Jacob Vecht - cp's OEIS frontend

A309312 Areas of Heronian triangles that uniquely define the lengths of the triangle's sides.

6, 30, 42, 54, 66, 72, 114, 132, 150, 198, 204, 234, 294, 306, 378, 390, 408, 468, 486, 522, 570, 594, 648, 684, 726, 744, 750, 798, 816, 876, 984, 1014, 1026, 1110, 1170, 1218, 1254, 1290, 1332, 1350, 1368, 1428, 1494, 1518, 1548, 1554, 1590, 1734, 1770, 1782, 1806, 1938, 1950, 1974

Offset: 1

Jacob Vecht, Jul 22 2019

These are not necessarily primitive Heronian triangles. The corresponding sequence for primitive Heronian triangles is neither a subset nor a superset of this sequence.

			A triangle with integer sides and an area of 42 square units necessarily has sides 7,15,20, so 42 is a term. 12 is not a term because an area of 12 could have sides 6,5,5 or 8,5,5.

Giovanni Resta, Table of n, a(n) for n = 1..10000 (first 500 terms from Jacob Vecht)
Wikipedia, Heronian triangle

Subsequence of A188158.

Corrected 37th term "1248" to "1254".

A249836 Numbers n for which tan(n) > n.

Original entry on oeis.org

1, 260515, 37362253, 122925461, 534483448, 3083975227, 902209779836, 74357078147863, 214112296674652, 642336890023956, 18190586279576483, 248319196091979065, 1108341089274117551, 118554299812338354516058, 1428599129020608582548671, 4285797387061825747646013

Offset: 1

Jacob Vecht, Nov 07 2014

nonn

Supersequence of A079332, hence 3083975227 and 214112296674652 are members. - Charles R Greathouse IV, Nov 07 2014
This sequence consists of all positive-valued terms of A088306. (Of the first 1000 terms in A088306, 518 are positive.) - Jon E. Schoenfield, Nov 07 2014
From Daniel Forgues, May 27 2015, Jun 12 2005: (Start)
Numbers n for which tanc(n) > 1, where tanc(n) = tan(n)/n, tanc(0) = 1, where n are radians; cf. Weisstein link.
It is an open problem whether tan(n) > n for infinitely many integer n.
Jan Kristian Haugland found a(3) = 37362253, Bob Delaney found a(6) = 3083975227.
For n <= tan(n) < n+1, or floor(tan(n)) = n, we get a fixed point of the iterated floor(tan(n)). Currently, the only known fixed points are 0 and 1. (Cf. A258024.)
It is proved that |tan n| > n for infinitely many n, and that tan n > n/4 for infinitely many n. (Bellamy, Lagarias, Lazebnik) (End)
Since tan(n) has a transcendental period, namely Pi, it seems very likely that not only tan(n) > n for infinitely many integers n, but also that tan(n) > kn for infinitely many integers n, for any integer k. It even seems likely that not only n < tan(n) < n+1 for infinitely many integers n (not just for n = 1), but also that kn < tan(n) < kn + 1 for infinitely many integers n, for any integer k. It seems that we are bound to stumble upon the requisite positive delta such that n mod Pi = Pi/2 - delta. - Daniel Forgues, Jun 15 2015
It appears that we need {n / Pi} = 0.5 - delta, with delta < k/n, for some k, where {.} denotes the fractional part: we have, 260515/Pi = 82924.49999917..., 37362253/Pi = 11892774.4999999915... etc. - Daniel Forgues, Jun 18 2015, edited by M. F. Hasler, Aug 19 2015
Indeed, from the graph of the function we see that tan(n) > n for numbers of the form n = (m + 1/2)*Pi - epsilon (i.e., n/Pi = m + 1/2 - epsilon/Pi) with small epsilon > 0, for which tan(n) = tan((m + 1/2)*Pi - epsilon) = tan(Pi/2-epsilon) ~ 1/epsilon, using tan(Pi/2-x) = sin(Pi/2-x)/cos(Pi/2-x) = cos(x)/sin(x) ~ 1/x as x -> 0. Thus tan(n) > n if epsilon < 1/n, or delta = epsilon/Pi < k/n with k = 1/Pi. - M. F. Hasler, Aug 19 2015
The first prime term is a(28). - Jacob Vecht, Aug 09 2020

			tan(1) = 1.557... > 1 so 1 is a member.

