David J. Rusin - cp's OEIS frontend

User: David J. Rusin

David J. Rusin has authored 4 sequences.

A108310 Successive maxima of log(n#)/n where n# is the product of the primes less than n.

2, 3, 5, 7, 13, 19, 43, 47, 73, 103, 107, 109, 113, 199, 283, 467, 661, 887, 1063, 1069, 1097, 1103, 1109, 1123, 1129, 1303, 1307, 1321, 1327, 1621, 1627, 2803, 3931, 3947, 4273, 4289, 4297, 5867, 5869, 5881, 6373, 6379, 9439, 9473, 9479, 9497, 9551, 9859

Offset: 1

David J. Rusin, Jun 29 2005

Every entry must be a prime.
Note that log(n#)=theta(n) (the Chebyshev function) for which bounds are known (e.g. Rosser and Schoenfeld have an estimate |theta(n)-n| < n/(40 log n).) In particular, log(n#)/n tends to 1, which allows a proof of the Prime Number Theorem. I suspect log(n#) can be greater than n for some n, which would make the sequence finite, but I do not know an example of such an n. (When n=30337841, 0.9999 < log(n#)/n < 1.)
When n=3745619057, 0.99999312926590387432389345880435140945170798255514 < log(n#)/n < 1. - Robert G. Wilson v, Jul 01 2005
Computational experiments show that it may be true that n > log(n#) for all n. In fact, it appears that, for any k, n > log(n#) + k*log(n) except for a finite number of small primes. For k=1, only 5, 7 and 19 are the exceptional n. This inequality is still consistent with 1 being the limiting value of log(n#)/n. - T. D. Noe, Apr 17 2006
Apparently in the long run (n-theta(n))/(Li(n)-Pi(n)) goes to log(n), so if Li(n)Martin Raab, May 13 2008
Sequence is finite since psi(x) - x is greater than sqrt x * log log log x infinitely often, and hence theta(x) > x infinitely often [but theta(x) - x = o(x), see Rosser & Schoenfeld]. See Hardy & Littlewood section 5. - Charles R Greathouse IV, Aug 02 2012

			13 follows 7 because log(7#)/7 = log(210)/7 = 0.7638, while log(8#)/8 and so on are smaller but log(13#)/13= 0.7931 is larger. A larger entry is 3445943 since log(n#)<0.99978 n for smaller n but log(3445943#)=3445185.8713457=(0.999780284)(3445943).

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
G. H. Hardy and J. E. Littlewood, Contributions to the theory of the Riemann Zeta-Function and the theory of the distribution of primes, Acta Mathematica 41 (1916), pp. 119-196.
J. Barkley Rosser and Lowell Schoenfeld, Sharper bounds for the Chebyshev functions theta(x) and psi(x), Collection of articles dedicated to Derrick Henry Lehmer on the occasion of his seventieth birthday. Math. Comp. 29 (1975), 243-269.

Cf. A034386, A215013.

A:=[]:b:=0:S:=0:n:=1: while true do n:=nextprime(n): S:=S+evalf(log(n)): if S>b*n then A:=[op(A),n]: b:= S/n: fi: od: #Program must be terminated manually! Array "A" is the sequence.

lmt = slp = 0; lst = {}; Do[p = Prime[n]; slp = slp + N[Log[p], 12]; If[slp/p > lmt, lmt = slp/p; AppendTo[lst, p]], {n, 1224}]; lst (* Robert G. Wilson v, Jul 01 2005 *)

r=th=0; forprime(p=2, 1e6, th+=log(p); t=th/p; if(t>r, r=t; print1(p", "))) \\ Charles R Greathouse IV, Dec 17 2014

More terms from Robert G. Wilson v, Jul 01 2005

A028419 Congruence classes of triangles which can be drawn using lattice points in n X n grid as vertices.

