcd
from sympy.ntheory import digits
def fd(t, b): return sum(t[-1-i]*b**i for i in range(len(t)))
def dr(k, b): return fd(digits(k, b)[1:][::-1], b)
def a(n, startk=1): return next(k for k in count(startk) if all(gcd(dr(k, b), k) > 1 for b in range(2, n+1)))
print([a(n) for n in range(2, 66)]) # Michael S. Branicky, Feb 06 2026
(Python) # use with above for initial segment of sequence
from itertools import islice
def agen(): # generator of terms
    an = 1
    yield from (an for n in count(2) if (an:=a(n, startk = an)))
print(list(islice(agen(), 64))) # Michael S. Branicky, Feb 06 2026

Crossrefs

Cf. A030101, A030102, A030103, A030104, A071249, A393075, A391057.

Sequence in context: A358548 A155067 A094011 * A351116 A295558 A295559

Adjacent sequences: A393173 A393174 A393175 * A393177 A393178 A393179

Keyword

nonn,base

Author

Scott R. Shannon, Feb 04 2026

Status

approved