Michel Marcus, Table of n, a(n) for n = 1..518 (from A088306 b-file).
David P. Bellamy, Jeffrey C. Lagarias, and Felix Lazebnik, Proposed Problem: Large Values of Tan n
David P. Bellamy, Jeffrey C. Lagarias, Felix Lazebnik and Stephen M. Gagola, Jr., Large Values of Tangent: 10656, The American Mathematical Monthly, Vol. 106, No. 8 (Oct. 1999), pp. 782-784.
Matt Parker, What is the biggest tangent of a prime?, Channel Stand-up Maths, YouTube, Aug 19 2020.
Eric Weisstein's World of Mathematics, Tanc Function.

Subsequence of positive terms of A088306. Supersequence of A079332.

Cf. A000503(n) = floor(tan(n)).

a249836[n_Integer] := Select[Range[n], Tan[#] > # &]; a249836[270000] (* Michael De Vlieger, Nov 23 2014 *)

is(n)=tan(n)>n \\ Charles R Greathouse IV, Nov 07 2014

log(a(n)) / n ~ Pi, conjectured. - M. F. Hasler, Sep 10 2020 [corrected thanks to Vaclav Kotesovec, Feb 22 2021]

A249780 Product of lowest and highest prime factors of 2^n-1.

Original entry on oeis.org

9, 49, 15, 961, 21, 16129, 51, 511, 93, 2047, 39, 67092481, 381, 1057, 771, 17179607041, 219, 274876858369, 123, 2359, 2049, 8388607, 723, 55831, 24573, 1838599, 381, 486737, 993, 4611686014132420609, 196611, 4196353, 393213, 3810551, 327, 137438953471, 1572861, 849583, 185043

Offset: 2

Jacob Vecht, Nov 05 2014

nonn

			The lowest and higest prime factors of 2^6-1 are 3 and 7, so A(6) = 21

Chai Wah Wu, Table of n, a(n) for n = 2..200

a:= proc(n) local F; F:= numtheory:-factorset(2^n-1); min(F)*max(F) end proc:
seq(a(n),n=2..50); # Robert Israel, Nov 05 2014

plhpf[n_]:=Module[{fn=FactorInteger[n]},fn[[1,1]]fn[[-1,1]]]; Table[plhpf [2^n-1],{n,2,40}] (* Harvey P. Dale, May 23 2020 *)

for(n=2, 50, p=2^n-1; print1(factor(p)[1, 1]*factor(p)[#factor(p)[, 1], 1], ", ")) \\ Derek Orr, Nov 05 2014

from sympy import primefactors
A249780_list, x = [], 1
for n in range(2,10):
    x = 2*x + 1
    p = primefactors(x)
    A249780_list.append(max(p)*min(p)) # Chai Wah Wu, Nov 05 2014

a(n) = A005420(n) * A049479(n)

More terms from Derek Orr, Nov 05 2014

A249627 Product of the lowest and highest prime factors of n-th repunit A002275(n).

Original entry on oeis.org

121, 111, 1111, 11111, 111, 1111111, 1507, 1001001, 100001, 11111111111, 29703, 14064697609, 10000001, 8718483, 64705883, 11111111111111111, 1001001, 1234567901234567900987654320987654321, 307571, 32516067, 5645629, 123456790123456790123454320987654320987654321

Offset: 2

Jacob Vecht, Nov 02 2014

			Highest and lowest prime factors of 111111 are 3 and 37, so the sixth term is 111.

Chai Wah Wu, Table of n, a(n) for n = 2..322
P. De Geest, Repunits and Their Factors

Cf. A002275.

plh[n_]:=Module[{fi=FactorInteger[FromDigits[PadRight[{},n,1]]]},fi[[1,1]]fi[[-1,1]]]; Array[plh,30,2] (* Harvey P. Dale, May 04 2021 *)

for(n=2,20,p=(10^n-1)/9;print1(factor(p)[1,1]*factor(p)[#factor(p)[,1],1],", ")) \\ Derek Orr, Nov 02 2014

from sympy import factorint
A249627_list = [min(fs)*max(fs) for fs in (factorint((10**n-1)//9) for n in range(2,31))] # Chai Wah Wu, Nov 04 2019

More terms from Derek Orr, Nov 02 2014

A117853 Decimal expansion of 2^30402457-1, the 43th Mersenne prime A000668(43).

