A133653 -id:A133653 - cp's OEIS frontend

A003261 Woodall (or Riesel) numbers: n*2^n - 1.

1, 7, 23, 63, 159, 383, 895, 2047, 4607, 10239, 22527, 49151, 106495, 229375, 491519, 1048575, 2228223, 4718591, 9961471, 20971519, 44040191, 92274687, 192937983, 402653183, 838860799, 1744830463, 3623878655, 7516192767, 15569256447, 32212254719, 66571993087

Offset: 1

N. J. A. Sloane

For n>1, a(n) is base at which zero is reached for the function "write f(j) in base j, read as base j+1 and then subtract 1 to give f(j+1)" starting from f(n) = n^2 - 1. - Henry Bottomley, Aug 06 2000

Sequence corresponds also to the maximum chain length of the classic puzzle whereby, under agreed commercial terms, an asset of unringed golden chain, when judiciously fragmented into as few as n pieces and n-1 opened links (through n-1 cuts), might be used to settle debt sequentially, with a golden link covering for unit cost. Here beside the n-1 opened links, the n fragmented pieces have lengths n, 2*n, 4*n, ..., 2^(n-1)*n. For instance, the chain of original length a(5)=159, if segregated by 4 cuts into 5+1+10+1+20+1+40+1+80, may be used to pay sequentially, i.e., a link-cost at a time, for an equivalent cost up to 159 links, to the same creditor. - Lekraj Beedassy, Feb 06 2003

			G.f. = x + 7*x^2 + 23*x^3 + 63*x^4 + 159*x^5 + 383*x^6 + 895*x^7 + ... - _Michael Somos_, Nov 04 2018

A. Brousseau, Number Theory Tables. Fibonacci Association, San Jose, CA, 1973, p. 159.
K. R. Bhutani and A. B. Levin, "The Problem of Sawing a Chain", Journal of Recreational Mathematics 2002-3 31(1) 32-35.
G. Everest, A. van der Poorten, I. Shparlinski and T. Ward, Recurrence Sequences, Amer. Math. Soc., 2003; see esp. p. 255.
M. Gardner, Martin Gardner's Sixth Book of Mathematical Diversions from Scientific American, "Gold Links", Problem 4, pp. 50-51; 57-58, University of Chicago Press, 1983.
O. O'Shea, Mathematical Brainteasers with Surprising Solutions, Problem 76, pp. 183-185, Prometheus Books, Guilford, Connecticut, 2020.
Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 241.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vladimir Pletser, Table of n, a(n) for n = 1..3000 (terms 1..300 from T. D. Noe).
Ray Ballinger, Woodall Primes: Definition and Status.
Attila Bérczes, István Pink, and Paul Thomas Young, Cullen numbers and Woodall numbers in generalized Fibonacci sequences, J. Num. Theor. (2024) Vol. 262, 86-102.
Alfred Brousseau, Fibonacci and Related Number Theoretic Tables, Fibonacci Association, San Jose, CA, 1972. See p. 159.
C. K. Caldwell, Woodall Numbers.
Orhan Eren and Yüksel Soykan, Gaussian Generalized Woodall Numbers, Arch. Current Res. Int'l (2023) Vol. 23, Iss. 8, Art. No. ACRI.108618, 48-68. See p. 50.
Orhan Eren and Yüksel Soykan, On Dual Hyperbolic Generalized Woodall Numbers, Arch. Current Res. Int'l (2024) Vol. 24, Iss. 11, Art. No. ACRI.126420, 398-423. See p. 401.
Ignas Gasparavičius, Andrius Grigutis, and Juozas Petkelis, Picturesque convolution-like recurrences and partial sums' generation, arXiv:2507.23619 [math.NT], 2025. See p. 27.
Paul Leyland, Factors of Cullen and Woodall numbers.
Paul Leyland, Generalized Cullen and Woodall numbers.
D. Marques, On Generalized Cullen and Woodall Numbers That are Also Fibonacci Numbers, Journal of Integer Sequences, 17 (2014), #14.9.4.
Hisanori Mishima, Factorizations of many number sequences: Riesel numbers, n=1..100, n=101..200, n=201..300, n=301..323.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
T. Sillke, Using Chains Links To Pay For A Room.
Amelia Carolina Sparavigna, On the generalized sums of Mersenne, Fermat, Cullen and Woodall Numbers, Politecnico di Torino (Italy, 2019).
Amelia Carolina Sparavigna, Composition Operations of Generalized Entropies Applied to the Study of Numbers, International Journal of Sciences (2019) Vol. 8, No. 4, 87-92.
Amelia Carolina Sparavigna, The groupoids of Mersenne, Fermat, Cullen, Woodall and other Numbers and their representations by means of integer sequences, Politecnico di Torino, Italy (2019), [math.NT].
Amelia Carolina Sparavigna, Some Groupoids and their Representations by Means of Integer Sequences, International Journal of Sciences (2019) Vol. 8, No. 10.
Eric Weisstein's World of Mathematics, Woodall Number.
Wikipedia, Woodall number.
Index entries for linear recurrences with constant coefficients, signature (5,-8,4).

