A159353 -id:A159353 - cp's OEIS frontend

A330718 a(n) = numerator(Sum_{k=1..n} (2^k-2)/k).

0, 1, 3, 13, 25, 137, 245, 871, 517, 4629, 8349, 45517, 83317, 1074679, 1992127, 7424789, 13901189, 78403447, 147940327, 280060651, 531718651, 11133725681, 21243819521, 40621501691, 15565330735, 388375065019, 248882304985, 479199924517, 923951191477, 2973006070891

Offset: 1

Amiram Eldar and Thomas Ordowski, Dec 28 2019

If p > 3 is prime, then p^2 | a(p).
Note the similarity to Wolstenholme's theorem.
Conjecture: for n > 3, if n^2 | a(n), then n is prime.
Are there the weak pseudoprimes m such that m | a(m)?
Primes p such that p^3 | a(p) are probably A088164.
If p is an odd prime, then a(p+1) == A330719(p+1) (mod p).
If p > 3 is a prime, then p^2 | numerator(Sum_{k=1..p+1} F(k)), where F(n) = Sum_{k=1..n} (2^(k-1)-1)/k. Cf. A027612 (a weaker divisibility).

			Numerators of 0, 1, 3, 13/2, 25/2, 137/6, 245/6, ...

G. C. Greubel, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Wolstenholme's Theorem

Cf. A000265, A001008, A007663, A088164, A159353, A225101, A279683, A290347, A330719.

[Numerator( &+[(2^k -2)/k: k in [1..n]] ): n in [1..30]]; // G. C. Greubel, Dec 28 2019

seq(numer(add((2^k -2)/k, k = 1..n)), n = 1..30); # G. C. Greubel, Dec 28 2019

Numerator @ Accumulate @ Array[(2^# - 2)/# &, 30]
Table[Numerator[Simplify[-(2^(n+1)*LerchPhi[2,1,n+1] +Pi*I +2*HarmonicNumber[n])]], {n,30}] (* G. C. Greubel, Dec 28 2019 *)

a(n) = numerator(sum(k=1, n, (2^k-2)/k)); \\ Michel Marcus, Dec 28 2019

[numerator( sum((2^k -2)/k for k in (1..n)) ) for n in (1..30)] # G. C. Greubel, Dec 28 2019

a(n) = numerator(Sum_{k=1..n} (2^(k-1)-1)/k).
a(n+1) = numerator(a(n)/A330719(n) + A225101(n+1)/(2*A159353(n+1))).
a(p) = a(p-1) + A007663(n)*A330719(p-1) for p = prime(n) > 2.
a(n) = numerator(-(2^(n+1)*LerchPhi(2,1,n+1) + Pi*i + 2*HarmonicNumber(n))). - G. C. Greubel, Dec 28 2019
a(n) = numerator(A279683(n)/n!) for n > 0. - Amiram Eldar and Thomas Ordowski, Jan 15 2020
For n > 1, a(n) = A000265(A290347(n)). - Thomas Ordowski, Mar 29 2025

A225101 Numerator of (2^n - 2)/n.

Original entry on oeis.org

0, 1, 2, 7, 6, 31, 18, 127, 170, 511, 186, 2047, 630, 8191, 10922, 32767, 7710, 131071, 27594, 524287, 699050, 2097151, 364722, 8388607, 6710886, 33554431, 44739242, 19173961, 18512790, 536870911, 69273666, 2147483647, 2863311530, 8589934591, 34359738366, 34359738367, 3714566310

Offset: 1

Alonso del Arte, Apr 28 2013

That (2^n - 2)/n is an integer when n is prime can easily be proved as a simple consequence of Fermat's little theorem.

It was believed long ago that (2^n - 2)/n is an integer only when n = 1 or a prime. In 1819, Frédéric Sarrus found the smallest counterexample, 341; these pseudoprimes are now sometimes called "Sarrus numbers" (A001567).

			a(4) = 7 because (2^4 - 2)/4 = 7/2.
a(5) = 6 because (2^5 - 2)/5 = 6.
a(6) = 31 because (2^6 - 2)/6 = 31/3.

Alkiviadis G. Akritas, Elements of Computer Algebra With Application. New York: John Wiley & Sons (1989): 66.
George P. Loweke, The Lore of Prime Numbers. New York: Vantage Press, 1982, p. 22.

