A057951 -id:A057951 - cp's OEIS frontend

A366708 Number of prime factors of 12^n - 1 (counted with multiplicity).

1, 2, 2, 4, 2, 5, 3, 6, 4, 4, 4, 8, 3, 6, 6, 9, 3, 9, 2, 8, 5, 6, 4, 12, 4, 8, 6, 10, 5, 13, 5, 11, 8, 6, 9, 14, 3, 6, 7, 14, 4, 14, 5, 12, 12, 8, 3, 18, 5, 10, 6, 13, 7, 16, 8, 13, 7, 8, 4, 19, 4, 8, 8, 13, 8, 17, 5, 10, 7, 14, 4, 21, 3, 7, 11, 11, 11, 18, 4

Offset: 1

Sean A. Irvine, Oct 17 2023

nonn

Max Alekseyev, Table of n, a(n) for n = 1..310

Cf. A024140, A001222, A046051, A057951, A057952, A057953, A057954, A057955, A057956, A057957, A057958, A250288, A252170, A366707, A366709, A366682.

Mathematica
```
PrimeOmega[12^Range[70]-1]
```
PARI
```
a(n)=bigomega(12^n-1)
```

a(n) = bigomega(12^n-1) = A001222(A024140(n)).

A366682 Number of prime factors of 11^n - 1 (counted with multiplicity).

Original entry on oeis.org

2, 5, 4, 7, 4, 9, 4, 9, 5, 8, 4, 13, 4, 8, 7, 12, 3, 12, 3, 11, 10, 11, 5, 17, 8, 10, 6, 13, 4, 15, 5, 15, 9, 9, 8, 17, 6, 10, 12, 15, 9, 17, 4, 15, 9, 12, 5, 24, 7, 14, 9, 13, 6, 16, 10, 19, 8, 10, 5, 21, 5, 12, 16, 19, 8, 22, 6, 15, 10, 19, 7, 24, 3, 11, 15

Offset: 1

Sean A. Irvine, Oct 16 2023

nonn

Max Alekseyev, Table of n, a(n) for n = 1..316

Cf. A024127, A001222, A366681, A366683, A366684, A366685, A366687.

Cf. A046051, A057958, A057957, A057956, A057955, A057954, A057953, A057952, A057951, A366708.

Mathematica
```
PrimeOmega[11^Range[70]-1]
```
PARI
```
a(n)=bigomega(11^n-1)
```

a(n) = bigomega(11^n-1) = A001222(A024127(n)).

A109472 Cumulative sum of primes p such that 2^p - 1 is a Mersenne prime.

Original entry on oeis.org

2, 5, 10, 17, 30, 47, 66, 97, 158, 247, 354, 481, 1002, 1609, 2888, 5091, 7372, 10589, 14842, 19265, 28954, 38895, 50108, 70045, 91746, 114955, 159452, 245695, 356198, 488247, 704338, 1461177, 2320610, 3578397, 4976666, 7952887, 10974264, 17946857, 31413774, 52409785, 76446368, 102411319, 132813776, 165396433, 202553100, 245196901, 288309510

Offset: 1

Jonathan Vos Post, Aug 28 2005

nonn

Prime cumulative sum of primes p such that 2^p - 1 is a Mersenne prime include: a(1) = 2, a(2) = 5, a(4) = 17, a(6) = 47, a(8) = 97, a(14) = 1609, a(18) = 10589. After 1, all such indices x of prime a(x) must be even.

			a(1) = 2, since 2^2-1 = 3 is a Mersenne prime.
a(2) = 2 + 3 = 5, since 2^3-1 = 7 is a Mersenne prime.
a(3) = 2 + 3 + 5 = 10, since 2^5-1 = 31 is a Mersenne prime.
a(4) = 2 + 3 + 5 + 7 = 17, since 2^7-1 = 127 is a Mersenne prime; 17 itself is prime (in fact a p such that 2^p-1 is a Mersenne prime).
a(18) = 2 + 3 + 5 + 7 + 13 + 17 + 19 + 31 + 61 + 89 + 107 + 127 + 521 + 607 + 1279 + 2203 + 2281 + 3217 = 10589 (which is prime).

Amiram Eldar, Table of n, a(n) for n = 1..48 (terms 1..47 from Gord Palameta)

Cf. A000043, A000668 for the Mersenne primes, A001348, A046051, A057951-A057958.