Cf. A002234, A002064, A005849, A050918, A006127.
a(n) = A036289(n) - 1 = A002064(n) - 2.
Cf. A133653.

a003261 = (subtract 1) . a036289  -- Reinhard Zumkeller, Mar 05 2012

[n*2^n - 1: n in [1..30]]; // G. C. Greubel, Nov 04 2018

for n from 1 to 3000 do n, n*2^n -1; end do; # Vladimir Pletser, Dec 30 2022

Table[n*2^n-1,{n,3*4!}] (* Vladimir Joseph Stephan Orlovsky, Apr 25 2010 *)
LinearRecurrence[{5,-8,4},{1,7,23},30] (* Harvey P. Dale, Mar 13 2022 *)

A003261(n)=n*2^n-1  \\ M. F. Hasler, Oct 31 2012

[n*2**n - 1 for n in range(1, 29)] # Michael S. Branicky, Jan 07 2021

G.f.: x*(-1-2*x+4*x^2) / ( (x-1)*(-1+2*x)^2 ). - Simon Plouffe in his 1992 dissertation
Binomial transform of A133653 and double binomial transform of [1, 5, -1, 1, -1, 1, ...]. - Gary W. Adamson, Sep 19 2007
a(n) = -(2)^n * A006127(-n) for all n in Z. - Michael Somos, Nov 04 2018
E.g.f.: 1 + exp(x)*(2*exp(x)*x - 1). - Stefano Spezia, Nov 24 2024

A175553 Product of first k triangular numbers divided by the sum of first k triangular numbers is an integer.

Original entry on oeis.org

1, 4, 7, 8, 10, 12, 13, 14, 16, 18, 19, 20, 22, 23, 24, 25, 26, 28, 30, 31, 32, 33, 34, 36, 37, 38, 40, 42, 43, 44, 46, 47, 48, 49, 50, 52, 53, 54, 55, 56, 58, 60, 61, 62, 63, 64, 66, 67, 68, 70, 72, 73, 74, 75, 76, 78, 79, 80, 82, 83, 84, 85, 86, 88, 89, 90, 91, 92, 93, 94, 96

Offset: 1

Ctibor O. Zizka, Jun 26 2010

nonn

Numbers k such that (1*3*6*10* ... *(k*(k+1)/2)) / (1+3+6+10+ ... +(k*(k+1)/2)) is an integer. What if, instead of triangular numbers, we use squares, 1*4*...*(k*k) / (1+4+...+k*k); odd numbers, 1*3*...*(2*k-1) / (1+3+...+(2*k-1)); or Fibonacci numbers, F(1)* ... *F(k) / (F(1)+ ... + F(k))?
It appears that the corresponding sequence for the Fibonacci numbers is given in A133653. - John W. Layman, Jul 10 2010
k > 6 is in this sequence if and only if k+2 is composite. - Robert Israel, Nov 04 2021

			For k=4 we have 1*3*6*10 /(1+3+6+10) = 9 so k=4 belongs to the sequence.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000292, A006472.

