A000058 -id:A000058 - cp's OEIS frontend

A362253 a(n) is the unique index such that prime A362252(n) divides A231830(a(n)).

1, 4, 7, 2, 19, 25, 30, 38, 45, 4, 26, 33, 27, 46, 10, 59, 102, 38, 84, 37, 22, 77, 80, 37, 240, 57, 45, 240, 173, 38, 41, 100, 88, 44, 114, 39, 63, 24, 14, 121, 177, 12, 155, 270, 65, 109, 44, 391, 54, 22, 96, 320, 194, 347, 182, 226, 143, 290, 105, 135, 29, 198, 113, 302, 572, 53, 692, 168, 366

Offset: 1

Max Alekseyev, Apr 21 2023

nonn

Max Alekseyev, Table of n, a(n) for n = 1..8503

Cf. A000058, A007996, A180871, A231830, A231831, A362250, A362251, A362252.

A092667 a(n) = number of Egyptian fractions 1 = 1/x_1 + ... + 1/x_k (for any k), with max{x_i}=n.

Original entry on oeis.org

1, 1, 1, 4, 1, 44, 1, 235, 921, 4038, 1, 66147, 1, 304383, 6581754, 45353329, 1, 1100311690, 1, 44423279911, 1250831952086, 284120133400, 1, 71664788693247, 511162204140999, 55479698795314, 10715917223431762, 505603414069366830, 1, 28696102343693431631, 1, 857699266471525509621, 30399386408588668316839, 63063040603038091480

Offset: 1

Max Alekseyev, Mar 02 2004

nonn

			a(4) = 4 since there are four fractions 1=1/2+1/4+1/4, 1=1/4+1/2+1/4, 1=1/4+1/4+1/2 and 1=1/4+1/4+1/4+1/4.

Index entries for sequences related to Egyptian fractions

Cf. A092666, A038034, A092669, A002967, A000058.

a(n) = A038034(n) - A038034(n-1).

a(n) = 1 if n is prime.

A225669 Slowest-growing sequence of odd primes whose reciprocals sum to 1.

Original entry on oeis.org

3, 5, 7, 11, 13, 17, 19, 23, 967, 101419, 2000490719, 106298338760698351, 586903266015193517540253132922939, 3494365451928289992209032562272585187947069047023572601254975717

Offset: 1

Jonathan Sondow, May 11 2013

nonn

See comments, references, and links in A075442 = slowest-growing sequence of primes whose reciprocals sum to 1.

a(n) = 3, 5, 7, 11, 13, 17, 19, 23, 967, ..., so A225671(2) = 23.

			Since 1/3 + 1/5 + 1/7 + 1/11 + 1/13 + 1/17 + 1/19 + 1/23 < 1, the first eight odd primes are members. The ninth is not, because adding 1/29 pushes the sum over 1.

Popular Computing (Calabasas, CA), Problem 175: A Sum of a Different Kind, Vol. 5 (No. 50, May 1977), p. PC50-8.

Amiram Eldar, Table of n, a(n) for n = 1..18

Cf. A000058, A075442, A046689, A136616, A181503, A225670, A225671.

a[n_] := a[n] = Block[{sm = Sum[1/(a[i]), {i, n - 1}]}, NextPrime[ Max[ a[n - 1], 1/(1 - sm)]]]; a[0] = 2; Array[a, 14]

A001042 a(n) = a(n-1)^2 - a(n-2)^2.

Original entry on oeis.org

1, 2, 3, 5, 16, 231, 53105, 2820087664, 7952894429824835871, 63248529811938901240357985099443351745, 4000376523371723941902615329287219027543200136435757892789536976747706216384

Offset: 0

N. J. A. Sloane, R. K. Guy

The next term has 152 digits. - Franklin T. Adams-Watters, Jun 11 2009

Archimedeans Problems Drive, Eureka, 27 (1964), 6.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Reinhard Zumkeller, Table of n, a(n) for n = 0..13
A. V. Aho and N. J. A. Sloane, Some doubly exponential sequences, Fibonacci Quarterly, Vol. 11, No. 4 (1973), pp. 429-437.
A. V. Aho and N. J. A. Sloane, Some doubly exponential sequences, Fibonacci Quarterly, Vol. 11, No. 4 (1973), pp. 429-437 (original plus references that F.Q. forgot to include - see last page!)
R. K. Guy, Letters to N. J. A. Sloane, June-August 1968
R. P. Loh, A. G. Shannon, A. F. Horadam, Divisibility Criteria and Sequence Generators Associated with Fermat Coefficients, Preprint, 1980.
Index entries for sequences of form a(n+1)=a(n)^2 + ...

