A344749 Numbers m with decimal expansion (d_k, ..., d_1) such that d_i = m ^ i mod 10 for i = 1..k.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 19, 42, 48, 55, 64, 66, 93, 97, 111, 248, 397, 464, 555, 666, 793, 842, 919, 1111, 1397, 1793, 1919, 5555, 6248, 6464, 6666, 6842, 11111, 26842, 31793, 46464, 55555, 66666, 71397, 86248, 91919, 111111, 191919, 426842, 486248

Offset: 1

Author

Rémy Sigrist, May 28 2021

Keywords

• nonn
• base
• easy

Comments

Positive terms are zeroless (A052382) and uniquely determined by their final digit (A010879) and the number of digits in their decimal expansion (A055642).

If m belongs to the sequence, then A217657(m) also belongs to the sequence.

Examples

			- 7^1 = 7 mod 10,
- 7^2 = 9 mod 10,
- 7^3 = 3 mod 10,
- 7^4 = 1 mod 10,
- so 1397 belongs to the sequence.
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 1..2251

Crossrefs

Cf. A010879, A052382, A055642, A217657, A344555, A344748, A344822.

Programs

PARI

is(n) = { my (r=n); for (k=1, oo, if (r==0, return (1), (n^k)%10!=r%10, return (0), r\=10)) }

PARI

print (setbinop((d,k) -> sum(i=1, k, 10^(i-1) * ((d^i)%10)), [1..9], [0..7])[1..50])

Python

def ok(m):
  d = str(m)
  return all(d[-i] == str((m**i)%10) for i in range(1, len(d)+1))
print(list(filter(ok, range(10**6)))) # Michael S. Branicky, May 29 2021

Python

def auptod(maxdigits):
  alst = [0]
  for k in range(1, maxdigits+1):
    aklst = []
    for d1 in range(1, 10):
      d = [(d1**i)%10 for i in range(k, 0, -1)]
      aklst.append(int("".join(map(str, d))))
    alst.extend(sorted(aklst))
  return alst
print(auptod(6)) # Michael S. Branicky, May 29 2021