Ken Takusagawa - cp's OEIS frontend

A216761 n * floor(log_2(n)) * floor(log_2(log_2(n))) * floor(log_2(log_2(log_2(n)))) ....

1, 2, 3, 8, 10, 12, 14, 24, 27, 30, 33, 36, 39, 42, 45, 128, 136, 144, 152, 160, 168, 176, 184, 192, 200, 208, 216, 224, 232, 240, 248, 320, 330, 340, 350, 360, 370, 380, 390, 400, 410, 420, 430, 440, 450, 460, 470, 480, 490, 500, 510, 520, 530, 540, 550

Offset: 1

Ken Takusagawa, Sep 15 2012

nonn

a(n) is the product of n, floor(log_2 n), floor (log_2(log_2 n)), ... with the base-2 logs iterated while the result remains greater than unity.
The sum of the reciprocals of a(n) diverge, but extremely slowly.
In particular, the sum of the reciprocals acts like lg* n asymptotically, where lg* x = 0 for x < 2 and lg* 2^x = 1 + lg* x. - Charles R Greathouse IV, Sep 25 2012

			a(0) is the product of 0 numbers, defined to be 1.
a(15) = 15 * floor(log_2 15) * floor(log_2 log_2 15) = 15 * 3 * 1 = 45.
a(17) = 17 * floor(log_2 17) * floor(log_2 log_2 17) * floor(log_2 log_2 log_2 17) = 17 * 4 * 2 * 1 = 136.

Cf. A216762 (ceiling instead of floor).

a = product . map floor . takeWhile (1<) . iterate log_2

Table[prod = 1; s = n; While[s > 1, prod = prod*Floor[s]; s = Log[2, s]]; prod, {n, 60}] (* T. D. Noe, Sep 24 2012 *)

a(n)=my(t=n);n+=1e-9;while(n>2,t*=floor(n=log(n)/log(2)));t \\ Charles R Greathouse IV, Sep 25 2012

A216762 a(n) = n * ceiling(log_2(n)) * ceiling(log_2(log_2(n))) * ceiling(log_2(log_2(log_2(n)))) ....

Original entry on oeis.org

1, 2, 6, 8, 30, 36, 42, 48, 72, 80, 88, 96, 104, 112, 120, 128, 510, 540, 570, 600, 630, 660, 690, 720, 750, 780, 810, 840, 870, 900, 930, 960, 1188, 1224, 1260, 1296, 1332, 1368, 1404, 1440, 1476, 1512, 1548, 1584, 1620, 1656, 1692, 1728, 1764, 1800

Offset: 1

Ken Takusagawa, Sep 15 2012

nonn

a(n) is the product of n, ceiling(log_2(n)), ceiling(log_2(log_2(n))), ... with the base-2 logs iterated while the result remains greater than unity.
The sum of the reciprocals of a(n) diverge, but extremely slowly.
In particular, the sum of the reciprocals acts like lg* n asymptotically, where lg* x = 0 for x < 2 and lg* 2^x = 1 + lg* x. - Charles R Greathouse IV, Sep 25 2012

			a(0) is the product of 0 numbers, defined to be 1.
a(15) = 15 * ceiling(log_2(15)) * ceiling(log_2(log_2(15))) * ceiling(log_2(log_2(log_2(15)))) = 15 * 4 * 2 * 1 = 120.
a(17) = 17 * ceiling(log_2(17)) * ceiling(log_2(log_2(17))) * ceiling(log_2(log_2(log_2(17)))) * ceiling(log_2(log_2(log_2(log_2(17))))) = 17 * 5 * 3 * 2 * 1 = 510.

Cf. A216761 (floor instead of ceiling).

a = product . map ceil . takeWhile (1<) . iterate log_2

Table[prod = 1; s = n; While[s > 1, prod = prod*Ceiling[s]; s = Log[2, s]]; prod, {n, 50}] (* T. D. Noe, Sep 24 2012 *)

a(n)=my(t=n);n-=1e-9;while(n>2,t*=ceil(n=log(n)/log(2)));t \\ Charles R Greathouse IV, Sep 25 2012

A181356 Smallest k such that 2^(2^n) - k is a safe prime.

Original entry on oeis.org

5, 29, 269, 209, 1469, 15449, 36113, 38117, 1093337, 1942289, 10895177, 43644929, 364486013, 718982153, 2356107437

Offset: 2

Ken Takusagawa, Jan 27 2011

These numbers are useful as Diffie-Hellman moduli.

			a(2)=5 because 2^(2^2) - 5 = 11 is the largest safe prime less than 16.

Mersenne Forum, CRUS-like sieving challenge

Cf. A058220, A005385, A057821, A335313.

a(n) = {k = 1; pow2 = 2^(2^n); while (! (isprime(pow2 - k) && isprime((pow2 - k - 1)/2)), k +=2;); k;} \\ Michel Marcus, Aug 16 2013

a(14) from Ken Takusagawa, May 31 2011
a(15) from Gary Barnes, Oct 26 2011
a(16) from Mark Andreas, Jan 23 2023

A124111 Primes with record large values of the second smallest positive primitive root.

