Sam Alexander - cp's OEIS frontend

A182660 a(2^(k+1)) = k; 0 everywhere else.

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

Offset: 0

Sam Alexander, Nov 27 2010

nonn

A surjection N->N designed to spite a guesser who is trying to guess whether it's a surjection, using the following naive guessing method: Guess that (n0,...,nk) is a subsequence of a surjection iff it contains every natural less than log_2(k+1).
This sequence causes the would-be guesser to change his mind infinitely often.
a(0)=0. Assume a(0),...,a(n) have been defined.
If the above guesser guesses that (a(0),...,a(n)) IS the beginning of a surjective sequence, then let a(n+1)=0. Otherwise let a(n+1) be the least number not in (a(0),...,a(n)).

Antti Karttunen, Table of n, a(n) for n = 0..65537
S. Alexander, On Guessing Whether A Sequence Has A Certain Property, arXiv:1011.6626 [math.LO], 2010-2012.
S. Alexander, On Guessing Whether A Sequence Has A Certain Property, J. Int. Seq. 14 (2011) # 11.4.4.

Cf. A082691, A135416, A209229.

[ exists(t){ k: k in [1..Ceiling(Log(n+1))] | n eq 2^(k+1) } select t else 0: n in [0..100] ];

A182660(n) = if(n<2,0,my(p = 0, k = isprimepower(n,&p)); if(2==p,k-1,0)); \\ Antti Karttunen, Jul 22 2018

A182659 A canonical permutation designed to thwart a certain naive attempt to guess whether sequences are permutations.

Original entry on oeis.org

0, 2, 3, 1, 5, 6, 7, 8, 9, 4, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 10, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 22, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68

Offset: 0

Sam Alexander, Nov 26 2010

A naive way to guess whether a function f:N->N is a permutation, based on just an initial subsequence (f(0),...,f(n)), is to guess "no" if (f(0),...,f(n)) contains a repeated entry or if there is some i in {0,...,n} such that i is not in {f(0),...,f(n)} and 2 i<=n; and guess "yes" otherwise. a(n) thwarts that method, causing it to change its mind infinitely often as n->infinity.
a(0)=0. Suppose a(0),...,a(n) have been defined.
1. If the above method guesses that (a(0),...,a(n)) is NOT an initial subsequence of a permutation, then unmark any "marked" numbers.
2. If the above method guesses that (a(0),...,a(n)) IS an initial subsequence of a permutation, then "mark" the smallest number not in {a(0),...,a(n)}.
3. Let a(n+1) be the least unmarked number not in {a(0),...,a(n)}.
A030301 can be derived by a similar method, where instead of trying to guess whether sequences are permutations, the naive victim is trying to guess whether sequences contain infinitely many 0s.

S. Alexander, On Guessing Whether A Sequence Has A Certain Property, arXiv:1011.6626 [math.LO], 2010-2012, J. Int. Seq. 14 (2011) # 11.4.4.

Cf. A030301, A082691, A182660, A068156, A033484.

a(0) = 0; if n = A068156(k+1) = 6*2^k - 3 for some k >= 0 then a(n) = A033484(k) = (n-1)/2; otherwise, a(n) = n+1. - Andrey Zabolotskiy, Feb 27 2025

a(22) corrected and further terms added by Andrey Zabolotskiy, Feb 27 2025

A174429 Collatz-Fibonacci numbers: a(1) = a(2) = 1; if n > 2, a(n) = a(C(n)) + a(C(C(n))), where C(n) = A006370(n).

Original entry on oeis.org

1, 1, 21, 2, 8, 34, 1597, 3, 6765, 13, 610, 55, 55, 2584, 2584, 5, 233, 10946, 10946, 21, 21, 987, 987, 89, 46368, 89, 114059301025943970552219, 4181, 4181, 4181, 10284720757613717413913, 8, 196418, 377, 377, 17711, 17711, 17711, 9227465, 34

Offset: 1

Sam Alexander, Mar 19 2010

			a(4) = a(2) + a(1) = 2. a(8) = a(4) + a(2) = 3. a(16) = a(8) + a(4) = 5. a(5) = a(16) + a(8) = 8.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Sam Alexander, Collatz Recursion

a174429 = a000045 . a008908  -- Reinhard Zumkeller, Nov 10 2011

collatz[n_] := If[EvenQ[n], n/2, 3n + 1]; collFibo[1] = 1; collFibo[2] = 1; collFibo[n_] := collFibo[n] = collFibo[collatz[n]] + collFibo[collatz[collatz[n]]]; Table[collFibo[n], {n, 100}] (* T. D. Noe, Jul 06 2010 *)

a(n) = Fibonacci(A008908(n)) = A000045(A008908(n)). - T. D. Noe, Jul 06 2010

Corrected by T. D. Noe, Jul 06 2010

A118874 A halting sequence: let f_n be the n-th recursive function, relative to the Godel numbering given in Cutland, then a(n) is f_n(n)+1 if the corresponding program halts on input n, 0 otherwise.

