A001676 -id:A001676 - cp's OEIS frontend

A228689 a(n) = A001676(4*n + 3).

1, 28, 992, 16256, 523264, 69524373504, 67100672, 7767211311104, 3014494287036416, 2303837503821447168, 341653284209033216, 798366828940770681028608, 11852230872517975212032, 91678339751618435453288448, 1986677733776616536315084668928, 284423744326342962334231917756416

Offset: 0

N. J. A. Sloane, Aug 30 2013

nonn

V. P. Snaith, A history of the Arf-Kervaire invariant problem, Notices Amer. Math. Soc., 60 (No. 8, 2013), 1040-1047.

Cf. A001676, A228690, A228691, A228692, A189995.

Term a(15) corrected by Andrey Zabolotskiy, Nov 27 2022

A228690 a(n) = A001676(4*n).

Original entry on oeis.org

1, 2, 1, 2, 24, 2, 2, 8, 6, 192, 8, 32, 24, 1, 4, 128, 24, 384, 80, 256

Offset: 1

N. J. A. Sloane, Aug 30 2013

Cf. A001676, A228689, A228691, A228692.

Term a(14) corrected, a(16)-a(20) added by Andrey Zabolotskiy, Nov 27 2022

A228691 a(n) = A001676(4*n + 1).

Original entry on oeis.org

1, 1, 8, 3, 16, 8, 4, 3, 32, 24, 32, 11520, 12, 32, 8, 1, 6144, 32, 128, 256, 9216

Offset: 0

N. J. A. Sloane, Aug 30 2013

Cf. A001676, A228689, A228690, A228692.

Term a(14) corrected, a(16)-a(20) added by Andrey Zabolotskiy, Nov 27 2022

A228692 a(n) = A001676(4n+2).

Original entry on oeis.org

1, 1, 6, 2, 16, 4, 12, 3, 32, 120, 96, 48, 24, 8, 4, 24, 512, 512, 384, 384, 10752

Offset: 0

N. J. A. Sloane, Aug 30 2013

Cf. A001676, A228689, A228690, A228691.

More terms from the b-file at A001676 added by Amiram Eldar, May 03 2024

A178579 Partial sums of A001676.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 34, 36, 44, 50, 1042, 1043, 1046, 1048, 17304, 17306, 17322, 17338, 540602, 540626, 540634, 540638, 69524914142, 69524914144, 69524914148, 69524914160, 69592014832, 69592014834, 69592014837, 69592014840, 7836803325944, 7836803325952, 7836803325984

Offset: 1

Jonathan Vos Post, Dec 24 2010

Amiram Eldar, Table of n, a(n) for n = 1..83 (calculated from the b-file at A001676)

Cf. A001676.

a(n) = Sum_{i=1..n} A001676(i).

More terms calculated from the b-file at A001676 and added by Amiram Eldar, May 03 2024

A187717 a(n) = A001676(n) / A187595(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 4, 6, 1, 1, 3, 2, 2, 2, 8, 16, 2, 24, 4, 4, 48, 2, 2, 12, 1, 2, 3, 3, 4, 8, 16, 32, 4, 6, 12, 120, 96, 192, 16, 96, 1, 8, 5760, 48, 96, 32, 6, 24, 16, 24, 16, 8, 3, 1, 4, 4, 4, 4, 1, 24, 16

Offset: 1

Paul Muljadi, Mar 18 2011

The quotient of |Theta_n| and |bP_(n+1)| in the 1961 Kervaire-Milnor theorem.

The quotient of |S_n| and |S_n^{bp}| in the 2011 Milnor survey (see Theorem 5, Further Details, and Table 2). - Jonathan Sondow, Jun 16 2011

Michel A. Kervaire and John W. Milnor, Groups of homotopy spheres: I, 1963
Michel A. Kervaire and John W. Milnor, Groups of homotopy spheres: I. Ann. of Math. (2) 77 1963, 504-537.
John W. Milnor, Differential Topology Forty-six Years Later, Notices Amer. Math. Soc. 58 (2011), 804-809.
John W. Milnor, Spheres, Abel Prize lecture (video), 2011.
Wikipedia, Exotic sphere

More terms from Jonathan Sondow, Jun 16 2011

Terms a(56), a(57), a(63) corrected by Andrey Zabolotskiy, Nov 27 2022

A358706 Erroneous version of A001676.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 28, 2, 8, 6, 992, 1, 3, 2, 16256, 2, 16, 16, 523264, 24, 8, 4, 69524373504, 2, 4, 12, 67100672, 2, 3, 3, 7767211311104, 8, 32, 32, 3014494287036416, 6, 24, 120, 2303837503821447168, 192, 32, 96, 341653284209033216, 8, 11520, 48, 798366828940770681028608, 32, 12, 24, 11852230872517975212032, 24, 32, 8, 91678339751618435453288448, 2, 16, 4, 1986677733776616536315084668928, 4, 1, 24, 142211872163171481167115958878208

Offset: 1

dead

Included in accordance with OEIS policy of including published but erroneous sequences to serve as pointers to the correct versions.

