A046051 -id:A046051 - cp's OEIS frontend

A057953 Number of prime factors of 8^n - 1 (counted with multiplicity).

1, 3, 2, 5, 3, 6, 4, 7, 3, 7, 4, 10, 4, 8, 6, 10, 5, 9, 4, 13, 7, 9, 4, 14, 7, 8, 6, 14, 6, 13, 3, 13, 8, 11, 11, 15, 6, 9, 9, 17, 5, 14, 5, 15, 10, 9, 6, 19, 7, 14, 8, 18, 8, 16, 10, 19, 7, 11, 6, 24, 5, 8, 10, 16, 8, 17, 6, 20, 9, 22, 7, 21, 7, 13, 14, 17, 10, 16, 8, 23, 10, 12, 5, 24

Offset: 1

Patrick De Geest, Nov 15 2000

nonn

Max Alekseyev, Table of n, a(n) for n = 1..500 (first 402 terms from Amiram Eldar)
S. S. Wagstaff, Jr., The Cunningham Project

bigomega(b^n-1): A057951 (b=10), A057952 (b=9), this sequence (b=8), A057954 (b=7), A057955 (b=6), A057956 (b=5), A057957 (b=4), A057958 (b=3), A046051 (b=2).

Cf. A001222, A024088, A085033, A274908.

f:=func; [f(8^n - 1):n in [1..90]]; // Marius A. Burtea, Feb 02 2020

PrimeOmega/@(8^Range[90]-1) (* Harvey P. Dale, May 24 2018 *)

Mobius transform of A085033. - T. D. Noe, Jun 19 2003

a(n) = A001222(A024088(n)) = A046051(3*n). - Amiram Eldar, Feb 02 2020

A057955 Number of prime factors of 6^n - 1 (counted with multiplicity).

Original entry on oeis.org

1, 2, 2, 3, 3, 4, 2, 4, 4, 6, 3, 7, 3, 6, 6, 6, 5, 7, 3, 9, 4, 5, 5, 9, 6, 7, 6, 9, 2, 11, 3, 9, 6, 8, 7, 13, 6, 6, 6, 12, 3, 10, 5, 9, 11, 8, 4, 13, 5, 10, 9, 11, 4, 11, 7, 14, 7, 6, 4, 20, 4, 5, 10, 12, 9, 12, 3, 11, 8, 18, 2, 18, 5, 10, 12, 9, 6, 15, 4, 17, 8, 7, 8, 17, 10, 7, 7, 12, 4, 18, 6

Offset: 1

Patrick De Geest, Nov 15 2000

nonn

			6^10 - 1 = 60466175 = 5^2 * 7 * 11 * 101 * 311 and a(10) = bigomega(60466175) = 2+1+1+1+1 = 6. - _Bernard Schott_, Feb 02 2020

Max Alekseyev, Table of n, a(n) for n = 1..430 (terms 1..420 from Amiram Eldar)
S. S. Wagstaff, Jr., The Cunningham Project

bigomega(b^n-1): A057951 (b=10), A057952 (b=9), A057953 (b=8), A057954 (b=7), this sequence (b=6), A057956 (b=5), A057957 (b=4), A057958 (b=3), A046051 (b=2).

Cf. A001222, A024062, A085031, A274907.

f:=func; [f(6^n - 1):n in [1..90]]; // Marius A. Burtea, Feb 02 2020

PrimeOmega[6^Range[100]-1] (* Harvey P. Dale, Dec 14 2015 *)

Möbius transform of A085031. - T. D. Noe, Jun 19 2003

a(n) = A001222(A024062(n)). - Amiram Eldar, Feb 02 2020

A057952 Number of prime factors of 9^n - 1 (counted with multiplicity).

Original entry on oeis.org

3, 5, 5, 7, 6, 8, 5, 10, 8, 10, 7, 11, 5, 9, 11, 12, 8, 12, 7, 13, 11, 11, 6, 17, 10, 9, 13, 13, 9, 17, 8, 14, 12, 12, 11, 16, 8, 11, 15, 18, 8, 18, 6, 16, 19, 10, 10, 21, 12, 18, 15, 13, 8, 18, 15, 19, 15, 13, 7, 24, 7, 13, 19, 16, 12, 18, 8, 17, 15, 20, 9, 24, 9, 13, 22, 17, 13, 22

Offset: 1

Patrick De Geest, Nov 15 2000

nonn

Max Alekseyev, Table of n, a(n) for n = 1..690 (first 330 terms from Amiram Eldar)
S. S. Wagstaff, Jr., The Cunningham Project

bigomega(b^n-1): A046051 (b=2), A057958 (b=3), A057957 (b=4), A057956 (b=5), A057955 (b=6), A057954 (b=7), A057953 (b=8), this sequence (b=9), A057951 (b=10), A366682 (b=11), A366708 (b=12).

