A305461 The number of one-digit numbers, k, in base n such that k^2 and k^3 end in the same digit.

1, 2, 2, 3, 2, 4, 2, 3, 4, 4, 2, 6, 2, 4, 4, 5, 2, 8, 2, 6, 4, 4, 2, 6, 6, 4, 4, 6, 2, 8, 2, 5, 4, 4, 4, 12, 2, 4, 4, 6, 2, 8, 2, 6, 8, 4, 2, 10, 8, 12, 4, 6, 2, 8, 4, 6, 4, 4, 2, 12, 2, 4, 8, 9, 4, 8, 2, 6, 4, 8, 2, 12, 2, 4, 12, 6, 4, 8, 2, 10, 10, 4, 2, 12, 4

Offset

1,2

Comments

It appears that a(n) is equal to the number of factors of n, except when n cannot be divided by its multiplicative projection (A230542). - Ian Newman, Jun 01 2018

Number of solutions to x^3 - x^2 == 0 (mod n). - Andrew Howroyd, Jul 22 2018

Links

Matthew Scroggs, Table of n, a(n) for n = 1..1000

Matthew Scroggs, Square and cube endings.

Formula

Multiplicative with a(p^e) = p^floor(e/2) + 1 for prime p. - Andrew Howroyd, Jul 22 2018

Example

In base 4,
  0^2 =  0, 0^3 =   0,
  1^2 =  1, 1^3 =   1,
  2^2 = 10, 2^3 =  20,
  3^2 = 21, 3^3 = 123.
Three of these pairs have the same final digit, so a(4)=3.

Mathematica

Table[Count[Range@ n, _?(PowerMod[#, 2, n] == PowerMod[#, 3, n] &)], {n, 85}] (* Michael De Vlieger, Jul 30 2018 *)
f[p_, e_] := p^Floor[e/2] + 1; a[n_] := Times @@ f @@@ FactorInteger[n]; a[1] = 1; Array[a, 100] (* Amiram Eldar, Sep 07 2023 *)

Prog

(Python) #
for base in range(1, 101):
....n = 0
....for j in range(base):
........if (j**2)%base == (j**3)%base:
............n += 1
....print(base, n)
(Haskell)
a305461 n = length $ filter (\i -> (i^3 - i^2) `mod` n == 0) [0..n-1]
-- Peter Kagey, Jun 10 2018
(PARI) a(n) = sum(k=0, n-1, mk = Mod(k, n); mk^2 == mk^3); \\ Michel Marcus, Jul 03 2018
(PARI) a(n)={my(f=factor(n)); prod(i=1, #f~, my(p=f[i, 1], e=f[i, 2]); p^(e\2) + 1)} \\ Andrew Howroyd, Jul 22 2018

Crossrefs

A034444 is the number of one-digit numbers, k, in base n such that k and k^2 end in the same digit.

Cf. A230542.

Sequence in context: A369890 A334762 A358040 * A043261 A157986 A025479

Adjacent sequences: A305458 A305459 A305460 * A305462 A305463 A305464

Keyword

nonn,easy,mult

Author

Matthew Scroggs, Jun 01 2018

Extensions

a(1) inserted by Andrew Howroyd, Jul 22 2018

Status

approved