Cf. A001146, A000278, A000283, A058182, A007018.

Cf. A064236 (numbers of digits).

a001042 n = a001042_list !! n
a001042_list = 1 : 2 : zipWith (-) (tail xs) xs
               where xs = map (^ 2) a001042_list
-- Reinhard Zumkeller, Dec 16 2013

RecurrenceTable[{a[0]==1,a[1]==2,a[n]==a[n-1]^2-a[n-2]^2},a,{n,0,12}] (* Harvey P. Dale, Jan 11 2013 *)

a(n) ~ c^(2^n), where c = 1.1853051643868354640833201434870139866230288004895868726506278977814490371... . - Vaclav Kotesovec, Dec 17 2014

More terms from James Sellers, Sep 19 2000.

A001056 a(n) = a(n-1)*a(n-2) + 1, a(0) = 1, a(1) = 3.

Original entry on oeis.org

1, 3, 4, 13, 53, 690, 36571, 25233991, 922832284862, 23286741570717144243, 21489756930695820973683319349467, 500426416062641238759467086706254193219790764168482, 10754042042885415070816603338436200915110904821126871858491675028294447933424899095

Offset: 0

N. J. A. Sloane, R. K. Guy

Archimedeans Problems Drive, Eureka, 19 (1957), 13.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..17
A. V. Aho and N. J. A. Sloane, Some doubly exponential sequences, Fibonacci Quarterly, Vol. 11, No. 4 (1973), pp. 429-437.
A. V. Aho and N. J. A. Sloane, Some doubly exponential sequences, Fibonacci Quarterly, Vol. 11, No. 4 (1973), pp. 429-437 (original plus references that F.Q. forgot to include - see last page!)
Index entries for sequences of form a(n+1)=a(n)^2 + ...

Cf. A001622 (phi), A258112.

a:=[1,3];; for n in [3..13] do a[n]:=a[n-1]*a[n-2]+1; od; a; # G. C. Greubel, Sep 19 2019

a001056 n = a001056_list !! n
a001056_list = 1 : 3 : (map (+ 1 ) $
               zipWith (*) a001056_list $ tail a001056_list)
-- Reinhard Zumkeller, Aug 15 2012

I:=[1,3]; [n le 2 select I[n] else Self(n-1)*Self(n-2) + 1: n in [1..13]]; // G. C. Greubel, Sep 19 2019

a:= proc (n) option remember;
if n=0 then 1
elif n=1 then 3
else a(n-1)*a(n-2) + 1
end if
end proc;
seq(a(n), n = 0..13); # G. C. Greubel, Sep 19 2019

RecurrenceTable[{a[0]==1,a[1]==3,a[n]==a[n-1]*a[n-2]+1},a,{n,0,14}] (* Harvey P. Dale, Jul 17 2011 *)
t = {1, 3}; Do[AppendTo[t, t[[-1]] * t[[-2]] + 1], {n, 2, 14}] (* T. D. Noe, Jun 25 2012 *)

m=13; v=concat([1,3], vector(m-2)); for(n=3, m, v[n]=v[n-1]*v[n-2] +1 ); v \\ G. C. Greubel, Sep 19 2019

def a(n):
    if (n==0): return 1
    elif (n==1): return 3
    else: return a(n-1)*a(n-2) + 1
[a(n) for n in (0..13)] # G. C. Greubel, Sep 19 2019

a(n) ~ c^(phi^n), where c = A258112 = 1.7978784900091604813559508837..., phi = (1+sqrt(5))/2 = A001622. - Vaclav Kotesovec, Dec 17 2014

A005831 a(n+1) = a(n) * (a(n-1) + 1).