A083420 a(n) = 2*4^n - 1.

Original entry on oeis.org

1, 7, 31, 127, 511, 2047, 8191, 32767, 131071, 524287, 2097151, 8388607, 33554431, 134217727, 536870911, 2147483647, 8589934591, 34359738367, 137438953471, 549755813887, 2199023255551, 8796093022207, 35184372088831, 140737488355327, 562949953421311

Offset: 0

Paul Barry, Apr 29 2003

Sum of divisors of 4^n. - Paul Barry, Oct 13 2005
Subsequence of A000069; A132680(a(n)) = A005408(n). - Reinhard Zumkeller, Aug 26 2007
If x = a(n), y = A000079(n+1) and z = A087289(n), then x^2 + 2*y^2 = z^2. - Vincenzo Librandi, Jun 09 2014
It seems that a(n) divides A001676(3+4n). Several other entries apparently have this sequence embedded in them, e.g., A014551, A168604, A213243, A213246-8, and A279872. - Tom Copeland, Dec 27 2016
To elaborate on Librandi's comment from 2014: all these numbers, even if prime in Z, are sure not to be prime in Z[sqrt(2)], since a(n) can at least be factored as ((2^(2n + 1) - 1) - (2^(2n) - 1)*sqrt(2))((2^(2n + 1) - 1) + (2^(2n) - 1)*sqrt(2)). For example, 7 = (3 - sqrt(2))(3 + sqrt(2)), 31 = (7 - 3*sqrt(2))(7 + 3*sqrt(2)), 127 = (15 - 7*sqrt(2))(15 + 7*sqrt(2)). - Alonso del Arte, Oct 17 2017
Largest odd factors of A147590. - César Aguilera, Jan 07 2020

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Roudy El Haddad, Recurrent Sums and Partition Identities, arXiv:2101.09089 [math.NT], 2021.
Roudy El Haddad, A generalization of multiple zeta value. Part 1: Recurrent sums. Notes on Number Theory and Discrete Mathematics, 28(2), 2022, 167-199, DOI: 10.7546/nntdm.2022.28.2.167-199.
A. J. Macfarlane, Generating functions for integer sequences defined by the evolution of cellular automata with even rule numbers, Fig 11.
Robert Schneider, Partition zeta functions, Research in Number Theory, 2(1):9, 2016.
Eric Weisstein's World of Mathematics, Rule 220
Index entries for linear recurrences with constant coefficients, signature (5,-4).

Cf. A083421, A000668 (primes in this sequence), A004171, A000244.
Cf. A001676, A014551, A168604, A213243, A213246, A213247, A213248, A279872.
Cf. A000302.

a083420 = subtract 1 . (* 2) . (4 ^)  -- Reinhard Zumkeller, Dec 22 2015

[2*4^n-1 : n in [0..30]]; // Wesley Ivan Hurt, Mar 14 2015

seq(2*4^n-1, n = 0..22); # Peter Luschny, Aug 17 2011

2 * 4^Range[0, 31] - 1 (* Alonso del Arte, Oct 17 2017 *)

a(n)=2*4^n-1 \\ Charles R Greathouse IV, Sep 24 2015

def A083420(n): return (1<<(n<<1|1))-1 # Chai Wah Wu, Mar 10 2025

G.f.: (1+2*x)/((1-x)*(1-4*x)).
E.g.f.: 2*exp(4*x)-exp(x).
With a leading zero, this is a(n) = (4^n - 2 + 0^n)/2, the binomial transform of A080925. - Paul Barry, May 19 2003
From Benoit Cloitre, Jun 18 2004: (Start)
a(n) = (-16^n/2)*B(2n, 1/4)/B(2n) where B(n, x) is the n-th Bernoulli polynomial and B(k) = B(k, 0) is the k-th Bernoulli number.
a(n) = 5*a(n-1) - 4*a(n-2).
a(n) = (-4^n/2)*B(2*n, 1/2)/B(2*n). (End)
a(n) = A099393(n) + A020522(n) = A000302(n) + A024036(n). - Reinhard Zumkeller, Feb 07 2006
a(n) = Stirling2(2*(n+1), 2). - Zerinvary Lajos, Dec 06 2006
a(n) = 4*a(n-1) + 3 with n > 0, a(0) = 1. - Vincenzo Librandi, Dec 30 2010
a(n) = A001576(n+1) - 2*A001576(n). - Brad Clardy, Mar 26 2011
a(n) = 6*A002450(n) + 1. - Roderick MacPhee, Jul 06 2012
a(n) = A000203(A000302(n)). - Michel Marcus, Jan 20 2014
a(n) = Sum_{i = 0..n} binomial(2n+2, 2i). - Wesley Ivan Hurt, Mar 14 2015
a(n) = (1/4^n) * Sum_{k = 0..n} binomial(2*n+1,2*k)*9^k. - Peter Bala, Feb 06 2019
a(n) = A147590(n)/A000079(n). - César Aguilera, Jan 07 2020
a(n) = numerator(zeta_star({2}(n + 1))/zeta(2*n + 2)) where zeta_star is the multiple zeta star values and ({2}_n) represents (2, ..., 2) where the multiplicity of 2 is n. - _Roudy El Haddad, Feb 22 2022

