A010705 -id:A010705 - cp's OEIS frontend

A366654 a(n) = phi(8^n-1), where phi is Euler's totient function (A000010).

6, 36, 432, 1728, 27000, 139968, 1778112, 6635520, 113467392, 534600000, 6963536448, 26121388032, 465193834560, 2427720325632, 28548223200000, 109586090557440, 1910296842179040, 9618417501143040, 123523151337020736, 406467072000000000, 7713001620195508224

Offset: 1

Sean A. Irvine, Oct 15 2023

nonn

Max Alekseyev, Table of n, a(n) for n = 1..500

phi(k^n-1): A053287 (k=2), A295500 (k=3), A295501 (k=4), A295502 (k=5), A366623 (k=6), A366635 (k=7), this sequence (k=8), A366663 (k=9), A295503 (k=10), A366685 (k=11), A366711 (k=12).

Cf. A000010, A010705, A024088, A057953, A059890, A274908, A366651, A366652, A366653.

Mathematica
```
EulerPhi[8^Range[30] - 1]
```
PARI
```
{a(n) = eulerphi(8^n-1)}
```

from sympy import totient
def A366654(n): return totient((1<<3*n)-1) # Chai Wah Wu, Oct 15 2023

a(n) = A053287(3*n). - Max Alekseyev, Jan 09 2024

A235089 a(n)*Pi is the total length of irregular spiral (center points: 2, 1, 3, 4) after n rotations.

Original entry on oeis.org

3, 10, 13, 20, 23, 30, 33, 40, 43, 50, 53, 60, 63, 70, 73, 80, 83, 90, 93, 100, 103, 110, 113, 120, 123, 130, 133, 140, 143, 150, 153, 160, 163, 170, 173, 180, 183, 190, 193, 200, 203, 210, 213, 220, 223, 230, 233, 240, 243, 250, 253, 260, 263, 270, 273, 280, 283, 290, 293, 300, 303, 310, 313, 320, 323, 330, 333

Offset: 1

Kival Ngaokrajang, Jan 03 2014

nonn

Let points 2, 1, 3 & 4 be placed on a straight line at intervals of 1 unit. At point 1 make a half unit circle then at point 2 make another half circle and maintain continuity of circumferences. Continue using this procedure at point 3, 4, 1, ... and so on. The form is non-expanded loop.
The alternative point order [2, 3, 1, 4] gives the same pattern with reflection, but the sequence will be 2*A047215(n). See illustration in links.
Conjecture: Numbers equivalent 0 or 3 modulo 10. - Ralf Stephan, Jan 13 2014

Kival Ngaokrajang, Illustration of initial terms

Cf. A014105*Pi (total spiral length, 2 inline center points). A234902*Pi, A234903*Pi, A234904*Pi (total spiral length, 3 inline center points).

Conjectured partial sums of A010705.

Conjecture: a(n) = -1+(-1)^n+5*n. a(n) = a(n-1)+a(n-2)-a(n-3). G.f.: x*(7*x+3) / ((x-1)^2*(x+1)). - Colin Barker, Jan 16 2014

A364855 Initial digit of 3^(3^n) (A055777(n)).

Original entry on oeis.org

3, 2, 1, 7, 4, 8, 6, 2, 2, 1, 3, 3, 6, 2, 1, 3, 3, 4, 6, 2, 2, 1, 1, 1, 5, 1, 2, 1, 1, 7, 4, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 7, 4, 8, 6, 2, 1, 2, 1, 3, 4, 1, 1, 1, 4, 8, 6, 2, 2, 1, 2, 2, 1, 5, 1, 6, 3, 3, 4, 1, 1, 2, 1, 5, 1, 4, 1

Offset: 0

Marco Ripà, Aug 10 2023

This sequence corresponds to the initial digit of 3vvn (since 3^(3^n) = ((((3^3)^3)^...)^3) n-times), where vv indicates weak tetration (see links).
The author conjectures that the distribution of the initial digits of the present sequence obey Benford's law or Zipf's law (see links).
The corresponding final digit of 3^(3^n) is A010705(n) = 3 if n even or 7 if n odd.

			a(2) = 1, since 3^(3^2) = 3^9 = 19683.

A. Iorliam, Natural Laws (Benford's Law and Zipf's Law) For Network Traffic Analysis, In: Cybersecurity in Nigeria. SpringerBriefs in Cybersecurity. Springer, Cham (2019), 3-22. DOI: 10.1007/978-3-030-15210-9_2

Pointless Large numbers stuff by Cookiefonster, 2.03 The Weak Hyper-Operators.
Wikipedia, Benford's law.
Wikipedia, Zipf's law.

Cf. A000030, A010705 (last digit), A055777, A364789, A364837.

Join[{3},Table[Floor[3^(3^n)/10^Floor[Log10[3^(3^n)]]],{n,16}]]

a(n) = floor(3^(3^n)/10^floor(log_10(3^(3^n)))).

a(n) = A000030(A055777(n)).

