A244415 -id:A244415 - cp's OEIS frontend

A054705 Number of powers of 4 modulo n.

1, 2, 1, 2, 2, 2, 3, 3, 3, 3, 5, 2, 6, 4, 2, 3, 4, 4, 9, 3, 3, 6, 11, 3, 10, 7, 9, 4, 14, 3, 5, 4, 5, 5, 6, 4, 18, 10, 6, 4, 10, 4, 7, 6, 6, 12, 23, 3, 21, 11, 4, 7, 26, 10, 10, 5, 9, 15, 29, 3, 30, 6, 3, 4, 6, 6, 33, 5, 11, 7, 35, 5, 9, 19, 10, 10, 15, 7, 39, 4, 27, 11, 41, 4, 4, 8, 14, 7, 11

Offset: 1

Henry Bottomley, Apr 20 2000

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from David W. Wilson)

Cf. A007735, A244415.

Cf. A054703 (base 2), A054704 (3), A054706 (5), A054707 (6), A054708 (7), A054709 (8), A054717 (9), A054710 (10), A351524 (11), A054712 (12), A054713 (13), A054714 (14), A054715 (15), A054716 (16).

a[n_] := IntegerExponent[2*n, 4] + MultiplicativeOrder[4, n/2^IntegerExponent[n, 2]]; Array[a, 100] (* Amiram Eldar, Aug 25 2024 *)

a(n) = A007735(n) + A244415(n). - Amiram Eldar, Aug 25 2024

A244417 Exponents of 6 in appearing in the 6-adic value of 1/n, n>=1 (A244416).

Original entry on oeis.org

0, 1, 1, 2, 0, 1, 0, 3, 2, 1, 0, 2, 0, 1, 1, 4, 0, 2, 0, 2, 1, 1, 0, 3, 0, 1, 3, 2, 0, 1, 0, 5, 1, 1, 0, 2, 0, 1, 1, 3, 0, 1, 0, 2, 2, 1, 0, 4, 0, 1, 1, 2, 0, 3, 0, 3, 1, 1, 0, 2, 0, 1, 2, 6, 0, 1, 0, 2, 1, 1, 0, 3, 0, 1, 1, 2, 0, 1, 0, 4, 4, 1, 0, 2, 0, 1, 1, 3, 0, 2, 0, 2, 1, 1, 0, 5, 0, 1, 2, 2

Offset: 1

Wolfdieter Lang, Jul 02 2014

For the definition of 'g-dic value of 1/n' see a comment on A244416. In the Mahler reference, p. 7, the present exponent of 6 is there called f = f(1/n) for g = 6.

			See A244416.

Kurt Mahler, p-adic numbers and their functions, second ed., Cambridge University Press, 1981.

Antti Karttunen, Table of n, a(n) for n = 1..19683

Cf. A122841, A244416, A007814 (g=2), A007949 (g=3), A244415 (g=4), A112765 (g=5), A051903, A065331.

Cf. also A322026, A322316.

a[n_] := Max[IntegerExponent[n, {2, 3}]]; Array[a, 100] (* Amiram Eldar, Aug 19 2024 *)

A244417(n) = max(valuation(n,2), valuation(n,3)); \\ Antti Karttunen, Dec 04 2018

a(n) = 0 if n is congruent 1 or 5 (mod 6). a(n) = max(A007814(n), A007949(n)) if n == 0 (mod 6). a(n) = A007814(n) if n == 2 or 4 (mod 6) and a(n) = A007949(n) if n == 3 (mod 6).
a(n) = max(A007814(n), A007949(n)), in all cases. - Antti Karttunen, Dec 04 2018
From Amiram Eldar, Aug 19 2024: (Start)
a(n) = A051903(A065331(n)).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 13/10. (End)

A240226 4-adic value of 1/n, n >= 1.

