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A305461 The number of one-digit numbers, k, in base n such that k^2 and k^3 end in the same digit. 1
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 (list; graph; refs; listen; history; text; internal format)
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 *)

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: A282446 A049599 A334762 * A043261 A157986 A025479

Adjacent sequences:  A305458 A305459 A305460 * A305462 A305463 A305464

KEYWORD

nonn,mult

AUTHOR

Matthew Scroggs, Jun 01 2018

EXTENSIONS

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

STATUS

approved

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Last modified August 7 12:19 EDT 2020. Contains 336276 sequences. (Running on oeis4.)