Original entry on oeis.org

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

Offset: 9152052

Jacob Vecht, May 02 2006

The decimal expansion of 2^30402457-1. This is the 43rd known Mersenne prime, found at Dec 15, 2005 by the GIMPS project / Curtis Cooper and Steven Boone. The number has 9152052 digits; as a string, it occupies 9.24 MB.

			A000668(43) = 3154164756188460809363030286645451701265196562623...7411652943871.

Landon Curt Knoll, 2^30402457-1 is prime.
George Woltman, GIMPS GIMPS Home Page.

Cf. A000043 (Mersenne exponents), A000668.

RealDigits[10^N[30402457Log[10, 2] - 9152051, 111]][[1]]

Edited by Georg Fischer, Jul 19 2021

A117805 Start with 3. Square the previous term and subtract it.

Original entry on oeis.org

3, 6, 30, 870, 756030, 571580604870, 326704387862983487112030, 106735757048926752040856495274871386126283608870, 11392521832807516835658052968328096177131218666695418950023483907701862019030266123104859068030

Offset: 0

Jacob Vecht, Apr 29 2006

The next term is too large to include.
a(n) = A005267(n+1)+1. - R. J. Mathar, Apr 22 2007. This is true by induction. - M. F. Hasler, May 04 2007<
For any a(0) > 2, the sequence a(n) = a(n-1) * (a(n-1) - 1) gives a constructive proof that there exists integers with at least n + 1 distinct prime factors, e.g., a(n). As a corollary, this gives a constructive proof of Euclid's theorem stating that there are an infinity of primes. - Daniel Forgues, Mar 03 2017

			Start with 3; 3^2 - 3 = 6; 6^2 - 6 = 30; etc.

Cf. A007018.

f:=proc(n) option remember; if n=0 then RETURN(3); else RETURN(f(n-1)^2-f(n-1)); fi; end;

k=3;lst={k};Do[k=k^2-k;AppendTo[lst,k],{n,9}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 19 2010 *)
RecurrenceTable[{a[0]==3, a[n]==a[n-1]*(a[n-1] - 1)}, a, {n, 0, 10}] (* Vaclav Kotesovec, Dec 17 2014 *)
NestList[#^2-#&,3,10] (* Harvey P. Dale, Oct 11 2023 *)

a(0) = 3, a(n) = (a(n-1))^2 - a(n-1).

a(n) ~ c^(2^n), where c = 2.330283023986140936420341573975137247354077600883596774023675490739568138... . - Vaclav Kotesovec, Dec 17 2014

A117293 Mersenne primes written in binary.

Original entry on oeis.org

11, 111, 11111, 1111111, 1111111111111, 11111111111111111, 1111111111111111111, 1111111111111111111111111111111, 1111111111111111111111111111111111111111111111111111111111111

Offset: 1

Jacob Vecht, Apr 23 2006

The number of bits in a(n) is equal to A000043(n). - Omar E. Pol, Feb 22 2008

			The third Mersenne prime is 31, i.e., 11111 in binary.

Amiram Eldar, Table of n, a(n) for n = 1..14
Wikipedia, Mersenne prime.

Cf. A000043, A000668, A007088.

Map[(10^# - 1)/9 &, MersennePrimeExponent[Range[9]]] (* Amiram Eldar, Oct 22 2024 *)

apply(p->fromdigits(binary(p)), select(isprime, vector(20,n,2^prime(n)-1))) \\ Charles R Greathouse IV, Sep 14 2016

a(n) = A007088(A000668(n)). - Amiram Eldar, Oct 22 2024

cp's OEIS Frontend

User: Jacob Vecht

A309312 Areas of Heronian triangles that uniquely define the lengths of the triangle's sides.

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A249836 Numbers n for which tan(n) > n.

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A249780 Product of lowest and highest prime factors of 2^n-1.

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A249627 Product of the lowest and highest prime factors of n-th repunit A002275(n).

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A117853 Decimal expansion of 2^30402457-1, the 43th Mersenne prime A000668(43).

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A117805 Start with 3. Square the previous term and subtract it.

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Maple

Mathematica

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A117293 Mersenne primes written in binary.

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Mathematica