Original entry on oeis.org

5, 7, 11, 23, 31, 191, 409, 1151, 2161, 2689, 3361, 4729, 9601, 14281, 23209, 31249, 33049, 55441, 71761, 110881, 2116921, 3384481, 5109721, 222855361, 288940681, 567719161, 831143041, 899978641

Offset: 1

Ken Takusagawa, Nov 27 2006

Sequence begins at the first prime with at least 2 primitive roots, namely 5.

			The smallest two positive primitive roots of a(26)=567719161 are 43 and 172. No integer less than 567719161 has a second root greater than or equal to 172.

Cf. A002230.

A113767 Values of the exponents in A084435.

Original entry on oeis.org

0, 1, 2, 1, 5, 1, 1, 29, 3, 37, 31, 227, 835, 115, 7615, 6071, 218431

Offset: 1

Ken Takusagawa, Jan 19 2006

If A084435 ever hits a Sierpinski number, the sequence is finite.

a(18) > 10^6. - Harsh R. Aggarwal, Sep 14 2024

Eric Weisstein's World of Mathematics, Sierpinski Number of the Second Kind.

Cf. A084435, A113334, A121799.

a=2;for(x=1,16,for(k=0,1000000,p=2^k*a+1;if(isprime(p),print(k);break));a=p)

a(16)-a(17) from Harsh R. Aggarwal, Apr 11 2018

A096393 Least primitive root of n-th Mersenne prime.

Original entry on oeis.org

2, 3, 3, 3, 17, 3, 3, 7, 37, 3, 3, 43, 3, 5, 5, 3, 3, 23, 3, 7

Offset: 1

Ken Takusagawa, Aug 05 2004

			a(5)=17 because 17 is the least primitive root of the 5th Mersenne prime, 8191.

Joerg Arndt, Matters Computational (The Fxtbook)
GIMPS, Great Internet Mersenne Prime Search
S. S. Wagstaff, Jr., The Cunningham Project

Cf. A000668, A000043.

a(16), a(18)-a(20) from Max Alekseyev, Feb 07 2024

A090659 Odd composites with increasing proportion of nontrivial non-witnesses of compositeness by the Miller-Rabin primality test.

Original entry on oeis.org

25, 91, 703, 1891, 12403, 38503, 79003, 88831, 146611, 188191, 218791, 269011, 286903, 385003, 497503, 597871, 736291, 765703, 954271, 1056331, 1314631, 1869211, 2741311, 3270403, 3913003, 4255903, 4686391, 5292631, 6186403, 6969511, 8086231, 9080191

Offset: 1

Ken Takusagawa, Dec 14 2003

nonn

Rabin has shown that the proportion has an upper bound of 0.25. If the trivial non-witnesses are counted, this upper bound is reached at 9. If the conjecture is true that the later terms are always the product of two primes p and (2*p-1), then the sequence continues 188191 218791 269011 286903 385003 497503 597871 736291 765703 954271 1056331 1314631 1869211 2741311 3270403 3913003 4255903 4686391 5292631.
Dickson's conjecture implies that this sequence is infinite. Can this be proved unconditionally? - Charles R Greathouse IV, Mar 10 2011
Higgins' conjecture 2 is implied by his conjecture 1, which is true by the general form of the number of non-witnesses of an odd number. - Charles R Greathouse IV, Mar 10 2011

			25 has 2 nontrivial non-witnesses (NTNW), namely (7,18), for a proportion of 2/22=0.0909. The denominator is 22 because the non-witnesses are selected from 2..23 (as 1 and 24 are trivial non-witnesses).
49 has 4 NTNW, namely (18,19,30,31) for a proportion of 4/46=0.0870. This is a smaller proportion than 0.0909 for 25.
91=7*13 has 16 NTNW in the range [2..89], namely [9, 10, 12, 16, 17, 22, 29, 38, 53, 62, 69, 74, 75, 79, 81, 82], for a proportion of 16/88=0.182. It also has two trivial non-witnesses 1 and 90, which are not counted. The next integer with a higher proportion is 703, with 160 nontrivial non-witnesses and proportion 0.229.

Charles R Greathouse IV, Table of n, a(n) for n = 1..5411
Brian C. Higgins, The Rabin-Miller Primality Test: Some Results on the Number of Non-witnesses to Compositeness, ALLEMO Spring 1996 Meeting at IUP, Proceedings Volume 1.
S. Narayanan, Improving the Speed and Accuracy of the Miller-Rabin Primality Test, MIT PRIMES-USA, 2015.
Michael O. Rabin, Probabilistic algorithm for testing primality, Journal of Number Theory 12:1 (1980), pp. 128-138.

Subsequence of A141768.

Extended and edited by Charles R Greathouse IV, Mar 09 2011

A057962 Number of points (x,y) in square lattice with (x-1/2)^2+(y-1/2)^2 <= n.

