Hollie L. Buchanan II - cp's OEIS frontend

A182020 Numbers that are squares and have a majority of one digit.

0, 1, 4, 9, 100, 121, 144, 225, 400, 441, 484, 676, 900, 1444, 10000, 11881, 14161, 17161, 22201, 27225, 28224, 29929, 40000, 44944, 46656, 55225, 66564, 69696, 88804, 90000, 97969, 160000, 250000, 360000, 490000

Offset: 1

Hollie L. Buchanan II, Apr 06 2012

			165^2=27225 and more than half its digits are 2.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

These are the squares of A060811.

Select[Range[0,700]^2,Max[DigitCount[#]]>IntegerLength[#]/2&] (* Harvey P. Dale, Nov 02 2015 *)

is(n)=if(issquare(n),my(v=vecsort(eval(Vec(Str(n)))), m=v[(#v+1)\2]); sum(i=1,#v,v[i]==m)*2>#v,0) \\ Charles R Greathouse IV, Apr 07 2012

A086119 Numbers of the form p^3 + q^3, p, q primes.

Original entry on oeis.org

16, 35, 54, 133, 152, 250, 351, 370, 468, 686, 1339, 1358, 1456, 1674, 2205, 2224, 2322, 2540, 2662, 3528, 4394, 4921, 4940, 5038, 5256, 6244, 6867, 6886, 6984, 7110, 7202, 8190, 9056, 9826, 11772, 12175, 12194, 12292, 12510, 13498, 13718, 14364

Offset: 1

Hollie L. Buchanan II, Jul 11 2003

nonn

			133 belongs to the sequence because it can be written as 2^3 + 5^3.

T. D. Noe, Table of n, a(n) for n=1..10000

Cf. A001235, A003325, A045636, A045699, A086120, A086121.

sumList[x_List, y_List] := Module[{t = {}}, Do[t = Union[t, x[[i]] + y], {i, Length[x]}];  t]; nn = 10; Select[sumList[Prime[Range[nn]]^3, Prime[Range[nn]]^3], # < Prime[nn]^3 &]

More terms from Alexander Adamchuk, Nov 10 2006

A083659 Denominator of fraction equal to the continued fraction [p(n); p(n-1),...,5,3,2].

Original entry on oeis.org

1, 2, 7, 37, 266, 2963, 38785, 662308, 12622637, 290982959, 8451128448, 262275964847, 9712661827787, 398481410904114, 17144413330704689, 806185907954024497, 42744997534894003030, 2522761040466700203267, 153931168466003606402317, 10315911048262708329158506

Offset: 1

Hollie L. Buchanan II, Jun 14 2003

			The 5th term is 266 because 11+1/(7+1/(5+1/(3+1/2))) = 2963/266.

Alois P. Heinz, Table of n, a(n) for n = 1..351

Cf. A036247, A036248.

b:= proc(n) option remember;
      `if`(n=1, 0, 1/b(n-1)) + ithprime(n)
    end:
a:= n-> denom(b(n)):
seq(a(n), n=1..20);  # Alois P. Heinz, Nov 03 2018

Table[Denominator[FromContinuedFraction[Prime[Range[n, 1, -1]]]], {n, 1, 20}]

One term corrected and extended by Alois P. Heinz, Nov 03 2018

A084962 Iterations of the Fibonacci sequence starting at 6.

Original entry on oeis.org

6, 8, 21, 10946

Offset: 0

Hollie L. Buchanan II, Jun 14 2003

nonn

The next term, a(4) = 1.695... * 10^2287, has 2288 digits and is too large to display.

This sequence is of interest because the sequences with this recurrence and a(0) in {0, 1, 2, 3, 4} all converge to 1 and the sequence with a(0) = 5 is constant.

			a(3) = Fibonacci(a(2)) = Fibonacci(21) = 10946.

Alonso del Arte, Table of a(n) for n=0..4, with digits of a(4) grouped in hundreds

Cf. A000045, A007097, A010716 (starting with 5), A084963 (starting with 7).

a:= proc(n) option remember; `if`(n=0, 6,
      (<<0|1>, <1|1>>^a(n-1))[1,2])
    end:
seq(a(n), n=0..4);  # Alois P. Heinz, May 09 2020

fibonaccieth6[m_] := Module[{ex = 6}, Do[ex = Fibonacci[ex], {m}]; ex] Table[fibonaccieth6[m], {m, 0, 4}]
NestList[Fibonacci[#] &, 6, 4] (* Alonso del Arte, Apr 30 2020 *)

val fiboLimited: LazyList[Int] = 0 #:: 1 #:: fiboLimited.zip(fiboLimited.tail).map { n => n._1 + n._2 }
def fibonaccieth(start: Int): LazyList[Int] = LazyList.iterate(start)(fiboLimited)
fibonaccieth(6).takeWhile( > 0).toList // _Alonso del Arte, Apr 30 2020

a(0) = 6, a(n) = Fibonacci(a(n-1)) for n>0.