A053381 Maximal number of linearly independent smooth nowhere-zero vector fields on a (2n+1)-sphere.

Original entry on oeis.org

1, 3, 1, 7, 1, 3, 1, 8, 1, 3, 1, 7, 1, 3, 1, 9, 1, 3, 1, 7, 1, 3, 1, 8, 1, 3, 1, 7, 1, 3, 1, 11, 1, 3, 1, 7, 1, 3, 1, 8, 1, 3, 1, 7, 1, 3, 1, 9, 1, 3, 1, 7, 1, 3, 1, 8, 1, 3, 1, 7, 1, 3, 1, 15, 1, 3, 1, 7, 1, 3, 1, 8, 1, 3, 1, 7, 1, 3, 1, 9, 1, 3, 1, 7, 1, 3, 1, 8, 1, 3, 1, 7, 1, 3, 1, 11, 1, 3, 1, 7, 1, 3

Offset: 0

Warren D. Smith, Jan 06 2000

The corresponding terms for a 2n-sphere are all 0 ("you can't comb the hair on a billiard ball"). The "3" and "7" come from the quaternions and octonions.
b(n) = a(n-1): b(2^e) = ((e+1) idiv 4) + 2^((e+1) mod 4) - 1, b(p^e) = 1, p>2. - Christian G. Bower, May 18 2005
a(n-1) is multiplicative. - Christian G. Bower, Jun 03 2005

T. D. Noe, Table of n, a(n) for n = 0..10000
J. Frank Adams, Vector fields on spheres, Topology, 1 (1962), 63-65.
J. Frank Adams, Vector fields on spheres, Bull. Amer. Math. Soc. 68 (1962) 39-41.
J. Frank Adams, Vector fields on spheres, Annals of Math. 75 (1962) 603-632.
A. Hurwitz, Uber die Komposition der quadratischen formen, Math. Annalen 88 (1923) 1-25.
M. Kervaire, Non-parallelizability of the sphere for n > 7, Proc. Nat. Acad. Sci. USA 44 (1958) 280-283.
J. Milnor, Some consequences of a theorem of Bott, Annals Math. 68 (1958) 444-449.
J. Radon, Lineare Scharen Orthogonaler Matrizen, Abh. Math. Sem. Univ. Hamburg 1 (1922) 1-14.

For another version see A003484. Cf. A189995, A001676.

Cf. A047530, A220466.

int MaxLinInd(int n){ /* Returns max # linearly indep smooth nowhere zero * vector fields on S^{n-1}, n=1,2,... */ int b,c,d,rho; b = 0; while((n & 1)==0){ n /= 2; b++; } c = b & 3; d = (b - c)/4; rho = (1 << c) + 8*d; return( rho - 1); }

int MaxLinInd(int n) { int b = _builtin_ctz(n); return (1<<b%4) + b/4*8 - 1; } /* _Jeremy Tan, Apr 09 2021 */

with(numtheory): for n from 1 to 601 by 2 do c := irem(ifactors(n+1)[2,1,2],4): d := iquo(ifactors(n+1)[2,1,2],4): printf(`%d,`, 2^c+8*d-1) od:
nmax:=101: A047530 := proc(n): ceil(n/4) + 2*ceil((n-1)/4) + 4*ceil((n-2)/4) + ceil((n-3)/4) end: for p from 0 to ceil(simplify(log[2](nmax))) do for n from 0 to ceil(nmax/(p+2))+1 do A053381((2*n+1)*2^p-1) := A047530(p+1): od: od: seq(A053381(n), n=0..nmax); # Johannes W. Meijer, Jun 07 2011, revised Jan 29 2013

a[n_] := Module[{b, c, d, rho, n0}, n0 = 2*n; b = 0; While[BitAnd[n0, 1] == 0, n0 /= 2; b++]; c = BitAnd[b, 3]; d = (b - c)/4; rho = 2^c + 8*d; Return[rho - 1]]; Table[a[n], {n, 1, 102}] (* Jean-François Alcover, May 16 2013, translated from C *)

