A344749 -id:A344749 - cp's OEIS frontend

A344748 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, 21, 26, 42, 47, 63, 68, 84, 89, 147, 284, 321, 468, 505, 642, 789, 826, 963, 2468, 2963, 4321, 4826, 6284, 6789, 8147, 8642, 50505, 52963, 54321, 56789, 58147, 208642, 258147, 406284, 456789, 604826, 654321, 802468, 852963

Offset: 1

Rémy Sigrist, May 28 2021

Positive terms have no trailing zero in decimal representation (A067251), and are uniquely determined by their final digit d (A010879) and the number of digits, say k, in their decimal expansion (A055642); d*k cannot be a multiple of 10.

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

			- 6 * 1 = 6 mod 10,
- 6 * 2 = 2 mod 10,
- 6 * 3 = 8 mod 10,
- 6 * 4 = 4 mod 10,
- so 4826 belongs to the sequence.

Rémy Sigrist, Table of n, a(n) for n = 1..1826

Cf. A010879, A055642, A067251, A217657, A344749, A344822.

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

print (setbinop((d,k) -> sum(i=1, k, 10^(i-1) * ((d*i)%10)), [1..9], [0..6]))

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

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)]
      if d[0] != 0: aklst.append(int("".join(map(str, d))))
    alst.extend(sorted(aklst))
  return alst
print(auptod(6)) # Michael S. Branicky, May 29 2021

A344823 Numbers m with decimal expansion (d_1, ..., d_k) such that d_i = m ^ i mod 10 for i = 1..k.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 55, 66, 111, 464, 555, 666, 919, 1111, 5555, 6666, 11111, 24862, 39713, 46464, 55555, 66666, 79317, 84268, 91919, 111111, 555555, 666666, 1111111, 4646464, 5555555, 6666666, 9191919, 11111111, 55555555, 66666666, 111111111

Offset: 1

Rémy Sigrist, May 29 2021

This sequence is infinite as it contains d * A002275(k) for any d in {1, 5, 6} and k > 0.

Also contains terms with patterns 4(64)^k, 9(19)^k, 2(4862)^k, 3(9713)^k, 7(9317)^k, 8(4268)^k for k >= 0, where ^ denotes repeated concatenation; all terms have first and last digits the same. - Michael S. Branicky, May 29 2021

			- 4^1 = 4 mod 10,
- 4^2 = 6 mod 10,
- 4^3 = 4 mod 10,
- so 464 belongs to the sequence.

Rémy Sigrist, Table of n, a(n) for n = 1..1253
Rémy Sigrist, PARI program for A344823

Cf. A002275, A344555, A344749, A344822.

is(n) = { my (d=digits(n), m=Mod(n,10)); for (k=1, #d, if (d[k] != m^k, return (0))); return (1) }

PARI
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See Links section.
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def ok(m):
  d = str(m)
  return all(d[i-1] == 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

def auptod(maxdigits):
  alst = [0]
  for k in range(1, maxdigits+1):
    for d1 in range(1, 10):
      d = [(d1**i)%10 for i in range(1, k+1)]
      if d1 == d[-1]: alst.append(int("".join(map(str, d))))
  return alst
print(auptod(9)) # Michael S. Branicky, May 29 2021

cp's OEIS Frontend

A344748 Numbers m with decimal expansion (d_k, ..., d_1) such that d_i = m * i mod 10 for i = 1..k.

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A344823 Numbers m with decimal expansion (d_1, ..., d_k) such that d_i = m ^ i mod 10 for i = 1..k.

Views

Author

Keywords

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Examples

Links

Crossrefs

Programs

PARI

PARI

Python

Python