A086120 Natural numbers of the form p^3 - q^3, where p and q are primes.

Original entry on oeis.org

19, 98, 117, 218, 316, 335, 866, 988, 1206, 1304, 1323, 1854, 1946, 2072, 2170, 2189, 2716, 3582, 4570, 4662, 4788, 4886, 4905, 5308, 5402, 5528, 6516, 6734, 6832, 6851, 7254, 9970, 10586, 10836, 11824, 12042, 12140, 12159, 12222, 17530, 17624, 18268

Offset: 1

Hollie L. Buchanan II, Jul 11 2003

nonn

To find all differences p^3 - q^3 less than N, it is required that all primes p and q up to sqrt(N/6) be tested.

			117 belongs to the sequence because it can be written as 5^3 - 2^3.

T. D. Noe, Table of n, a(n) for n=1..10000

Cf. A086119, A086121. Also see A045636, A045699.

sumList[x_List, y_List] := (punchline = {}; Do[punchline = Union[punchline, x[[i]] + y], {i, Length[x]}]; punchline); posPart[x_List] := (punchline = {}; Do[If[x[[i]] > 0, punchline = Union[punchline, {x[[i]]}]], {i, Length[x]}]; punchline); posPart[sumList[Prime[Range[10]]^3, - Prime[Range[10]]^3]]
nn=10^5; Union[Reap[Do[n=Prime[i]^3-Prime[j]^3; If[n<=nn, Sow[n]], {i,PrimePi[Sqrt[nn/6]]}, {j,i-1}]][[2,1]]] (* T. D. Noe, Oct 04 2010 *)
With[{upto=20000},Select[Abs[#[[1]]-#[[2]]]&/@Subsets[Prime[ Range[ Sqrt[ upto/6]]]^3,{2}]//Union,#<=upto&]] (* Harvey P. Dale, Dec 10 2017 *)

Corrected by T. D. Noe, Oct 04 2010

A086121 Positive sums or differences of two cubes of primes.

Original entry on oeis.org

16, 19, 35, 54, 98, 117, 133, 152, 218, 250, 316, 335, 351, 370, 468, 686, 866, 988, 1206, 1304, 1323, 1339, 1358, 1456, 1674, 1854, 1946, 2072, 2170, 2189, 2205, 2224, 2322, 2540, 2662, 2716, 3528, 3582, 4394, 4570, 4662, 4788, 4886, 4905, 4921, 4940, 5038

Offset: 1

Hollie L. Buchanan II, Jul 11 2003

nonn

			117 and 133 each belong to the (set) sequence because can be written as 117 = 5^3 - 2^3 and 133 = 5^3 + 2^3.

Hans Havermann and T. D. Noe, Table of n, a(n) for n = 1..20000.

Cf. A086119, A086120. Also see A045636, A045699.

nn=10^6; td=Reap[Do[n=Prime[i]^3-Prime[j]^3; If[n<=nn, Sow[n]], {i,PrimePi[Sqrt[nn/6]]}, {j,i-1}]][[2,1]]; ts=Reap[Do[n=Prime[i]^3+Prime[j]^3; If[n<=nn, Sow[n]], {i,PrimePi[nn^(1/3)]}, {j,i}]][[2,1]]; Union[td,ts] (* T. D. Noe, Oct 04 2010 *)
n = 100; Select[Sort@Flatten@ Table[Prime[i]^3 + (-1)^k Prime[j]^3, {i, n}, {j, i}, {k, 2}], 0 < # < (Prime[n] + 2)^3 - Prime[n]^3 &] (* Ray Chandler, Oct 05 2010 *)

Edited by N. J. A. Sloane, Oct 05 2010 to remove a discrepancy between the terms of the sequence and the b-file. The old Mma program and b-file were wrong.

A084963 a(0) = 7, a(n+1) = Fibonacci(a(n)).

Original entry on oeis.org

7, 13, 233, 2211236406303914545699412969744873993387956988653

Offset: 0

Hollie L. Buchanan II, Jun 14 2003

nonn

Iteration of the Fibonacci sequence.

a(3) = 2.2112... * 10^48.

			a(2) = F(a(1)) = F(13) = 233.