Original entry on oeis.org

1, 4, 1, 4, 1, 4, 1, 16, 1, 4, 1, 4, 1, 4, 1, 16, 1, 4, 1, 4, 1, 4, 1, 16, 1, 4, 1, 4, 1, 4, 1, 64, 1, 4, 1, 4, 1, 4, 1, 16, 1, 4, 1, 4, 1, 4, 1, 16, 1, 4, 1, 4, 1, 4, 1, 16, 1, 4, 1, 4, 1, 4, 1, 64, 1, 4, 1, 4, 1, 4, 1, 16, 1, 4, 1, 4, 1, 4, 1, 16, 1, 4, 1, 4, 1, 4, 1, 16, 1, 4

Offset: 1

Wolfdieter Lang, Jun 28 2014

For the definition of g-adic value of x, called |x|_g with g an integer >= 2, see the Mahler reference, p. 7. Sometimes also called g-adic absolute value of x. If g is not a prime then this is called a non-archimedean pseudo-valuation. See Mahler, p. 10.

			n = 2: A006519(2) = 1, 2 divides 4^1, hence f(1/2) = 1 and a(2) = 4^1 = 4.
n = 4: A006519(4) = 2^2, 4 divides 4^1, hence f(1/4) = 1 and a(4) = 4.
n = 8: A006519(8) = 2^3, 8 does not divide 4^1 but 4^2, hence f(1/8) = 2 and a(8) = 4^2 = 16.

Kurt Mahler, p-adic numbers and their functions, second ed., Cambridge University Press, 1981.

Antti Karttunen, Table of n, a(n) for n = 1..16383

Cf. A001620, A006519, A038500 (3-adic value of 1/n), A244415.

Array[4^IntegerExponent[2 #, 4] &, 90] (* Michael De Vlieger, Nov 06 2018 *)

a(n) = 4^ceil(valuation(n, 2)/2); \\ Andrew Howroyd, Jul 31 2018

def A240226(n): return 1<<((~n&n-1).bit_length()+1&-2) # Chai Wah Wu, Jul 09 2023

a(n) = 1 if n is odd. a(n) = 4^f(1/n) if n is even, where f(1/n) is the smallest positive integer such that the highest power of 2 in n (that is A006519(n)) divides 4^f(1/n). The f(1/n) values are given in A244415(n).
From Andrew Howroyd, Jul 31 2018: (Start)
a(n) = 4^valuation(2*n, 4) = 4^A244415(n).
Multiplicative with a(2^e) = 4^ceiling(e/2), a(p^e) = 1 for odd prime p. (End)
From Amiram Eldar, Oct 24 2023: (Start)
Dirichlet g.f.: zeta(s)*(2^s-1)*(2^s+4)/(4^s-4).
Sum_{k=1..n} a(k) ~ (3/(4*log(2))) * n * (log(n) + gamma + 4*log(2)/3 - 1), where gamma is Euler's constant (A001620). (End)

A363228 Exponent of 4 in 9^n - 1.

Original entry on oeis.org

1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 2, 1, 4, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 2, 1, 4, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 2, 1, 4, 1, 2, 1, 2

Offset: 1

Ruud H.G. van Tol, May 21 2023

Not the same as A147648-without-zeros.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A007814, A090739, A147648, A244415.

Cf. A024101, A235127.

a[n_] := IntegerExponent[2*n, 4] + 1; Array[a, 100] (* Amiram Eldar, May 22 2023 *)

PARI
```
a(n) = valuation(2*n, 4) + 1;
```

def A363228(n): return (~n&n-1).bit_length()+3>>1 # Chai Wah Wu, Jul 09 2023

a(n) = floor(A090739(n)/2).
a(n) = A244415(n) + 1.
a(n) = A235127(A024101(n)). - Michel Marcus, May 21 2023
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 5/3. - Amiram Eldar, Jul 13 2023
Conjecture: a(n) = A235127(A000045(6*n)), all other 4-adic 6-sections A235127(A000045(.))=0. - R. J. Mathar, Jun 28 2025

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A054705 Number of powers of 4 modulo n.

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A244417 Exponents of 6 in appearing in the 6-adic value of 1/n, n>=1 (A244416).

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A240226 4-adic value of 1/n, n >= 1.

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A363228 Exponent of 4 in 9^n - 1.

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