Original entry on oeis.org

0, 1, 1, 2, 4, 12, 60, 780, 47580, 37159980, 1768109008380, 65702897157329640780, 116169884340604934905464739377180, 7632697963609645128663145969343357330533515068777580, 886689639639303288926299195509965193299034793881606681727875910370940270908216401980

Offset: 0

N. J. A. Sloane, Jeffrey Shallit

A discrete analog of the derivative of t(x) = tetration base e, since t'(x) = t(x) * t(x-1) * t(x-2) * ... y = y * exp(y) * exp(exp(y)) * ... * t(x) This sequence satisfies almost the same equation but the derivative is replaced by a difference, comparable to the relations between differential equations and their associated difference equations. - Raes Tom (tommy1729(AT)hotmail.com), Aug 06 2008

			a(5) = 12 since 12 = 1*2*4 + 4.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..19
M. E. Mays, Iterating the division algorithm, Fib. Quart., 25 (1987), 204-213.

Cf. A007807.

Cf. A000058, A001622, A001113, A102575, A096436, A111129.

a005831 n = a005831_list !! n
a005831_list = 0:1:zipWith (*) (tail a005831_list) (map succ a005831_list)
-- Reinhard Zumkeller, Mar 19 2011

a=0;b=1;lst={a,b};Do[c=a*b+b;AppendTo[lst,c];a=b;b=c,{n,18}];lst (* Vladimir Joseph Stephan Orlovsky, Sep 13 2009 *)
RecurrenceTable[{a[0]==0,a[1]==1,a[n]==a[n-1](a[n-2]+1)},a,{n,15}] (* Harvey P. Dale, Aug 17 2013 *)

a(0) = a(1) = 1, a(2) = 2; a(n) = a(n-1)*a(n-2)*a(n-3)*... + a(n-1). - Raes Tom (tommy1729(AT)hotmail.com), Aug 06 2008

The sequence grows like a doubly exponential function, similar to Sylvester's sequence. In fact we have the asymptotic form : a(n) ~ e ^ (Phi ^ n) where e and Phi are the best possible constants. - Raes Tom (tommy1729(AT)hotmail.com), Aug 06 2008

Edited by N. J. A. Sloane, Aug 29 2008 at the suggestion of R. J. Mathar

A067686 a(n) = a(n-1) * a(n-1) - B * a(n-1) + B, a(0) = 1 + B for B = 7.

Original entry on oeis.org

8, 15, 127, 15247, 232364287, 53993160246468367, 2915261353400811631533974206368127, 8498748758632331927648392184620600167779995785955324343380396911247

Offset: 0

Drastich Stanislav (drass(AT)spas.sk), Feb 05 2002

This is the special case k=7 of sequences with exact mutual k-residues. In general, a(1)=k+1 and a(n)=min{m | m>a(n-1), mod(m,a(i))=k, i=1,...,n-1}. k=1 gives Sylvester's sequence A000058 and k=2 Fermat sequence A000215. - Seppo Mustonen, Sep 04 2005

Alois P. Heinz, Table of n, a(n) for n = 0..10
A. V. Aho and N. J. A. Sloane, Some doubly exponential sequences, Fibonacci Quarterly, Vol. 11, No. 4 (1973), pp. 429-437.
A. V. Aho and N. J. A. Sloane, Some doubly exponential sequences, Fibonacci Quarterly, Vol. 11, No. 4 (1973), pp. 429-437 (original plus references that F.Q. forgot to include - see last page!)
Stanislav Drastich, Rapid growth sequences, arXiv:math/0202010 [math.GM], 2002.
S. W. Golomb, On certain nonlinear recurring sequences, Amer. Math. Monthly 70 (1963), 403-405.
S. Mustonen, On integer sequences with mutual k-residues
Seppo Mustonen, On integer sequences with mutual k-residues [Local copy]
Index entries for sequences of form a(n+1)=a(n)^2 + ....

Cf. B=1: A000058 (Sylvester's sequence), B=2: A000215 (Fermat numbers), B=3: A000289, B=4: A000324, B=5: A001543, B=6: A001544.

Column k=7 of A177888.