Cf. A133653. - John W. Layman, Jul 10 2010

A006472 := proc(n) n!*(n-1)!/2^(n-1) ; end proc:
A000292 := proc(n) binomial(n+2,3) ; end proc:
for n from 1 to 200 do a := A006472(n+1)/A000292(n) ; if type(a,'integer') then printf("%d,",n) ; end if; end do: # R. J. Mathar, Jun 28 2010

fQ[n_] := Mod[6n!(n - 1)!, (n + 2)2^n ] == 0; Select[Range@ 96, fQ@# &] (* Robert G. Wilson v, Jun 29 2010 *)

{k: A006472(k+1)/A000292(k) in Z}. - R. J. Mathar, Jun 28 2010

More terms from R. J. Mathar and Robert G. Wilson v, Jun 28 2010

A239977 a(n) = -Sum_{k=0..n} binomial(n, k)*A226158(k).

Original entry on oeis.org

0, 1, 3, 6, 9, 10, 9, 14, 33, 18, -135, 22, 2097, 26, -38199, 30, 929601, 34, -28820583, 38, 1109652945, 42, -51943281687, 46, 2905151042529, 50, -191329672483911, 54, 14655626154768753, 58, -1291885088448017655, 62, 129848163681107302017, 66

Offset: 0

Paul Curtz, Mar 30 2014

Let T(n, k) denote the difference table of a(n).
(-1)^(k+1)*T(3, 3+k) = T(k+3, 3) for k >= 0.
Without the first two rows and the first two columns we have the core of the Genocchi numbers, like A240581(n)/A239315(n) for the Bernoulli numbers. See A226158(n).
   0,   1,   3,   6,   9,  10, ...
   1,   2,   3,   3,   1,  -1, ...
   1,   1,   0,  -2,  -2,   6, ...
   0,  -1,  -2,   0,   8,   8, ...
  -1,  -1,   2,   8,   0, -56, ...
   0,   3,   6,  -8, -56,   0, ...
The definition reflects the identity 2*((1-2^n)*B(n,1) + n) =
2*Sum_{k=0..n} C(n,k)*(2^k-1)*B(k,1) where B(n,x) denotes the Bernoulli polynomials. - Peter Luschny, Apr 16 2014

B. Sury, The value of Bernoulli Polynomials at rational numbers, Bull. London Math. Soc. 25 (1993), 327-29.

Cf. A083007.

[0,1] cat [2*((1 - 2^n)*Bernoulli(n) + n): n in [2..40]]; // Vincenzo Librandi, Mar 03 2015

A239977 := n -> 2*((1-2^n)*bernoulli(n,1) + n):
seq(A239977(n), n=0..33); # Peter Luschny, Mar 08 2015

a[n_] := (EulerE[n-1, 0]+2)*n; a[0] = 0; a[1] = 1; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Apr 02 2014 *)

a(n) = 2*n + A226158(n).
a(2n) is divisible by 3.
a(2n+1) = A133653(n).
a(n) = 2*((1 - 2^n)*B(n, 1) + n), B(n, x) the Bernoulli polynomial. - Peter Luschny, Apr 16 2014
From Peter Bala, Mar 02 2015: (Start)
a(n) = (-2)^n * ( B(n,-1/2) - B(n,0) ), where B(n,x) denotes the n-th Bernoulli polynomial. More generally, for any nonzero integer k, k^n*( B(n,1/k) - B(n,0) ) is an integer for n >= 0. Cf. A083007.
a(0) = 0 and for n >= 1, a(n) = 1 - 1/(n + 1)*Sum_{k = 1..n-1} (-2)^(n-k)*binomial(n+1,k)*a(k).
E.g.f.: 2*x*exp(2*x)/(1 + exp(x)) = x + 3*x^2/2! + 6*x^3/3! + .... (End)
a(n) = Sum_{k=0..n-1} 2^k*binomial(n, k)*Bernoulli(k, 1). - Peter Luschny, Aug 17 2021

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A003261 Woodall (or Riesel) numbers: n*2^n - 1.

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A175553 Product of first k triangular numbers divided by the sum of first k triangular numbers is an integer.

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A239977 a(n) = -Sum_{k=0..n} binomial(n, k)*A226158(k).

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