Christian Schroeder - cp's OEIS frontend

User: Christian Schroeder

Christian Schroeder has authored 9 sequences.

A309107 A member of a family of generalizations of van Eck's sequence as defined below.

0, 0, 0, 2, 0, 2, 2, 3, 0, 4, 0, 2, 5, 0, 3, 7, 0, 3, 3, 4, 10, 0, 5, 10, 3, 6, 0, 5, 5, 6, 4, 11, 0, 6, 4, 4, 5, 8, 0, 6, 6, 7, 26, 0, 5, 8, 8, 9, 0, 5, 5, 6, 11, 21, 0, 6, 4, 21, 4, 2, 48, 0, 7, 21, 6, 9, 18, 0, 6, 4, 11, 18, 5, 22, 0, 7, 13, 0, 3, 54, 0, 3, 3, 4, 14, 0, 5, 14, 3, 6, 21, 27, 0, 7, 18, 23, 0, 4, 14, 11

Offset: 1

Christian Schroeder, Jul 12 2019

For n >= 1, if there exists an m < n-1 such that a(m) = a(n), take the largest such m and set a(n+1) = n-m; otherwise a(n+1) = 0. Start with a(1) = a(2) = 0.

T: let 0 <= k < l. For n > k, if there exists an m <= n-l such that a(m) = a(n-k), take the largest such m and set a(n+1) = n-m; otherwise a(n+1) = 0. Start with a(1) = ... = a(l) = 0. Setting k = 0, l = 1 produces van Eck's sequence A181391; setting k = 0, l = 2 produces this sequence.

Cf. A181391.

function VEg = VE_generalized(N, k, l)
    assert(l > k);
    VEg = zeros(1, l);
    for n = l:(N - 1)
        prev = VEg(n - k);
        VEg(n + 1) = 0;
        for j = (n - l):-1:1
            if VEg(j) == prev
                VEg(n + 1) = n - j;
                break
            end
        end
    end
end

A274326 Number of distinct irregular n-gon flexagons.

Original entry on oeis.org

0, 0, 1, 2, 11, 40

Offset: 2

Christian Schroeder, Jun 18 2016

			a(6) = 11, because there are 11 distinct irregular hexaflexagons.

Les Pook, Flexagons inside out, Cambridge University Press, 2003, p. 84.

Cf. A001683.

A260680 Peaceable coexisting armies of queens: number of inequivalent configurations with maximum number of queens as given in A250000.

Original entry on oeis.org

1, 1, 1, 10, 3, 35, 19, 71, 18, 380

Offset: 1

Christian Schroeder, Nov 15 2015

From Rob Pratt, Apr 05 2019: (Start)
Two solutions are regarded as equivalent if one can be obtained from the other by rotations, reflections, interchanging the colors (a group of order 16).
I used two computational methods, both implemented via PROC OPTMODEL from SAS:
One round of constraint programming, with LEXICO constraints to account for symmetry and an option to generate all solutions.  This method returns only the lexicographically smallest representative of each equivalence class.
Multiple rounds of integer linear programming, with 16 additional cuts (one per group element) after each solution is found, to avoid generating an equivalent solution.  This method terminates when the resulting cuts make the problem infeasible.
The attached text files are from the second method. (End)

			For n = 3, a(3) = 1 because the following solution is unique up to equivalence:
-----
|W..|
|...|
|.B.|
-----
From _Rob Pratt_ in A250000, Nov 30 2014 thru Jul 29 2015: (Start)
n=4:
----------------------------------------------------------
|..B.||.B..||.B..||....||.BB.||..B.||...W||..B.|..B.|..W.|
|....||.B..||...B||.B.B||....||.B..||.B..||...B|B...|B...|
|...B||....||....||....||....||...W||..B.||.W..|...W|...B|
|WW..||W.W.||W.W.||W.W.||W..W||W...||W...||W...|.W..|.W..|
----------------------------------------------------------
n=5:
---------------------
|W...W||..B.B||.W.W.|
|..B..||W....||..W..|
|.B.B.||..B.B||B...B|
|..B..||W....||..W..|
|W...W||.W.W.||B...B|
---------------------
(End)
From _Rob Pratt_, Mar 18 2019, additional solution for n=6 (not covered in attached pdf):
--------
|....W.|
|...W.W|
|B.....|
|B.B...|
|....WW|
|B.B...|
--------

Luca Petrone, Graphic illustrations of a(6) and a(7)
Rob Pratt, Solutions for n = 3
Rob Pratt, Solutions for n = 4
Rob Pratt, Solutions for n = 5
Rob Pratt, Solutions for n = 6
Rob Pratt, Solutions for n = 7
Rob Pratt, Solutions for n = 8
Rob Pratt, Solutions for n = 9
Rob Pratt, Solutions for n = 10

Cf. A250000.

a(6)-a(8) from Luca Petrone, Mar 11 2016
a(4), a(6), and a(8) corrected by Rob Pratt, Mar 18 2019
a(9) and a(10) from Rob Pratt, Mar 19 2019

A175000 Number of stable n-celled patterns ("still lifes") in the HighLife (B36/S23) cellular automaton.