More terms from Jinyuan Wang, Aug 11 2023

A176106 Decimal expansion of (21+5*sqrt(21))/14.

Original entry on oeis.org

3, 1, 3, 6, 6, 3, 4, 1, 7, 6, 7, 6, 9, 9, 4, 2, 8, 5, 9, 4, 9, 5, 7, 3, 1, 1, 4, 0, 6, 1, 7, 1, 4, 5, 8, 8, 8, 9, 2, 3, 0, 2, 0, 2, 0, 5, 9, 8, 8, 5, 6, 1, 3, 9, 3, 7, 8, 8, 3, 0, 0, 7, 5, 8, 5, 3, 8, 1, 6, 7, 2, 9, 4, 8, 3, 8, 4, 8, 9, 6, 0, 2, 3, 5, 8, 7, 0, 0, 5, 5, 6, 9, 6, 1, 9, 4, 4, 6, 5, 4, 5, 2, 7, 7, 1

Offset: 1

Klaus Brockhaus, Apr 10 2010

Continued fraction expansion of (21+5*sqrt(21))/14 is A010705.

			(21+5*sqrt(21))/14 = 3.13663417676994285949...

Daniel Starodubtsev, Table of n, a(n) for n = 1..10000

Cf. A010477 (decimal expansion of sqrt(21)), A010705 (repeat 3, 7).

RealDigits[(21+5*Sqrt[21])/14,10,120][[1]] (* Harvey P. Dale, Apr 27 2017 *)

A176435 Decimal expansion of (21+5*sqrt(21))/6.

Original entry on oeis.org

7, 3, 1, 8, 8, 1, 3, 0, 7, 9, 1, 2, 9, 8, 6, 6, 6, 7, 2, 1, 5, 6, 7, 0, 5, 9, 9, 4, 7, 7, 3, 3, 4, 0, 4, 0, 7, 4, 8, 7, 0, 4, 7, 1, 4, 7, 3, 0, 6, 6, 4, 3, 2, 5, 2, 1, 7, 2, 7, 0, 1, 7, 6, 9, 9, 2, 2, 3, 9, 0, 3, 5, 4, 6, 2, 3, 1, 4, 2, 4, 0, 5, 5, 0, 3, 6, 3, 4, 6, 3, 2, 9, 1, 1, 2, 0, 4, 1, 9, 3, 8, 9, 8, 0, 0

Offset: 1

Klaus Brockhaus, Apr 19 2010

Continued fraction expansion of (21+5*sqrt(21))/6 is A010705 preceded by 7.

			(21+5*sqrt(21))/6 = 7.31881307912986667215...

Daniel Starodubtsev, Table of n, a(n) for n = 1..10000

Cf. A010477 (decimal expansion of sqrt(21)), A010705 (repeat 3, 7).

RealDigits[(21+5*Sqrt[21])/6,10,120][[1]] (* Harvey P. Dale, Feb 20 2022 *)

A359632 Sequence of gaps between deletions of multiples of 7 in step 4 of the sieve of Eratosthenes.

Original entry on oeis.org

12, 7, 4, 7, 4, 7, 12, 3, 12, 7, 4, 7, 4, 7, 12, 3, 12, 7, 4, 7, 4, 7, 12, 3, 12, 7, 4, 7, 4, 7, 12, 3, 12, 7, 4, 7, 4, 7, 12, 3, 12, 7, 4, 7, 4, 7, 12, 3, 12, 7, 4, 7, 4, 7, 12, 3, 12, 7, 4, 7, 4, 7, 12, 3, 12, 7, 4, 7, 4, 7, 12, 3, 12, 7, 4, 7, 4, 7, 12, 3

Offset: 1

Alexandre Herrera, Jan 08 2023

This sequence is a repeating cycle 12, 7, 4, 7, 4, 7, 12, 3 of length A005867(4) = 8 = (prime(1)-1)*(prime(2)-1)*(prime(3)-1).
The mean of the cycle is prime(4) = 7.
The cycle is constructed from the sieve of Eratosthenes as follows.
In the first 2 steps of the sieve, the gaps between the deleted numbers are constant: gaps of 2 in step 1 when we delete multiples of 2, and gaps of 3 in step 2 when we delete multiples of 3.
In step 3, when we delete all multiples of 5, the gaps are alternately 7 and 3 (i.e., cycle [7,3]).
For this sequence, we look at the interesting cycle from step 4 (multiples of 7).
Excluding the final 3, the cycle has reflective symmetry: 12, 7, 4, 7, 4, 7, 12. This is true for every subsequent step of the sieve too.
The central element is 7 (BUT not all steps have their active prime number as the central element).
a(1) is A054272(4).
a(8) = 3, the first appearance of the last element of the cycle, corresponds to deletion of 217 = A002110(4)+7.

			After sieve step 3, multiples of 2,3,5 have been eliminated leaving
  7,11,13,17,19,23,29,31,37,41,43,47,49,53, ...
  ^                                   ^
The first two multiples of 7 are 7 itself and 49 and they are distance 12 apart in the list so that a(1) = 12.
For n = 2, a(n) = 7, because the third multiple of 7 that is not a multiple of 2, 3 or 5 is 77 = 7 * 11, which is located 7 numbers after 49 = 7*7 in the list of numbers without the multiples of 2, 3 and 5.

Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,1).

Cf. A002110, A005867, A054272, A236175.

Equivalent sequences for steps 1..3: A007395, A010701, A010705 (without the initial 3).

PadRight[{}, 100, {12, 7, 4, 7, 4, 7, 12, 3}] (* Paolo Xausa, Jul 01 2024 *)

numbers = []
for i in range(2,880):
    numbers.append(i)
gaps = []
step = 4
current_step = 1
while current_step <= step:
    prime = numbers[0]
    new_numbers = []
    gaps = []
    gap = 0
    for i in range(1,len(numbers)):
        gap += 1
        if numbers[i] % prime != 0:
            new_numbers.append(numbers[i])
        else:
            gaps.append(gap)
            gap = 0
    current_step += 1
    numbers = new_numbers
print(gaps)

a(n) = A236175(n)+1. - Peter Munn, Jan 21 2023

A368811 a(n) = period length of the sequence A020639(n^k - 1), k >= 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 12, 1, 10, 1, 1, 1, 60, 1, 10, 1, 1, 1, 18, 1, 2, 1, 1, 1, 660, 1, 66, 1, 1, 1, 1, 1, 10, 1, 1, 1, 4620, 1, 6, 1, 1, 1, 660, 1, 2, 1, 1, 1, 31878, 1, 2, 1, 1, 1, 197340, 1, 5742, 1, 1, 1, 1, 1, 52026, 1, 1, 1, 440220, 1, 28014, 1, 1, 1, 4, 1, 2610, 1, 1, 1, 28014, 1, 2, 1, 1, 1, 3693690, 1, 2, 1, 1, 1, 1, 1, 7590, 1, 1, 1, 1642460820

Offset: 3

Max Alekseyev, Jan 06 2024

nonn

For n = 2, the sequence A020639(n^k - 1) is not periodic (see A049479), but it is such for any n >= 3.

a(n) divides A058254(A000720(A020639(n-1))).

			a(8) = 2 is the period length of A010705.
a(12) = 12 is the period length of A366717.

Max Alekseyev, Table of n, a(n) for n = 3..10000

Cf. A010705, A020639, A049479, A058254, A366717.

{ a368811(n) = my(r=[], z); forprime(p=2, factor(n-1)[1, 1], if(n%p==0, next); z=znorder(Mod(n, p)); if(!#r || vecmin(apply(x->z%x,r)), r=concat(r,[z])) ); lcm(r); }

For odd n >= 3, a(n) = 1.

A131757 Period 10: repeat 3, 3, 3, -7, 3, 3, -7, 3, 3, -7.

Original entry on oeis.org

3, 3, 3, -7, 3, 3, -7, 3, 3, -7, 3, 3, 3, -7, 3, 3, -7, 3, 3, -7, 3, 3, 3, -7, 3, 3, -7, 3, 3, -7, 3, 3, 3, -7, 3, 3, -7, 3, 3, -7, 3, 3, 3, -7, 3, 3, -7, 3, 3, -7, 3, 3, 3, -7, 3, 3, -7, 3, 3, -7, 3, 3, 3, -7, 3, 3, -7, 3, 3, -7, 3, 3, 3, -7, 3, 3, -7, 3, 3, -7, 3, 3, 3, -7, 3, 3, -7, 3, 3, -7, 3, 3, 3, -7, 3, 3, -7, 3, 3, -7, 3, 3, 3, -7, 3, 3, -7, 3, 3, -7, 3

Offset: 0

Paul Curtz, Oct 04 2007

Index entries for linear recurrences with constant coefficients, signature (-1,-1,-1,-1,-1,-1,-1,-1,-1).

Cf. A010705, A096915.

PadRight[{},120,{3,3,3,-7,3,3,-7,3,3,-7}] (* Harvey P. Dale, Aug 07 2018 *)

a(n)=[3, 3, 3, -7, 3, 3, -7, 3, 3, -7][n%10+1] \\ Charles R Greathouse IV, Jul 13 2016

cp's OEIS Frontend

A366654 a(n) = phi(8^n-1), where phi is Euler's totient function (A000010).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula

A235089 a(n)*Pi is the total length of irregular spiral (center points: 2, 1, 3, 4) after n rotations.

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

A364855 Initial digit of 3^(3^n) (A055777(n)).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A176106 Decimal expansion of (21+5*sqrt(21))/14.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A176435 Decimal expansion of (21+5*sqrt(21))/6.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A359632 Sequence of gaps between deletions of multiples of 7 in step 4 of the sieve of Eratosthenes.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Python

Formula

A368811 a(n) = period length of the sequence A020639(n^k - 1), k >= 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A131757 Period 10: repeat 3, 3, 3, -7, 3, 3, -7, 3, 3, -7.

Views

Author

Keywords