Let f(n) be the number of linearly independent smooth nowhere-zero vector fields on an n-sphere. Then f(n) = 2^c + 8d - 1 where n+1 = (2a+1) 2^b and b = c+4d and 0 <= c <= 3. f(n) = 0 if n is even.
a((2*n+1)*2^p-1) = A047530(p+1), p >= 0 and n >= 0. a(2*n) = 1, n >= 0, and a(2^p-1) = A047530(p+1), p >= 0. - Johannes W. Meijer, Jun 07 2011
a(n) = A209675(n+1) - 1. - Reinhard Zumkeller, Mar 11 2012
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 10/3. - Amiram Eldar, Nov 29 2022

More terms from James Sellers, Jun 01 2000

A048648 Order of n-th stable homotopy group of spheres.

Original entry on oeis.org

2, 2, 24, 1, 1, 2, 240, 4, 8, 6, 504, 1, 3, 4, 960, 4, 16, 16, 528, 24, 4, 4, 3144960, 4, 4, 12, 24, 2, 3, 6, 65280, 16, 32, 32, 114912, 6, 12, 120, 1267200, 384, 32, 96, 552, 8, 5760, 48, 12579840, 64, 12, 24, 384, 24, 16, 8, 20880, 2, 8, 4, 687456, 4, 1

Offset: 1

Stephen A. Silver

Proved by Serre to be finite for all positive n.

The best current reference is Isaksen-Wang-Xu, Table 1. - Charles Rezk, Aug 22 2020

			Pi_1^S = Pi_4(S^3) = Z/2Z, so a(1) = |Z/2Z| = 2.

D. B. Fuks, "Spheres, homotopy groups of the", Encyclopaedia of Mathematics, Vol. 8.
S. O. Kochman, Stable homotopy groups of spheres. A computer-assisted approach. Lecture Notes in Mathematics, 1423. Springer-Verlag, Berlin, 1990. 330 pp. ISBN: 3-540-52468-1. [Math. Rev. 91j:55016]
Douglas C. Ravenel, Complex cobordism and stable homotopy groups of spheres, AMS Chelsea Publishing, 2003.
Hirosi Toda, Composition Methods in Homotopy Groups of Spheres, Princeton University Press, 1962.

Andrey Zabolotskiy, Table of n, a(n) for n = 1..83 (terms 1..81 from Charles Rezk, terms 82..83 using data from Isaksen, Wang & Xu (2023))
Kevin Hartnett, An Old Conjecture Falls, Making Spheres a Lot More Complicated, Quanta Magazine (2023)
A. Hatcher, Stable Homotopy Groups of Spheres
Daniel C. Isaksen, Guozhen Wang and Zhouli Xu, Stable homotopy groups of spheres: from dimension 0 to 90, Publications mathématiques de l'IHÉS, 137 (2023), 107-243; arXiv:2001.04511 [math.AT], 2020-2023.
S. O. Kochman and M. E. Mahowald, On the computation of stable stems, The Cech Centennial (Boston, MA, 1993), 299-316, Contemp. Math., 181, Amer. Math. Soc., Providence, RI, 1995. [Math. Rev. 96j:55018]
John W. Milnor, Differential Topology Forty-six Years Later, Notices Amer. Math. Soc. 58 (2011), 804-809.
Robert Scharein's program sphere-link.c linked from the Linked Spheres page [has incorrect a(23) and a(29)-a(33)]
Wikipedia, Homotopy groups of spheres

Cf. A001676.

a(n) = |Pi_n^S| = |Pi_{k+n}(S^k)| for k > n+1.

More terms from Alex Fink (finka(AT)math.ucalgary.ca), Aug 10 2006
a(23) and a(29)-a(33) corrected by Charles Rezk, Aug 22 2020
More terms from Charles Rezk, Aug 25 2020

cp's OEIS Frontend

A228689 a(n) = A001676(4*n + 3).

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A228690 a(n) = A001676(4*n).

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A228691 a(n) = A001676(4*n + 1).

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A228692 a(n) = A001676(4n+2).

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A178579 Partial sums of A001676.

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A187717 a(n) = A001676(n) / A187595(n).

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A358706 Erroneous version of A001676.

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A083420 a(n) = 2*4^n - 1.

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A053381 Maximal number of linearly independent smooth nowhere-zero vector fields on a (2n+1)-sphere.

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A048648 Order of n-th stable homotopy group of spheres.

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