Original entry on oeis.org

4, 12, 16, 24, 32, 44, 52, 60, 68, 76, 80, 88, 96, 112, 120, 124, 140, 148, 156, 164, 172, 180, 188, 192, 208, 216, 232, 240, 248, 256, 268, 276, 284, 300, 308, 316, 332, 348, 360, 368, 376, 384, 392, 400, 408, 424, 432, 440, 448, 460, 468, 484, 492, 500

Offset: 1

Ken Takusagawa, Oct 15 2000

Always a multiple of 4. Useful for rasterizing circles.

			a(2)=12 because (-1,0); (-1,1); (0,-1); (0,0); (0,1); (0,2); (1,-1); (1,0); (1,1); (1,2); (2,0); (2,1) are covered by any disc of radius between sqrt(2.5) and sqrt(4.5) and centered at (0.5,0.5).

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 106.

Cf. A057961, A004018, A004020. Partial sums of A005883.

A057961 Number of points in square lattice covered by a disc centered at (0,0) as its radius increases.

Original entry on oeis.org

1, 5, 9, 13, 21, 25, 29, 37, 45, 49, 57, 61, 69, 81, 89, 97, 101, 109, 113, 121, 129, 137, 145, 149, 161, 169, 177, 185, 193, 197, 213, 221, 225, 233, 241, 249, 253, 261, 277, 285, 293, 301, 305, 317, 325, 333, 341, 349, 357, 365, 373, 377, 385, 401, 405, 421

Offset: 1

Ken Takusagawa, Oct 15 2000

Useful for rasterizing circles.

Conjecture: the number of lattice points in a quadrant of the disk is equal to A000592(n-1). - L. Edson Jeffery, Feb 10 2014

			a(2)=5 because (0,0); (0,1); (0,-1); (1,0); (-1,0) are covered by any disc of radius between 1 and sqrt(2).

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 106.

T. D. Noe, Table of n, a(n) for n=1..1000
L. Edson Jeffery, Illustration of first few terms.

Cf. A004018, A004020, A005883, A057962. Distinct terms of A057655.

Cf. A000404, A001481, A232499.

max = 100; A001481 = Select[Range[0, 4*max], SquaresR[2, #] != 0 &]; Table[SquaresR[2, A001481[[n]]], {n, 1, max}] // Accumulate (* Jean-François Alcover, Oct 04 2013 *)

A045544 Odd values of n for which a regular n-gon can be constructed by compass and straightedge.

Original entry on oeis.org

3, 5, 15, 17, 51, 85, 255, 257, 771, 1285, 3855, 4369, 13107, 21845, 65535, 65537, 196611, 327685, 983055, 1114129, 3342387, 5570645, 16711935, 16843009, 50529027, 84215045, 252645135, 286331153, 858993459, 1431655765, 4294967295

Offset: 1

Ken Takusagawa

If there are no more Fermat primes, then 4294967295 is the last term in the sequence.
From Daniel Forgues, Jun 17 2011: (Start)
The 31 = 2^5 - 1 terms of this sequence are the nonempty products of distinct Fermat primes. The 5 known Fermat primes are in A019434.
Prepending the empty product, i.e., 1, to this sequence gives A004729.
The initial term for this sequence is thus a(1) (offset=1), since a(0) should correspond to the omitted empty product, term a(0) of A004729.
Rows 1 to 31 of Sierpiński's triangle, if interpreted as a binary number converted to decimal (A001317), give a(1) to a(31). (End)

Albert H. Beiler, Recreations in the theory of numbers, New York, Dover, (2nd ed.) 1966. See Table 73 at pp. 181-182.

Wilfrid Keller, Prime factors k.2^n + 1 of Fermat numbers F_m.
OEIS Wiki, Constructible odd-sided polygons.
OEIS Wiki, Sierpinski's triangle.

Cf. A019434. Essentially same as A004729.
Coincides with A001317 for the first 31 terms only. - Robert G. Wilson v, Dec 22 2001
Cf. A053576.

Union[Times@@@Rest[Subsets[{3,5,17,257,65537}]]] (* Harvey P. Dale, Sep 20 2011 *)

Each term is the product of distinct odd Fermat primes.

Sum_{n>=1} 1/a(n) = -1 + Product_{n>=1} (1+1/A019434(n)) = 0.7007354948... >= 1003212011/1431655765 = sigma(2^32-1)/(2^32-1) - 1, with equality if there are only five Fermat primes (A019434). - Amiram Eldar, Jan 22 2022

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User: Ken Takusagawa

A216761 n * floor(log_2(n)) * floor(log_2(log_2(n))) * floor(log_2(log_2(log_2(n)))) ....

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A216762 a(n) = n * ceiling(log_2(n)) * ceiling(log_2(log_2(n))) * ceiling(log_2(log_2(log_2(n)))) ....

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A181356 Smallest k such that 2^(2^n) - k is a safe prime.

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A124111 Primes with record large values of the second smallest positive primitive root.

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A113767 Values of the exponents in A084435.

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A096393 Least primitive root of n-th Mersenne prime.

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A090659 Odd composites with increasing proportion of nontrivial non-witnesses of compositeness by the Miller-Rabin primality test.

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Crossrefs

Extensions

A057962 Number of points (x,y) in square lattice with (x-1/2)^2+(y-1/2)^2 <= n.

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A057961 Number of points in square lattice covered by a disc centered at (0,0) as its radius increases.

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