Original entry on oeis.org

0, 1, 8, 29, 79, 172, 333, 587, 963, 1494, 2228, 3195, 4455, 6050, 8032, 10481, 13464, 17014, 21235, 26190, 31980, 38666, 46388, 55144, 65131, 76449, 89132, 103337, 119184, 136757, 156280, 177796, 201430, 227331, 255668, 286606, 320294, 356884, 396376, 439100, 485427, 535049, 588457, 645803

Offset: 0

David J. Rusin

nonn

Ron Knott, Table of n, a(n) for n = 0..60
D. Rusin, Lattice Problem (Triangles) [Dead link]
D. Rusin, Lattice Problem (Triangles) [Cached copy]

Cf. A028492, A189979, A190021, A190022, A189978, A190313, A189978, A190313.

a:=proc(n) local TriangleSet,i,j,k,l,A,B,C; TriangleSet:={}: for i from 0 to n do for j from 0 to n do for k from 0 to n do for l from 0 to n do A:=i^2+j^2: B:=k^2+l^2: C:=(i-k)^2+(j-l)^2: if A^2+B^2+C^2<>2*(A*B+B*C+C*A) then TriangleSet:={op(TriangleSet),sort([sqrt(A),sqrt(B),sqrt(C)])}: fi: od: od: od: od: return(nops(TriangleSet)); end: # Martin Renner, May 03 2011

More terms from Chris Cole (chris(AT)questrel.com), Jun 28 2003

a(36)-a(39) from Martin Renner, May 08 2011

A028492 Similarity classes of triangles which can be drawn using the lattice points in an n X n grid for vertices.

Original entry on oeis.org

0, 1, 6, 20, 55, 119, 229, 402, 667, 1019, 1536, 2216, 3049, 4168, 5546, 7203, 9278, 11755, 14597, 18054, 22138, 26625, 31958, 38120, 44821, 52764, 61663, 71204, 82250, 94479, 107680, 122740, 139224, 156699, 176450, 198176, 220883

Offset: 0

David J. Rusin

nonn

Sean A. Irvine, Java program (github)
D. Rusin, Lattice Problem (Triangles) [Broken link]
D. Rusin, Lattice Problem (Triangles) [Cached copy]

Cf. A028419.

Extended by Ray Chandler, May 03 2011

A048290 Numbers m that divide Sum_{k=1..m} phi(k).

Original entry on oeis.org

1, 2, 5, 6, 16, 25, 36, 249, 617, 1296, 13763, 76268, 189074, 783665, 1102394, 3258466, 3808854, 7971034, 15748051, 27746990, 41846733, 153673168, 195853251, 302167272, 402296412, 732683468, 807656448, 844492262, 848152352, 1122039882, 2258200198, 2438160726

Offset: 1

David J. Rusin

The odd terms of this sequence and A063986 are the same. - Jud McCranie, Jun 26 2005

			Euler sums are *1*, *2*, 4, 6, *10*, *12*, ..., *80*, ..., *510624*,... for n=1, 2, 3, 4, 5, 6, ..., 16, ...., 1296, ...

Donovan Johnson, Table of n, a(n) for n = 1..37 (terms < 10^12)
Edward A. Bender, Oren Patashnik and Howard Rumsey, Jr., Pizza Slicing, Phi's and the Riemann Hypothesis, American Mathematical Monthly, Vol. 101 (1994), pp. 307-317.
D. Rusin, Euler phi function

Cf. A000010, A002088. See A063986 for n divides Sum_{k=1..n} k-phi(k).

s = 0; Do[s = s + EulerPhi[n]; If[IntegerQ[s/n], Print[n]], {n, 1, 10^8}]

list(lim)=my(v=List(),s); for(k=1,lim, s+=eulerphi(k); if(s%k==0, listput(v, k))); Vec(v) \\ Charles R Greathouse IV, Feb 07 2017

Sum_{k=1..m} phi(k) is about (3/Pi^2)*m^2 [cf. A002088, first formula].

Not obviously infinite; rough heuristics predict about 3/2 log(N) terms less than N, log(N) even ones and log(N)/2 odd ones.

10 more terms computed by Dean Hickerson
One more term from Robert G. Wilson v, Sep 07 2001
More terms from Naohiro Nomoto, Mar 22 2002
5 more terms from Jud McCranie, Jun 21 2005

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User: David J. Rusin

A108310 Successive maxima of log(n#)/n where n# is the product of the primes less than n.

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A028419 Congruence classes of triangles which can be drawn using lattice points in n X n grid as vertices.

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A028492 Similarity classes of triangles which can be drawn using the lattice points in an n X n grid for vertices.

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A048290 Numbers m that divide Sum_{k=1..m} phi(k).

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Mathematica

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