Original entry on oeis.org

0, 0, 0, 2, 1, 4, 4, 9, 9, 25, 44, 111, 218

Offset: 1

Christian Schroeder, Apr 03 2010

Counts only distinct strict still lifes; reflections and rotations are removed. Pseudo-still lifes (e.g., patterns made up of two disjoint still lifes) are not counted.

			a(4)=2 because there are only two still-lives with four cells: the block (a 2x2 block) and the tub (four cells bordering all four sides of an empty spot).

LifeWiki, HighLife

Cf. A019473, A166476

A175002 Number of stable n-celled patterns ("still lifes") in the Maze (B3/S12345) cellular automaton.

Original entry on oeis.org

0, 2, 1, 5, 7, 18, 40, 197, 669

Offset: 1

Christian Schroeder, Apr 03 2010

			a(3)=1 because there is only one distinct still life with three cells: a diagonal line of length three.

LifeWiki, Maze

Cf. A019473, A166476, A175000, A175001

A175001 Number of stable n-celled patterns ("still lifes") in the Move (a.k.a. Morley; B368/S245) cellular automaton.

Original entry on oeis.org

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

Offset: 1

Christian Schroeder, Apr 03 2010

			a(4)=1 because there is only one still-life with four cells: the tub (four cells bordering all four sides of an empty spot).

LifeWiki, Move

Cf. A019473, A166476, A175000

A098820 Periodicity of entries in the first row of a Laver Table of size 2^n.

Original entry on oeis.org

1, 1, 2, 4, 4, 8, 8, 8, 8, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16

Offset: 0

Christian Schroeder, Oct 08 2004

nonn

All sequence elements are powers of 2. The first n for which a(n)=32 is at least A(9,A(8,A(8,255))), where A denotes the Ackermann function (R. Dougherty). If a rank-into-rank exists, then the sequence is diverging (R. Laver).

			a(4)=4 because the entries in the first row of the Laver table of size 4^2=16 are 2,12,14,16,2,12,14,16,2,12,14,16,2,12,14,16 (and thus repeat with a periodicity of 4).

Richard Laver, On the Algebra of Elementary Embeddings of a Rank into Itself, arXiv:math/9204204 [math.LO], 1992; Advances in Mathematics 110, p. 334, 1995
Wikipedia, Laver table

More terms from Adam P. Goucher, Dec 18 2013

A088165 NSW primes: NSW numbers that are also prime.

Original entry on oeis.org

7, 41, 239, 9369319, 63018038201, 489133282872437279, 19175002942688032928599, 123426017006182806728593424683999798008235734137469123231828679

Offset: 1

Christian Schroeder, Sep 21 2003

nonn

Next term a(9) is too large (99 digits) to include here. - Ray Chandler, Sep 21 2003
These primes are the prime RMS numbers (A140480): primes p such that (1+p^2)/2 is a square r^2. Then r is a Pell number, A000129. - T. D. Noe, Jul 01 2008
Also prime numerators with an odd index in A001333. - Ctibor O. Zizka, Aug 13 2008
r in the above note of T. D. Noe is a prime Pell number (A000129) with an odd index. - Ctibor O. Zizka, Aug 13 2008
General recurrence is a(n) = (a(1)-1)*a(n-1) - a(n-2), a(1) >= 4, lim_{n->infinity} a(n) = x*(k*x+1)^n, k = a(1)-3, x = (1+sqrt((a(1)+1)/(a(1)-3)))/2. Examples in the OEIS: a(1)=4 gives A002878, primes in it A121534. a(1)=5 gives A001834, primes in it A086386. a(1)=6 gives A030221, primes in it not in the OEIS {29, 139, 3191, ...}. a(1)=7 gives A002315, primes in it A088165. a(1)=8 gives A033890, primes in it not in the OEIS (do there exist any ?). a(1)=9 gives A057080, primes in it not in the OEIS {71, 34649, 16908641, ...}. a(1)=10 gives A057081, primes in it not in the OEIS {389806471, 192097408520951, ...}. - Ctibor O. Zizka, Sep 02 2008

Paulo Ribenboim, The New Book of Prime Number Records, 3rd edition, Springer-Verlag, New York, 1995, pp. 367-369.