Accumulate[Select[Range[3000], PrimeQ[2^# - 1] &]] (* Vladimir Joseph Stephan Orlovsky, Jul 08 2011 *)
Accumulate@ MersennePrimeExponent@ Range@ 45 (* Michael De Vlieger, Jul 22 2018 *)

a(n) = Sum_{i=1..n} A000043(i).

a(38)-a(47) from Gord Palameta, Jul 21 2018

A366165 a(n) is the least k > 0 such that 10^(2n-1) - k can be written as a product jm, where j and m have an equal number of decimal digits.

Original entry on oeis.org

1, 1, 1, 1, 10, 1, 3, 1, 5, 3, 1, 6, 1, 7, 1, 2, 2, 1, 4, 7, 5, 1, 1, 3, 2, 1, 1, 1, 1, 2, 1, 1, 10, 4, 3, 3, 10, 1, 2, 3, 1, 1, 1, 7, 1, 1

Offset: 1

Hugo Pfoertner, Oct 04 2023

a(n) <= 10 since 10^(2n-1)-10 = (10^(n-1)+1)(10^n-10). A consequence is that j and m in the product both have n decimal digits. - Chai Wah Wu, Oct 05 2023

			n a(n) 10^(2n-1)-a(n)       j       m
1  1   9                    1       9
2  1   999                 27      37
3  1   99999              123     813
4  1   9999999           2151    4649
5 10   999999990        10001   99990
6  1   99999999999     194841  513239
7  3   9999999999997  2769823 3610339
More than one pair (j,m) may exist, e.g., 9 = 1*9 = 3*3.

Cf. A002275, A003020, A004022, A057951, A327435.

A067272 are the solutions for even exponents of 10, corresponding to (j,m) = (9,9), (99,99), (999,999), ... .

a366165(n)={my (p10=10^(2*n-1)); for (dd=1, p10, my (d=p10-dd); fordiv (d, x, fordiv (d, y, if (x*y==d && #digits(x)==#digits(y), return(dd)))))};

from itertools import count, takewhile
from sympy import divisors
def A366165(n):
    a, l1, l2 = 10**((n<<1)-1), 10**(n-1), 10**n
    for k in count(1):
        b = a-k
        if any(l1<=db for d in takewhile(lambda m:m*m<=b, divisors(b))):
            return k # Chai Wah Wu, Oct 05 2023

a(33)-a(35) from Chai Wah Wu, Oct 05 2023

a(36)-a(46) from Chai Wah Wu, Oct 07 2023

A366922 a(n) is the exponent of 3 in the prime factorization of 10^n - 1.

Original entry on oeis.org

2, 2, 3, 2, 2, 3, 2, 2, 4, 2, 2, 3, 2, 2, 3, 2, 2, 4, 2, 2, 3, 2, 2, 3, 2, 2, 5, 2, 2, 3, 2, 2, 3, 2, 2, 4, 2, 2, 3, 2, 2, 3, 2, 2, 4, 2, 2, 3, 2, 2, 3, 2, 2, 5, 2, 2, 3, 2, 2, 3, 2, 2, 4, 2, 2, 3, 2, 2, 3, 2, 2, 4, 2, 2, 3, 2, 2, 3, 2, 2, 6, 2, 2, 3, 2, 2, 3, 2, 2, 4

Offset: 1

Hugo Pfoertner, Oct 28 2023

1

Cf. A002283, A003020, A007949, A051064, A057951, A102347, A366921.

a[n_]:=IntegerExponent[10^n-1,3]; Array[a,90] (* Stefano Spezia, Oct 28 2023 *)

PARI
```
a366922(n) = valuation(10^n-1,3)
```

def A366922(n):
    c = 0
    a, b = divmod(10**n-1, 3)
    while b == 0:
        a, b = divmod(a, 3)
        c += 1
    return c # Chai Wah Wu, Oct 29 2023

a(n) = A007949(10^n - 1).

a(n) = A007949(n) + 2 = A051064(n) + 1.

cp's OEIS Frontend

A366708 Number of prime factors of 12^n - 1 (counted with multiplicity).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A366682 Number of prime factors of 11^n - 1 (counted with multiplicity).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A109472 Cumulative sum of primes p such that 2^p - 1 is a Mersenne prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A366165 a(n) is the least k > 0 such that 10^(2*n-1) - k can be written as a product j*m, where j and m have an equal number of decimal digits.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

Python

Extensions

A366922 a(n) is the exponent of 3 in the prime factorization of 10^n - 1.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

PARI

Python

Formula

A366165 a(n) is the least k > 0 such that 10^(2n-1) - k can be written as a product jm, where j and m have an equal number of decimal digits.