Cf. A000045, A084962.

a:= proc(n) option remember; `if`(n=0, 7,
       (<<0|1>, <1|1>>^a(n-1))[1,2])
    end:
seq(a(n), n=0..3);  # Alois P. Heinz, May 09 2020

a[m_] := Module[{ex = 7}, Do[ex = Fibonacci[ex], {m}]; ex] Table[a[m], {m, 0, 3}]
NestList[Fibonacci[#]&,7,3] (* Harvey P. Dale, Mar 18 2018 *)

A084844 Denominators of the continued fraction n + 1/(n + 1/...) [n times].

Original entry on oeis.org

1, 2, 10, 72, 701, 8658, 129949, 2298912, 46866034, 1082120050, 27916772489, 795910114440, 24851643870041, 843458630403298, 30918112619119426, 1217359297034666112, 51240457936070359069, 2296067756927144738850, 109127748348241605689981

Offset: 1

Hollie L. Buchanan II, Jun 08 2003

The (n-1)-th term of the Lucas sequence U(n,-1). The numerator is the n-th term. Adjacent terms of the sequence U(n,-1) are relatively prime. - T. D. Noe, Aug 19 2004
From Flávio V. Fernandes, Mar 05 2021: (Start)
Also, the n-th term of the n-th metallic sequence (the diagonal through the array A073133, and its equivalents, which is rows formed by sequences beginning with A000045, A000129, A006190, A001076, A052918) as shown below (for n>=1):
   0    1    0    1     0      1  ... A000035
   0   [1]   1    2     3      5  ... A000045
   0    1   [2]   5    12     29  ... A000129
   0    1    3  [10]   33    109  ... A006190
   0    1    4   17   [72]   305  ... A001076
   0    1    5   26   135   [701] ... A052918. (End)

			a(4) = 72 since 4 + 1/(4 + 1/(4 + 1/4)) = 305/72.

Alois P. Heinz, Table of n, a(n) for n = 1..387
Eric Weisstein's World of Mathematics, Lucas Sequence

Cf. A084845 (numerators).

Cf. A000045, A097690, A097691, A117715, A290864 (primes in this sequence).

A084844 :=proc(n) combinat[fibonacci](n, n) end:
seq(A084844(n), n=1..30); # Zerinvary Lajos, Jan 03 2007

myList[n_] := Module[{ex = {n}}, Do[ex = {ex, n}, {n - 1}]; Flatten[ex]] Table[Denominator[FromContinuedFraction[myList[n]]], {n, 1, 20}]
Table[s=n; Do[s=n+1/s, {n-1}]; Denominator[s], {n, 20}] (* T. D. Noe, Aug 19 2004 *)
Table[Fibonacci[n, n], {n, 1, 20}] (* Vladimir Reshetnikov, May 07 2016 *)
Table[DifferenceRoot[Function[{y,m},{y[2+m]==n*y[1+m]+y[m],y[0]==0,y[1]==1}]][n],{n,1,20}] (* Benedict W. J. Irwin, Nov 03 2016 *)

from sympy import fibonacci
def a(n):
    return fibonacci(n, n)
print([a(n) for n in range(1, 31)]) # Indranil Ghosh, Aug 12 2017

a(n) = (s^n - (-s)^(-n))/(2*s - n), where s = (n + sqrt(n^2 + 4))/2. - Vladimir Reshetnikov, May 07 2016
a(n) = y(n,n), where y(m+2,n) = n*y(m+1,n) + y(m,n), with y(0,n)=0, y(1,n)=1 for all n. - Benedict W. J. Irwin, Nov 03 2016
a(n) ~ n^(n-1). - Vaclav Kotesovec, Jun 03 2017
a(n) = A117715(n,n). - Bobby Jacobs, Aug 12 2017
a(n) = [x^n] x/(1 - n*x - x^2). - Ilya Gutkovskiy, Oct 10 2017
a(n) == 0 (mod n) for even n and 1 (mod n) for odd n. - Flávio V. Fernandes, Dec 08 2020
a(n) == 0 (mod n) for even n and 1 (mod n^2) for odd n; see A065599. - Flávio V. Fernandes, Dec 25 2020
a(n) == 0 (mod 2*(n/2)^2) for even n and 1 (mod n^2) for odd n; see A129194. - Flávio V. Fernandes, Feb 06 2021

A084845 Numerators of the continued fraction n+1/(n+1/...) [n times].