Alois P. Heinz, Table of n, a(n) for n = 1..14
Morris Newman, Daniel Shanks, H. C. Williams, Simple groups of square order and an interesting sequence of primes, Acta Arith., 38 (1980/1981), pp. 129-140.
M. Newman, D. Shanks and L. L. Foster, Simple groups of square order (6176), The American Mathematical Monthly, Vol. 86, No. 4 (Apr., 1979), pp. 314-315.
The Prime Glossary, NSW numbers

Cf. A002315 (NSW numbers), A005850 (indices for NSW primes).

w=3+quadgen(32); forprime(p=2,1e3, if(ispseudoprime(t=imag((1+w)*w^p)), print1(t", "))) \\ Charles R Greathouse IV, Apr 29 2015

a(n) mod A005850(n) = 1. - Altug Alkan, Mar 17 2016

More terms from Ray Chandler, Sep 21 2003

A088164 Wolstenholme primes: primes p such that binomial(2p-1,p-1) == 1 (mod p^4).

Original entry on oeis.org

16843, 2124679

Offset: 1

Christian Schroeder, Sep 21 2003

McIntosh and Roettger showed that the next term, if it exists, must be larger than 10^9. - Felix Fröhlich, Aug 23 2014
When cb(m)=binomial(2m,m) denotes m-th central binomial coefficient then, obviously, cb(a(n))=2 mod a(n)^4. I have verified that among all naturals 1A246134). One might therefore wonder whether this is true in general. - Stanislav Sykora, Aug 26 2014
Romeo Mestrovic, Congruences for Wolstenholme Primes, Lemma 2.3, shows that the criterion for p to be a Wolstenholme prime is equivalent to p dividing A027641(p-3). In 1847 Cauchy proved that this was a necessary condition for the failure of the first case of Fermat's Last Theorem for the exponent p (see Ribenboim, 13 Lectures, p. 29). - John Blythe Dobson, May 01 2015
Primes p such that p^3 divides A001008(p-1) (Zhao, 2007, p. 18). Also: Primes p such that (p, p-3) is an irregular pair (cf. Buhler, Crandall, Ernvall, Metsänkylä, 1993, p. 152). Keith Conrad observes that for the two known (as of 2015) terms ord_p(H_p-1) = 3 is satisfied, where ord_p(H_p-1) gives the p-adic valuation of H_p-1 (cf. Conrad, p. 5). Romeo Mestrovic conjectures that p is a Wolstenholme prime if and only if S_(p-2)(p) == 0 (mod p^3), where S_k(i) denotes the sum of the k-th powers of the positive integers up to and including (i-1) (cf. Mestrovic, 2012, conjecture 2.10). - Felix Fröhlich, May 20 2015
Primes p that divide the Wolstenholme quotient W_p (A034602). Also, primes p such that p^2 divides the Babbage quotient b_p (A263882). - Jonathan Sondow, Nov 24 2015
The only known composite numbers n such that binomial(2n-1, n-1) is congruent to 1 mod n^2 are the numbers n = p^2 where p is a Wolstenholme prime: see A267824. - Jonathan Sondow, Jan 27 2016
The converse of Wolstenholme's theorem implies that if an integer n satisfies the congruence binomial(2*n-1, n-1) == 1 (mod n^4), then n is a term of this sequence, i.e., then n is necessarily prime, or, equivalently, A298946(i) > 1 for all i > 0. Whether this is true for all such n is an open problem. - Felix Fröhlich, Feb 21 2018
Primes p such that binomial(2*p-1, p-1) == 1-2*p*Sum_{k=1..p-1} 1/k - 2*p^2*Sum_{k=1..p-1} 1/k^2 (mod p^7) (cf. Mestrovic, 2011, Corollary 4). - Felix Fröhlich, Feb 21 2018
These are primes p such that p^2 divides A007406(p-1) (Mestrovic, 2015, p. 241, Lemma 2.3). - Amiram Eldar and Thomas Ordowski, Jul 29 2019
If a third Wolstenholme prime exists it is larger than 6*10^10 (cf. Hathi, Mossinghoff, Trudgian, 2021). - Felix Fröhlich, Apr 27 2021
Named after the English mathematician Joseph Wolstenholme (1829-1891). - Amiram Eldar, Jun 10 2021
Primes p such that tanh(Sum_{k=1..p-1} artanh(k/p)) == 0 (mod p^4). - Thomas Ordowski, Apr 17 2025

Richard K. Guy, Unsolved Problems in Number Theory, Sect. B31.
Paulo Ribenboim, 13 Lectures on Fermat's Last Theorem (Springer, 1979).
Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 23.