Cf. A001222, A002591, A024101, A074477, A085034, A133801, A274909, A366660, A366661, A366662.

PrimeOmega[Table[9^n - 1, {n, 1, 30}]] (* Amiram Eldar, Feb 02 2020 *)

Mobius transform of A085034. - T. D. Noe, Jun 19 2003
a(n) = A001222(A024101(n)) = A057958(2*n). - Amiram Eldar, Feb 02 2020
a(n) = A057941(n) + A057958(n). - Max Alekseyev, Jan 07 2024

A057956 Number of prime factors of 5^n - 1 (counted with multiplicity).

Original entry on oeis.org

2, 4, 3, 6, 4, 7, 3, 8, 5, 7, 3, 10, 3, 7, 7, 11, 4, 11, 5, 11, 6, 8, 4, 13, 8, 7, 9, 10, 5, 14, 4, 14, 6, 8, 9, 16, 5, 10, 6, 15, 4, 16, 4, 12, 12, 8, 3, 17, 4, 13, 8, 12, 5, 19, 10, 13, 7, 9, 4, 21, 5, 9, 11, 18, 8, 15, 7, 14, 9, 16, 4, 22, 5, 10, 16, 14, 7, 14, 5, 20, 11, 10, 5, 22, 9, 10

Offset: 1

Patrick De Geest, Nov 15 2000

nonn

Max Alekseyev, Table of n, a(n) for n = 1..502 (first 448 terms from Amiram Eldar)
S. S. Wagstaff, Jr., The Cunningham Project

bigomega(b^n-1): A057951 (b=10), A057952 (b=9), A057953 (b=8), A057954 (b=7), A057955 (b=6), this sequence (b=5), A057957 (b=4), A057958 (b=3), A046051 (b=2).

Cf. A001222, A024049, A074479, A085030.

PrimeOmega[5^Range[90]-1] (* Harvey P. Dale, Dec 16 2017 *)

Mobius transform of A085030. - T. D. Noe, Jun 19 2003

a(n) = A001222(A024049(n)). - Amiram Eldar, Feb 01 2020

A335885 The length of a shortest path from n to a power of 2, when applying the nondeterministic maps k -> k - k/p and k -> k + k/p, where p can be any of the odd prime factors of k, and the maps can be applied in any order.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 1, 0, 2, 1, 2, 1, 2, 1, 2, 0, 1, 2, 2, 1, 2, 2, 2, 1, 2, 2, 3, 1, 2, 2, 1, 0, 3, 1, 2, 2, 3, 2, 3, 1, 2, 2, 3, 2, 3, 2, 2, 1, 2, 2, 2, 2, 3, 3, 3, 1, 3, 2, 3, 2, 2, 1, 3, 0, 3, 3, 2, 1, 3, 2, 3, 2, 3, 3, 3, 2, 3, 3, 2, 1, 4, 2, 3, 2, 2, 3, 3, 2, 3, 3, 3, 2, 2, 2, 3, 1, 2, 2, 4, 2, 3, 2, 3, 2, 3

Offset: 1

Antti Karttunen, Jun 29 2020

nonn

The length of a shortest path from n to a power of 2, when using the transitions x -> A171462(x) and x -> A335876(x) in any order.

a((2^e)-1) is equal to A046051(e) = A001222((2^e)-1) when e is either a Mersenne exponent (in A000043), or some other number: 1, 4, 6, 8, 16, 32. For example, 32 is present because 2^32 - 1 = 4294967295 = 3*5*17*257*65537, a squarefree product of five known Fermat primes. - Antti Karttunen, Aug 11 2020

			A335876(67) = 68, and A171462(68) = 64 = 2^6, and this is the shortest path from 67 to a power of 2, thus a(67) = 2.
A171462(15749) = 15748, A335876(15748) = 15872, A335876(15872) = 16384 = 2^14, and this is the shortest path from 15749 to a power of 2, thus a(15749) = 3.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000043, A046051, A052126, A171462, A335876, A335875, A335881, A335884, A335904, A335907.

Cf. A000079, A335911, A335912 (positions of 0's, 1's and 2's in this sequence) and array A335910.

A335885(n) = { my(f=factor(n)); sum(k=1,#f~,if(2==f[k,1],0,f[k,2]*(1+min(A335885(f[k,1]-1),A335885(f[k,1]+1))))); };

\\ Or empirically as:
A171462(n) = if(1==n,0,(n-(n/vecmax(factor(n)[, 1]))));
A335876(n) = if(1==n,2,(n+(n/vecmax(factor(n)[, 1]))));
A209229(n) = (n && !bitand(n,n-1));
A335885(n) = if(A209229(n),0,my(xs=Set([n]),newxs,a,b,u); for(k=1,oo, newxs=Set([]); for(i=1,#xs,u = xs[i]; a = A171462(u); if(A209229(a), return(k)); b = A335876(u); if(A209229(b), return(k)); newxs = setunion([a],newxs); newxs = setunion([b],newxs)); xs = newxs));

Fully additive with a(2) = 0, and a(p) = 1+min(a(p-1), a(p+1)), for odd primes p.
For all n >= 1, a(n) <= A335875(n) <= A335881(n) <= A335884(n) <= A335904(n).
For all n >= 0, a(A000244(n)) = n, and these also seem to give records.