Original entry on oeis.org

1, 5, 33, 305, 3640, 53353, 927843, 18674305, 426938895, 10928351501, 309601751184, 9616792908241, 324971855514293, 11868363584907985, 465823816409224245, 19553538801258341377, 874091571490181406680

Offset: 1

Hollie L. Buchanan II, Jun 08 2003

The n-th term of the Lucas sequence U(n,-1). The denominator is the (n-1)-th term. Adjacent terms of the sequence U(n,-1) are relatively prime. - T. D. Noe, Aug 19 2004

			a(4) = 305 since 4+1/(4+1/(4+1/4)) = 305/72.

Alois P. Heinz, Table of n, a(n) for n = 1..386
Eric Weisstein's World of Mathematics, Lucas Sequence

Cf. A084844 (denominators).

Cf. A097690, A097691, A117715.

A084845 := proc(n)
    fibonacci(n+1,n) ;
end proc:
seq(A084845(n),n=1..20) ; # Zerinvary Lajos, Dec 01 2006

myList[n_] := Module[{ex = {n}}, Do[ex = {ex, n}, {n - 1}]; Flatten[ex]] Table[Numerator[FromContinuedFraction[myList[n]]], {n, 1, 20}]
Table[s=n; Do[s=n+1/s, {n-1}]; Numerator[s], {n, 20}] (* T. D. Noe, Aug 19 2004 *)

{a(n)=polcoeff(1/(1-n*x-x^2+x*O(x^n)),n)} \\ Paul D. Hanna, Dec 27 2012

from sympy import fibonacci
def a117715(n, m): return 0 if n==0 else fibonacci(n, m)
def a(n): return a117715(n + 1, n)
print([a(n) for n in range(1, 31)]) # Indranil Ghosh, Aug 12 2017

a(n) = Sum_{k=0..floor(n/2)}* binomial(n-k, k)*n^(n-2k). - Michel Lagneau
a(n) = [x^n] 1/(1 - n*x - x^2). - Paul D. Hanna, Dec 27 2012
a(n) = (s^(n+1) - (-s)^(-n-1))/(2*s - n), where s = (n + sqrt(n^2 + 4))/2. - Vladimir Reshetnikov, May 07 2016
a(n) = A117715(n+1,n). - Alois P. Heinz, Aug 12 2017

A076661 Index of first term of the harmonic sequence having the same denominator as the partial harmonic sequence beginning with 1/n.

Original entry on oeis.org

1, 2, 4, 9, 9, 10, 10, 14, 25, 27, 27, 27, 27, 27, 27, 27, 49, 49, 49, 49, 49, 49, 49, 49, 49, 50, 50, 125, 125, 125, 125, 125, 125, 125, 143, 143, 143, 143, 143, 136, 136, 136, 136, 136, 136, 136, 136, 136, 136, 136, 98, 98, 98, 133, 133, 133, 133, 125, 125, 125, 125

Offset: 1

Hollie L. Buchanan II, Oct 24 2002

nonn

Of more interest is the index of terms after which the denominators of the harmonic sequence always match the denominators of the partial harmonic sequence. Notice that 1/4+..1/21 has denominator 15519504, but 1/1+1/2+..1/21 has denominator 5173168.

			a(4) = firstHM[4] = 9 because 1/4+1/5+1/6+1/7+1/8+1/9 has the same denominator (2520) as 1/1+1/2+..+1/8+1/9 (and the sums to 4,5,6,7 and 8 do not).

Cf. A002805.

harmNumber[m_, n_] := HarmonicNumber[n] - HarmonicNumber[m - 1]; denH[n_] := Denominator[HarmonicNumber[n]]; denH[m_, n_] := Denominator[harmNumber[m, n]]; firstHM[m_] := Do[If[denH[k] == denH[m, k], Return[k], ], {k, m, 10^4}]

cp's OEIS Frontend

User: Hollie L. Buchanan II

A182020 Numbers that are squares and have a majority of one digit.

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A086119 Numbers of the form p^3 + q^3, p, q primes.

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A083659 Denominator of fraction equal to the continued fraction [p(n); p(n-1),...,5,3,2].

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Maple

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A084962 Iterations of the Fibonacci sequence starting at 6.

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A086120 Natural numbers of the form p^3 - q^3, where p and q are primes.

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A086121 Positive sums or differences of two cubes of primes.

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A084963 a(0) = 7, a(n+1) = Fibonacci(a(n)).

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Maple

Mathematica

A084844 Denominators of the continued fraction n + 1/(n + 1/...) [n times].

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