Ronald Bruck, Wolstenholme's Theorem, Stirling Numbers, and Binomial Coefficients.
Joe Buhler, Richard Crandall, Reijo Ernvall and Tauno Metsänkylä, Irregular primes and cyclotomic invariants to four million, Math. Comp., Vol. 61, No. 203 (1993), pp. 151-153.
Chris Caldwell, The Prime Glossary, Wolstenholme prime.
Leonardo Carofiglio, Luigi De Filpo, and Alessandro Gambini, p-adic valuation of harmonic sums and their connections with Wolstenholme primes, arXiv:2303.15010 [math.NT], 2023.
Keith Conrad, The p-adic growth of harmonic sums.
Shehzad Hathi, Michael J. Mossinghoff, and Timothy S. Trudgian, Wolstenholme and Vandiver primes, The Ramanujan Journal, (2021); arXiv version, 2101.11157 [math.NT], 2021.
Richard J. McIntosh, On the converse of Wolstenholme's theorem, Acta Arithmetica, Vol. 71, No. 4 (1995), pp. 381-389.
Richard J. McIntosh and Eric L. Roettger, A search for Fibonacci-Wieferich and Wolstenholme primes, Math. Comp. Vol 76, No. 260 (2007), pp. 2087-2094.
R. Mestrovic, Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862-2011), arXiv:1111.3057 [math.NT], 2011.
Romeo Meštrović, On a Congruence Modulo n^3 Involving Two Consecutive Sums of Powers, Journal of Integer Sequences, Vol. 17 (2014), Article 14.8.4.
Romeo Meštrović, Congruences for Wolstenholme Primes, arXiv:1108.4178 [math.NT], 2011.
Romeo Meštrović, Congruences for Wolstenholme Primes, Czechoslovak Mathematical Journal, Vol. 65 (2015), pp. 237-253.
Romeo Meštrović, A congruence modulo n^3 involving two consecutive sums of powers and its applications, arXiv:1211.4570 [math.NT], 2012.
Romeo Meštrović, Several generalizations and variations of Chu-Vandermonde identity, arXiv:1807.10604 [math.CO], 2018.
Jonathan Sondow, Extending Babbage's (non-)primality tests, in Combinatorial and Additive Number Theory II, Springer Proc. in Math. & Stat., Vol. 220, CANT 2015 and 2016, New York, 2017, pp. 269-277; arXiv:1812.07650 [math.NT], 2018.
Eric Weisstein's World of Mathematics, Wolstenholme Prime.
Eric Weisstein's World of Mathematics, Integer Sequence Primes.
Wikipedia, Wolstenholme prime.
Jianqiang Zhao, Bernoulli numbers, Wolstenholme's theorem, and p^5 variations of Lucas' theorem, J. Number Theory, Vol. 123 (2007), pp. 18-26.

Cf. A000984, A001008, A007406, A027641, A034602, A099908, A246130, A246132, A246133, A246134, A263882, A267824, A298946.

[p: p in PrimesUpTo(2*10^4)| (Binomial(2*p-1,p-1) mod (p^4)eq 1)]; // Vincenzo Librandi, May 02 2015

For[i = 2, i <= 20000, i++, {If[PrimeQ[i] && Mod[Binomial[2*i - 1, i - 1], i^4] == 1, Print[i]]}] (* Dylan Delgado, Mar 02 2021 *)

forprime(n=2, 10^9, if(Mod(binomial(2*n-1, n-1), n^4)==1, print1(n, ", "))); \\ Felix Fröhlich, May 18 2014

A000984(a(n)) = 2 mod a(n)^4. - Stanislav Sykora, Aug 26 2014
A099908(a(n)) == 1 mod a(n)^4. - Jonathan Sondow, Nov 24 2015
A034602(PrimePi(a(n))) == 0 mod a(n) and A263882(PrimePi(a(n))) == 0 mod a(n)^2. - Jonathan Sondow, Dec 03 2015

cp's OEIS Frontend

User: Christian Schroeder

A309107 A member of a family of generalizations of van Eck's sequence as defined below.

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A274326 Number of distinct irregular n-gon flexagons.

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A260680 Peaceable coexisting armies of queens: number of inequivalent configurations with maximum number of queens as given in A250000.

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Keywords

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Crossrefs

Extensions

A175000 Number of stable n-celled patterns ("still lifes") in the HighLife (B36/S23) cellular automaton.

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Author

Keywords

Comments

Examples

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Crossrefs

A175002 Number of stable n-celled patterns ("still lifes") in the Maze (B3/S12345) cellular automaton.

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Author

Keywords

Examples

Links

Crossrefs

A175001 Number of stable n-celled patterns ("still lifes") in the Move (a.k.a. Morley; B368/S245) cellular automaton.

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Author

Keywords

Examples

Links

Crossrefs

A098820 Periodicity of entries in the first row of a Laver Table of size 2^n.

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Extensions

A088165 NSW primes: NSW numbers that are also prime.

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PARI

Formula

Extensions

A088164 Wolstenholme primes: primes p such that binomial(2p-1,p-1) == 1 (mod p^4).

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Magma

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