Paul Muljadi - cp's OEIS frontend

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

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

A187595 Bernoulli-Kervaire-Milnor numbers.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 28, 1, 2, 1, 992, 1, 1, 1, 8128, 1, 2, 1, 261632, 1, 2, 1, 1448424448, 1, 2, 1, 67100672, 1, 1, 1, 1941802827776, 1, 2, 1, 753623571759104, 1, 2, 1, 23998307331473408, 1, 2, 1, 341653284209033216, 1, 2, 1, 8316321134799694594048, 1, 2, 1, 740764429532373450752, 1, 2, 1, 30559446583872811817762816, 1, 2, 1, 496669433444154134078771167232, 1, 1, 1, 17776484020396435145889494859776

Offset: 1

Paul Muljadi, Mar 11 2011

nonn

|bP_(n+1)| in 1961 Kervaire-Milnor theorem.
|b_n| in Milnor 2011 Table 2 (explained in Further Details p. 807). - Jonathan Sondow, Jun 16 2011
Named after the Swiss mathematician Jacob Bernoulli (1655-1705), the French mathematician Michel André Kervaire (1927-2007) and the American mathematician John Willard Milnor (b. 1931). - Amiram Eldar, Jun 22 2021

Michel A. Kervaire and John W. Milnor, Groups of homotopy spheres: I, December 1961.
Michel A. Kervaire and John W. Milnor, Groups of homotopy spheres: I, Annals of Mathematics, Second Series, Vol. 77, No. 3 (1963), pp. 504-537; alternative link.
John W. Milnor, Differential Topology Forty-six Years Later, Notices Amer. Math. Soc., Vol. 58, No. 6 (2011), pp. 804-809.
John W. Milnor, Spheres, Abel Prize lecture (video), 2011.
Georgios Dimitroglou Rizell and Jonathan David Evans, Exotic spheres and the topology of symplectomorphism groups, J. Topol., Vol. 8, No. 2 (2015), pp. 586-602.
Wikipedia, Exotic sphere.

Cf. A001676.

More terms from Jonathan Sondow, Jun 16 2011.

A187488 Decimal expansion of the Hausdorff dimension of the Feigenbaum attractor and repeller.

Original entry on oeis.org

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

Offset: 0

Paul Muljadi, Mar 10 2011

			0.53804514358054991167141556...

F. Christiansen, P. Cvitanovic, and H. H. Rugh, The spectrum of the period-doubling operator in terms of cycles, J. Phys A, 23 (14) (1990), pp. L713S-L717S.
P. Grassberger, On the Hausdorff dimension of fractal attractors, Journal of Statistical Physics 26 (1981), pp. 173-179.
J. Thurlby, Rigorous calculations of renormalisation fixed points and attractors, PhD thesis, U. Portsmouth, (2021), eq. (5.58)

Cf. A006890.

A187089 Decimal expansion of the fractal dimension of the Apollonian sphere packing.

Original entry on oeis.org

2, 4, 7, 3, 9, 4, 6

Offset: 1

Paul Muljadi, Mar 08 2011

			2.473946...

M. Borkovec, W. de Paris, and R. Peikert, The Fractal Dimension of the Apollonian Sphere Packing, Fractals, 2 (1994), pp. 521-526; Preprint.

Cf. A052483.

A184853 Decimal expansion of 9th Stieltjes constant, negated.

Original entry on oeis.org

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

Offset: 0

Paul Muljadi, Feb 01 2011

			-0.000034394774418088048177914623798227390620789538594442....

G. C. Greubel, Table of n, a(n) for n = 0..10000
Krzysztof Maślanka and Andrzej Koleżyński, The High Precision Numerical Calculation of Stieltjes Constants. Simple and Fast Algorithm, arXiv preprint (2022). arXiv:2210.04609 [math.NT]
Eric Weisstein's World of Mathematics, Stieltjes Constants
Wikipedia, Stieltjes constants

Cf. A001620, A082633, A086279, A086280, A086281, A086282, A183141, A183167, A183206, A184854.

evalf(gamma(9)); # R. J. Mathar, Feb 02 2011

Join[{0, 0, 0, 0}, RealDigits[ N[ -StieltjesGamma[9], 101]][[1]]] (* Jean-François Alcover, Nov 07 2012 *)

A184854 Decimal expansion of 10th Stieltjes constant.

Original entry on oeis.org

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

Offset: 0

Paul Muljadi, Feb 01 2011

			0.00020533281490906479468372228923706530295985377416676....

Steven R. Finch, "Stieltjes Constants." Section 2.21 in Mathematical Constants. Cambridge: Cambridge University Press (2003), 166 - 171.