Colin Barker, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Chinese Hypothesis

Cf. A001567, A064535, A159353 (denominators).

[Numerator((2^n - 2)/n): n in  [1..60]]; // Vincenzo Librandi, Nov 09 2014

A225101:=n->numer((2^n-2)/n): seq(A225101(n), n=1..50); # Wesley Ivan Hurt, Nov 10 2014

Mathematica
```
Table[Numerator[(2^n - 2)/n], {n, 50}]
```

vector(100, n, numerator((2^n - 2)/n)) \\ Colin Barker, Nov 09 2014

A146077 Numbers n such that gcd(n, 2^n-2) does not equal the least prime factor of n.

Original entry on oeis.org

28, 35, 45, 55, 70, 77, 95, 105, 112, 115, 119, 143, 153, 154, 155, 161, 165, 175, 186, 187, 196, 203, 209, 215, 221, 225, 231, 235, 238, 247, 253, 275, 276, 280, 285, 287, 295, 299, 319, 322, 323, 325, 329, 335, 341, 345, 355, 364, 371, 377, 385, 391, 395

Offset: 1

Ray Chandler, Apr 11 2009

nonn

Also, positive integers n where A159353(n) differs from A032742(n).

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A159353, A032742.

Select[Range[400],GCD[#,2^#-2]!=FactorInteger[#][[1,1]]&] (* Harvey P. Dale, Dec 14 2015 *)

Extended by Ray Chandler, Apr 11 2009

A091669 a(n) = (2^(n-1)/n!) * Product_{k=1..n-1} (2^k-1).

Original entry on oeis.org

1, 1, 2, 7, 42, 434, 7812, 248031, 14055090, 1436430198, 267176016828, 91151551074486, 57425477176926180, 67196011936600334340, 146782968474309770332296, 601204690999713530559792879

Offset: 1

Karol A. Penson, Jan 27 2004

nonn

Primes p such that 2^p-2 divides a(p) are A216838. - Amiram Eldar and Thomas Ordowski, Jan 16 2020
For odd n > 1, if a(n-1) divides a(n) and n does not divide a(n), then n is a prime (for which 2 is a primitive root, A001122). Composite numbers m such that a(m-1) divides a(m) are the pseudoprimes A001567 and A006935. Numbers n > 1 such that a(m) divides a(n) for all m < n are primes 2, 3, 5, 7, and 13. These are the primes p for which gpf(2^p-2) = p. - Thomas Ordowski, Jan 17 2020
If p is a prime with primitive root 2, A001122, then p | a(p-1) + 2^(p-2). Conjecture: (for n > 2), if n | a(n-1) + 2^(n-2), then n is a prime (A001122). Note that if p is an odd prime for which 2 is not a primitive root, A216838, then p | a(p-1). - Amiram Eldar and Thomas Ordowski, Jan 19 2020

Amiram Eldar, Table of n, a(n) for n = 1..86

Cf. A000142, A001122, A001567, A005329, A006935, A159353, A216838, A225101.

[1] cat [2^(n-1)/Factorial(n)*&*[(2^k-1):k in [1..n-1]]:n in [2..16]]; // Marius A. Burtea, Jan 16 2020

seq( (2^(n-1)/n!)*mul(2^j-1, j=1..n-1), n=1..20); # G. C. Greubel, Feb 05 2020

Table[QFactorial[n-1, 2] 2^(n-1)/n!, {n, 20}]

a(n) = (2^(n-1)/n!) * prod(k=1, n-1, 2^k-1); \\ Michel Marcus, Jan 16 2020

from sage.combinat.q_analogues import q_factorial
[2^(n-1)*q_factorial(n-1, 2)/factorial(n) for n in (1..20)] # G. C. Greubel, Feb 05 2020

a(n) = 2^(n-1)*A005329(n-1)/n!.

a(n) = Product_{k=2..n} (2^k-2)/k = Product_{k=2..n} A225101(k)/A159353(k). - Thomas Ordowski, Jan 16 2020

Corrected and edited by Thomas Ordowski, Jan 16 2020

cp's OEIS Frontend

A330718 a(n) = numerator(Sum_{k=1..n} (2^k-2)/k).

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A225101 Numerator of (2^n - 2)/n.

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A146077 Numbers n such that gcd(n, 2^n-2) does not equal the least prime factor of n.

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A091669 a(n) = (2^(n-1)/n!) * Product_{k=1..n-1} (2^k-1).

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