Original entry on oeis.org

1, 3, 1, 4, 2, 1, 1, 0, 1, 12, 2, 1, 1, 1, 1, 16, 0, 19, 1, 21, 3, 2, 2, 0, 1, 1, 1, 1, 1, 1, 1, 32, 1, 0, 0, 36, 2, 1, 1, 0, 2, 45, 3, 2, 2, 2, 2, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 64, 1, 67, 1, 0, 0, 0, 0, 0, 1, 76, 2, 1, 1, 1, 1, 81, 0, 84, 2, 86, 4, 3, 3, 0, 2, 2, 2, 2, 2, 2, 2, 1, 1, 0, 0

Offset: 0

Sam Alexander, May 24 2006

nonn

The prototypical example of a noncomputable sequence.

			Using Cutland's Godel numbering, 80 corresponds to the URM program "Z(1) J(1,1,1) S(1)", which clearly loops forever on any input, so a(80)=0. On the other hand, 17 corresponds to the URM program "S(1) T(1,1)", which, on input 17, produces 18. So a(17)=18+1=19.

Nigel Cutland, "Computability: An introduction to recursive function theory". Cambridge University Press, 1980. p. 78.

Ramin Naimi, URM Simulator

Cf. A004147, A028444, A052200, A060843, A081419.

A111003 Decimal expansion of Pi^2/8.

Original entry on oeis.org

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

Offset: 1

Sam Alexander, Oct 01 2005

According to Beckmann, Euler discovered the formula for this number as a sum of squares of reciprocals of odd numbers, along with similar formulas for Pi^2/6 and Pi^2/12. - Alonso del Arte, Apr 01 2013

Equals the asymptotic mean of the abundancy index of the odd numbers. - Amiram Eldar, May 12 2023

			1.23370055013616982735431137498451889191421242590509882830166867202...
1 + 1/9 + 1/25 + 1/49 + 1/81 + 1/121 + 1/169 + 1/225 + ... - _Bruno Berselli_, Mar 06 2017

F. Aubonnet, D. Guinin and B. Joppin, Précis de Mathématiques, Analyse 2, Classes Préparatoires, Premier Cycle Universitaire, Bréal, 1990, Exercice 908, pages 82 and 91-92.
Petr Beckmann, A History of Pi, 5th Ed. Boulder, Colorado: The Golem Press (1982): p. 153.
George Boros and Victor H. Moll, Irresistible integrals, Cambridge University Press (2006), p. 122.
Calvin C. Clawson, The Beauty and Magic of Numbers. New York: Plenum Press (1996): 98.
L. B. W. Jolley, Summation of Series, Dover (1961).
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 54.

Eric Weisstein's World of Mathematics, Dirichlet Lambda Function. See (5).
Index entries for transcendental numbers

Cf. A013661 (Pi^2/6), A002388, A102753, A091476, A309891.

RealDigits[Pi^2/8, 10, 105][[1]] (* Robert G. Wilson v *)