A085021 Number of prime factors of cyclotomic(n,2), which is A019320(n), the value of the n-th cyclotomic polynomial evaluated at x=2.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 1, 2, 1, 2, 1, 1, 2, 3, 1, 1, 1, 1, 1, 2, 2, 2, 1, 2, 1, 2, 1, 3, 2, 2, 1, 3, 2, 1, 2, 3, 3, 3, 2, 3, 1, 2, 2, 2, 2, 1, 1, 2, 2, 1, 2, 2, 3, 1, 2, 3, 2, 3, 2, 2, 3, 1, 1, 3, 1, 3, 2, 2, 2, 1, 1, 2, 2, 1, 1, 3, 4, 1, 2, 3, 2, 2, 1, 3, 4

Offset: 1

T. D. Noe, Jun 19 2003

nonn

The Mobius transform of this sequence yields A046051, the number of prime factors of Mersenne number 2^n-1.

The number of prime factors in the primitive part of 2^n-1. - T. D. Noe, Jul 19 2008

			a(11) = 2 because cyclotomic(11,2) = 2047, which has two factors: 23 and 89.

Max Alekseyev, Table of n, a(n) for n = 1..1206 (first 500 term from T. D. Noe)
Vjekoslav Kovač and Florian Luca, On the number of divisors of Mersenne numbers, arXiv:2506.04883 [math.NT], 2025. See Table 2 p. 9.
H. Mishima, Factorization of Cyclotomic Numbers
Eric Weisstein's World of Mathematics, Cyclotomic Polynomial
Eric Weisstein's World of Mathematics, Möbius Transform

omega(Phi(n,x)): this sequence (x=2), A085028 (x=3), A085029 (x=4), A085030 (x=5), A085031 (x=6), A085032 (x=7), A085033 (x=8), A085034 (x=9), A085035 (x=10).

Cf. A005420, A019320, A046051, A046801, A059499, A064078, A112927, A212953.

Join[{0}, Table[Plus@@Transpose[FactorInteger[Cyclotomic[n, 2]]][[2]], {n, 2, 100}]]

a(n) = #factor(polcyclo(n, 2))~; \\ Michel Marcus, Mar 06 2015

A086251 Number of primitive prime factors of 2^n - 1.

Original entry on oeis.org

0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 2, 3, 1, 1, 1, 1, 1, 2, 2, 2, 1, 2, 1, 2, 1, 3, 2, 2, 1, 3, 2, 1, 2, 3, 3, 3, 1, 3, 1, 2, 2, 2, 2, 1, 1, 2, 2, 1, 2, 2, 3, 1, 2, 3, 2, 3, 2, 2, 3, 1, 1, 3, 1, 3, 2, 2, 2, 1, 1, 2, 2, 1, 1, 3, 4, 1, 2, 3, 2, 2, 1, 3, 3, 2, 3, 2, 2, 3

Offset: 1

T. D. Noe, Jul 14 2003

A prime factor of 2^n - 1 is called primitive if it does not divide 2^r - 1 for any r < n. Equivalently, p is a primitive prime factor of 2^n - 1 if ord(2,p) = n. Zsigmondy's theorem says that there is at least one primitive prime factor for n > 1, except for n=6. See A086252 for those n that have a record number of primitive prime factors.
Number of odd primes p such that A002326((p-1)/2) = n. Number of occurrences of number n in A014664. - Thomas Ordowski, Sep 12 2017
The prime factors are not counted with multiplicity, which matters for a(364)=4 and a(1755)=6. - Jeppe Stig Nielsen, Sep 01 2020

			a(11) = 2 because 2^11 - 1 = 23*89 and both 23 and 89 have order 11.

Table of n, a(n) for n = 1..1206
Eric Weisstein's World of Mathematics, Zsigmondy Theorem

Cf. A046800, A046051 (number of prime factors, with repetition, of 2^n-1), A086252, A002588, A005420, A002184, A046801, A049093, A049094, A059499, A085021, A097406, A112927, A237043.