G. C. Greubel, Table of n, a(n) for n = 0..10000
Krzysztof Maślanka and Andrzej Koleżyński, The High Precision Numerical Calculation of Stieltjes Constants. Simple and Fast Algorithm, arXiv preprint arXiv:2210.04609 [math.NT], 2022.
Eric Weisstein, Stieltjes Constants from Mathworld
Wikipedia, Stieltjes constants

Cf. A001620, A082633, A086279, A086280, A086281, A086282, A183141, A183167, A183206, A184853

RealDigits[StieltjesGamma[10], 10, 100][[1]] (* Alonso del Arte, Feb 01 2011 *)

A183206 Decimal expansion of 8th Stieltjes constant, negated.

Original entry on oeis.org

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

Offset: 0

Paul Muljadi, Feb 01 2011

			-0.00035212335380303950960205216500120874172918053379235....

G. C. Greubel, Table of n, a(n) for n = 0..10000
Krzysztof Maślanka and Andrzej Koleżyński, The High Precision Numerical Calculation of Stieltjes Constants. Simple and Fast Algorithm, arXiv preprint (2022). arXiv:2210.04609 [math.NT]
Eric Weisstein's World of Mathematics, Stieltjes Constants
Wikipedia, Stieltjes constants

Cf. A001620, A082633, A086279-A086282, A183141, A183167, A184853, A184854.

Maple
```
evalf(gamma(8)) ;
```

RealDigits[StieltjesGamma[8], 10, 100][[1]] (* Alonso del Arte, Feb 01 2011 *)

A183167 Decimal expansion of 7th Stieltjes constant, negated.

Original entry on oeis.org

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

Offset: 0

Paul Muljadi, Feb 01 2011

			-0.000527289567057751046074097505478858281996253472969895...

G. C. Greubel, Table of n, a(n) for n = 0..10000
Krzysztof Maślanka and Andrzej Koleżyński, The High Precision Numerical Calculation of Stieltjes Constants. Simple and Fast Algorithm, arXiv preprint (2022). arXiv:2210.04609 [math.NT]
Eric Weisstein's World of Mathematics, Stieltjes Constants
Wikipedia, Stieltjes constants

Cf. A001620, A082633, A086279, A086280, A086281, A086282, A183141, A183206, A184853, A184854.

evalf(gamma(7)) ; # R. J. Mathar, Feb 02 2011

Join[{0, 0, 0}, RealDigits[StieltjesGamma[7], 10, 100][[1]]] (* Alonso del Arte, Feb 01 2011 *)

A183141 Decimal expansion of 6th Stieltjes constant, negated.

Original entry on oeis.org

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

Offset: 0

Paul Muljadi, Feb 01 2011

			-0.000238769345430199609872421841908004277783715156358....

G. C. Greubel, Table of n, a(n) for n = 0..10000
Krzysztof Maślanka and Andrzej Koleżyński, The High Precision Numerical Calculation of Stieltjes Constants. Simple and Fast Algorithm, arXiv preprint (2022). arXiv:2210.04609 [math.NT]
Eric Weisstein's World of Mathematics, Stieltjes Constants
Wikipedia, Stieltjes constants

Cf. A001620, A082633, A086279, A086280, A086281, A086282, A183167, A183206, A184853, A184854.

evalf(gamma(6)); # R. J. Mathar, Feb 02 2011

RealDigits[StieltjesGamma[6], 10, 100][[1]]

A179998 Numbers which are members of both A000140 and A039909.

Original entry on oeis.org

1, 2, 101, 573, 3836, 29228, 250749, 2409581, 3727542188, 50626553988, 738680521142, 11501573822788, 190418421447330, 3344822488498265, 24965661442811799655, 538134522243713149122, 12140037056605135928410

Offset: 1

Paul Muljadi, Jan 13 2011

nonn

Vincenzo Librandi, Table of n, a(n) for n = 1..97

PS:=PowerSeriesRing(Integers()); { Max(Coefficients(&*[&+[ x^i: i in [0..j] ]: j in [0..n-1] ])): n in [1..25] } meet { Max(Coefficients(&*[&+[ (-x)^i: i in [0..j] ]: j in [0..n] ])): n in [1..24] }; // Klaus Brockhaus, Jan 18 2011

cp's OEIS Frontend

User: Paul Muljadi

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

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A187595 Bernoulli-Kervaire-Milnor numbers.

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A187488 Decimal expansion of the Hausdorff dimension of the Feigenbaum attractor and repeller.

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A187089 Decimal expansion of the fractal dimension of the Apollonian sphere packing.

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A184853 Decimal expansion of 9th Stieltjes constant, negated.

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Mathematica

A184854 Decimal expansion of 10th Stieltjes constant.

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Mathematica

A183206 Decimal expansion of 8th Stieltjes constant, negated.

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Maple

Mathematica

A183167 Decimal expansion of 7th Stieltjes constant, negated.

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Maple

Mathematica

A183141 Decimal expansion of 6th Stieltjes constant, negated.

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Programs

Maple