Pi^2/8 \\ Charles R Greathouse IV, Dec 04 2016

sumpos(n=1,(2*n-1)^-2) \\ Charles R Greathouse IV, Mar 02 2018

Equals 1 + 1/(2*3) + (1/3)*(1*2)/(3*5) + (1/4)*(1*2*3)/(3*5*7) + ... [Jolley eq 276]
Equals Sum_{k >= 1} 1/(2*k - 1)^2 [Clawson and Wells]. - Alonso del Arte, Aug 15 2012
Equals 2*(Integral_{t=0..1} sqrt(1 - t^2) dt)^2. - Alonso del Arte, Mar 29 2013
Equals Sum_{k >= 1} 2^k/(k^2*binomial(2*k, k)). - Jean-François Alcover, Apr 29 2013
Equals Integral_{x=0..1} log((1+x^2)/(1-x^2))/x dx. - Bruno Berselli, May 13 2013
Equals limit_{p->0} Integral_{x=0..Pi/2} x*tan(x)^p dx. [Jean-François Alcover, May 17 2013, after Boros & Moll p. 230]
Equals A002388/8 = A102753/4 = A091476/2. - R. J. Mathar, Oct 13 2015
Equals Integral_{x>=0} x*K_0(x)*K_1(x)dx where K are modified Bessel functions [Gradsteyn-Ryzhik 6.576.4]. - R. J. Mathar, Oct 22 2015
Equals (3/4)*zeta(2) = (3/4)*A013661. - Wolfdieter Lang, Sep 02 2019
From Amiram Eldar, Jul 17 2020: (Start)
Equals -Integral_{x=0..1} log(x)/(1 - x^2) dx = Integral_{x>=1} log(x)/(x^2-1) dx.
Equals -Integral_{x=0..oo} log(x)/(1 - x^4) dx.
Equals Integral_{x=0..oo} arctan(x)/(1 + x^2) dx. (End)
Equals Integral_{x=0..1} log(1+x+x^2+x^3)/x dx (Aubonnet). - Bernard Schott, Feb 04 2022
Equals Sum_{n>=1} A309891(n)/n^2. - Friedjof Tellkamp, Jan 25 2025
Equals lambda(2), where lambda is the Dirichlet lambda function. - Michel Marcus, Aug 15 2025

More terms from Robert G. Wilson v, Oct 04 2005

A107484 Some numbers k such that the elliptic curve y^2 = x^3 + k*x has rank 2.

Original entry on oeis.org

14, 33, 34, 39, 46, -17, -56, -65, -77

Offset: 1

Sam Alexander, May 28 2005

Dale Husemoller, "Elliptic Curves", Springer-Verlag: New York, 1987, p. 35.

Cf. A002151.

This is the first few terms of A076329 followed by the first few negated terms of A309032.

A092698 Disk degeneracies for brane III in the O(K)->P^1 x P^1 geometry.

Original entry on oeis.org

0, 2, 40, 310, 1520, 5628

Offset: 0

Sam Alexander, Mar 05 2004

nonn

M. Aganagic, A. Klemm and C. Vafa, Disk Instantons, Mirror Symmetry and the Duality Web p. 41

A092699 Disk degeneracies for brane III in the O(K)->P^1 x P^1 geometry.

Original entry on oeis.org

0, 2, 100, 1520, 12908, 76488

Offset: 0

Sam Alexander, Mar 05 2004

nonn

M. Aganagic, A. Klemm and C. Vafa, Disk Instantons, Mirror Symmetry and the Duality Web p. 41

A092700 Disk degeneracies for brane III in the O(K)->P^1 x P^1 geometry.

Original entry on oeis.org

0, 2, 210, 5628, 91070, 680940

Offset: 0

Sam Alexander, Mar 05 2004

nonn

M. Aganagic, A. Klemm and C. Vafa, Disk Instantons, Mirror Symmetry and the Duality Web p. 41

A092701 Disk degeneracies for brane III in the O(K)->P^1 x P^1 geometry.

Original entry on oeis.org

0, 2, 392, 17184, 353316, 4515558

Offset: 0

Sam Alexander, Mar 05 2004

nonn

M. Aganagic, A. Klemm and C. Vafa, Disk Instantons, Mirror Symmetry and the Duality Web p. 41

cp's OEIS Frontend

User: Sam Alexander

A182660 a(2^(k+1)) = k; 0 everywhere else.

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A182659 A canonical permutation designed to thwart a certain naive attempt to guess whether sequences are permutations.

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A174429 Collatz-Fibonacci numbers: a(1) = a(2) = 1; if n > 2, a(n) = a(C(n)) + a(C(C(n))), where C(n) = A006370(n).

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A118874 A halting sequence: let f_n be the n-th recursive function, relative to the Godel numbering given in Cutland, then a(n) is f_n(n)+1 if the corresponding program halts on input n, 0 otherwise.

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A111003 Decimal expansion of Pi^2/8.

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PARI

PARI

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A107484 Some numbers k such that the elliptic curve y^2 = x^3 + k*x has rank 2.

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A092698 Disk degeneracies for brane III in the O(K)->P^1 x P^1 geometry.

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A092699 Disk degeneracies for brane III in the O(K)->P^1 x P^1 geometry.

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A092700 Disk degeneracies for brane III in the O(K)->P^1 x P^1 geometry.

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A092701 Disk degeneracies for brane III in the O(K)->P^1 x P^1 geometry.

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