Join[{0}, Table[cnt=0; f=Transpose[FactorInteger[2^n-1]][[1]]; Do[If[MultiplicativeOrder[2, f[[i]]]==n, cnt++ ], {i, Length[f]}]; cnt, {n, 2, 200}]]

a(n) = sumdiv(n, d, moebius(n/d)*omega(2^d-1)); \\ Michel Marcus, Sep 12 2017

a(n) = my(m=polcyclo(n, 2)); omega(m/gcd(m,n)) \\ Jeppe Stig Nielsen, Sep 01 2020

a(n) = Sum{d|n} mu(n/d) A046800(d), inverse Mobius transform of A046800.
a(n) <= A182590(n). - Thomas Ordowski, Sep 14 2017
a(n) = A001221(A064078(n)). - Thomas Ordowski, Oct 26 2017

Terms to a(500) in b-file from T. D. Noe, Nov 11 2010
Terms a(501)-a(1200) in b-file from Charles R Greathouse IV, Sep 14 2017
Terms a(1201)-a(1206) in b-file from Max Alekseyev, Sep 11 2022

A173898 Decimal expansion of sum of the reciprocals of the Mersenne primes.

Original entry on oeis.org

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

Offset: 0

Jonathan Vos Post, Mar 01 2010

We know this a priori to be strictly less than the Erdős-Borwein constant (A065442), which Erdős (1948) showed to be irrational. This new constant would also seem to be irrational.

			Decimal expansion of (1/3) + (1/7) + (1/31) + (1/127) + (1/8191) + (1/131071) + (1/524287) + ... = .5164541789407885653304873429715228588159685534154197.
This has continued fraction expansion 0 + 1/(1 + 1/(1 + 1/(14 + 1/(1 + ...)))) (see A209601).

Peter B. Borwein, On the Irrationality of Certain Series, Math. Proc. Cambridge Philos. Soc. 112, 141-146, 1992.
Paul Erdős, On Arithmetical Properties of Lambert Series, J. Indian Math. Soc. 12, 63-66, 1948.
Yoshihiro Tanaka, On the Sum of Reciprocals of Mersenne Primes, American Journal of Computational Mathematics, Vol. 7, No. 2 (2017), pp. 145-148.
Eric Weisstein's World of Mathematics, Erdos-Borwein Constant.
Marek Wolf, Computer experiments with Mersenne primes, arXiv preprint arXiv:1112.2412 [math.NT], 2011.

Cf. A209601, A000668, A065442 (decimal expansion of Erdos-Borwein constant), A000043, A001348, A046051, A057951-A057958, A034876, A124477, A135659, A019279, A061652, A000225.

Digits := 120 ; L := [ 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377, 6972593, 13466917 ] ;
x := 0 ; for i from 1 to 30 do x := x+1.0/(2^op(i,L)-1 ); end do ;

RealDigits[Sum[1/(2^p - 1), {p, MersennePrimeExponent[Range[14]]}], 10, 100][[1]] (* Amiram Eldar, May 24 2020 *)

isM(p)=my(m=Mod(4,2^p-1));for(i=1,p-2,m=m^2-2);!m
s=1/3;forprime(p=3,default(realprecision)*log(10)\log(2), if(isM(p), s+=1./(2^p-1)));s \\ Charles R Greathouse IV, Mar 22 2012

Sum_{i>=1} 1/A000668(i).

Entry revised by N. J. A. Sloane, Mar 10 2012

A193330 Number of prime factors of n^2 + 1, counted with multiplicity.

Original entry on oeis.org

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

Offset: 0

Vladimir Joseph Stephan Orlovsky, Jul 22 2011

nonn

a(n) is also the number of terms, counted with multiplicity, in a prime factorization of n + i in the ring of Gaussian integers. - Jason Kimberley, Dec 17 2011

Charles R Greathouse IV, Table of n, a(n) for n = 0..10000

Cf. A046051, A054992, A193295.

A193330[n_Integer] := PrimeOmega[n^2 + 1];A193330 /@ Range[0, 100]
PrimeOmega/@(Range[0,90]^2+1) (* Harvey P. Dale, Apr 03 2019 *)

a(n)=bigomega(n^2+1) \\ Charles R Greathouse IV, Jul 23 2011

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

Original entry on oeis.org

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)).

cp's OEIS Frontend

A057953 Number of prime factors of 8^n - 1 (counted with multiplicity).

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A057955 Number of prime factors of 6^n - 1 (counted with multiplicity).

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A057952 Number of prime factors of 9^n - 1 (counted with multiplicity).

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A057956 Number of prime factors of 5^n - 1 (counted with multiplicity).

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A335885 The length of a shortest path from n to a power of 2, when applying the nondeterministic maps k -> k - k/p and k -> k + k/p, where p can be any of the odd prime factors of k, and the maps can be applied in any order.

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A085021 Number of prime factors of cyclotomic(n,2), which is A019320(n), the value of the n-th cyclotomic polynomial evaluated at x=2.

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A086251 Number of primitive prime factors of 2^n - 1.

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A173898 Decimal expansion of sum of the reciprocals of the Mersenne primes.

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