RecurrenceTable[{a[0]==8, a[n]==a[n-1]*(a[n-1]-7)+7}, a, {n, 0, 10}] (* Vaclav Kotesovec, Dec 17 2014 *)
NestList[#^2-7#+7&,8,10] (* Harvey P. Dale, Jan 26 2025 *)

a(n) ~ c^(2^n), where c = 3.3333858371760195832345950846454963835549715770476958790043961891683146201... . - Vaclav Kotesovec, Dec 17 2014

A081461 Consider the mapping f(a/b) = (a^2+b^3)/(a^3+b^2) from rationals to rationals. Starting with 1/2 (a=1, b=2) and applying the mapping to each new (reduced) rational number gives 1/2, 9/5, 103/377, ... . Sequence gives values of the numerators.

Original entry on oeis.org

1, 9, 103, 26796621, 236092315725004393, 3561970421302126514421966146019939188025056477849165490630219227287

Offset: 1

Amarnath Murthy, Mar 22 2003

nonn

For the mapping g(a/b) = (a^2+b)/(a+b^2), starting with 1/2 the same procedure leads to the periodic sequence 1/2, 3/5, 1/2, 3/5, ...

Cf. A000058, A081462, A081463, A081465.

nxt[{a_,b_}]:=Module[{frac=(a^2+b^3)/(a^3+b^2)},{Numerator[frac], Denominator[ frac]}]; Transpose[NestList[nxt,{1,2},5]][[1]] (* Harvey P. Dale, Nov 09 2011 *)

{r=1/2; for(n=1,7,a=numerator(r); b=denominator(r); print1(a,","); r=(a^2+b^3)/(a^3+b^2))}

Edited and extended by Klaus Brockhaus, Mar 28 2003

A081462 Consider the mapping f(a/b) = (a^2+b^3)/(a^3+b^2) from rationals to rationals. Starting with 1/2 (a=1, b=2) and applying the mapping to each new (reduced) rational number gives 1/2, 9/5, 103/377, ... . Sequence gives values of the denominators.

Original entry on oeis.org

2, 5, 377, 617428, 19241552119440973526245, 6579843627298148620615676439841151690983233028443241

Offset: 1

Amarnath Murthy, Mar 22 2003

nonn

Cf. A000058, A081461, A081466.

{r=1/2; for(n=1,7,a=numerator(r); b=denominator(r); print1(b,","); r=(a^2+b^3)/(a^3+b^2))}

Edited and extended by Klaus Brockhaus, Mar 28 2003

A081465 Consider the mapping f(a/b) = (a^2+b^2)/(a^2-b^2) from rationals to rationals. Starting with 2/1 (a=2, b=1) and applying the mapping to each new (reduced) rational number gives 2/1, 5/3, 17/8, 353/225, ... . Sequence gives values of the numerators.

Original entry on oeis.org

2, 5, 17, 353, 87617, 9045146753, 60804857528809666817, 4138643330264389621194448797227488932353, 13864359953311401274177801350481278132199085263747363330276605034095638011503617

Offset: 1

Amarnath Murthy, Mar 22 2003

nonn

For the mapping g(a/b) = (a^2+b)/(a+b^2), starting with 2/1 the same procedure leads to the periodic sequence 2, 5/3, 2, 5/3, ...

Cf. A000058, A081461, A081462, A081466.

nxt[n_]:=Module[{a=Numerator[n],b=Denominator[n]}, (a^2+b^2)/(a^2-b^2)]; Numerator/@NestList[nxt,2/1,10]  (* Harvey P. Dale, Mar 19 2011 *)

{r=2; for(n=1,9,a=numerator(r); b=denominator(r); print1(a,","); r=(a^2+b^2)/(a^2-b^2))}

Edited and extended by Klaus Brockhaus, Mar 24 2003

cp's OEIS Frontend

A362253 a(n) is the unique index such that prime A362252(n) divides A231830(a(n)).

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A092667 a(n) = number of Egyptian fractions 1 = 1/x_1 + ... + 1/x_k (for any k), with max{x_i}=n.

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A225669 Slowest-growing sequence of odd primes whose reciprocals sum to 1.

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A001042 a(n) = a(n-1)^2 - a(n-2)^2.

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A001056 a(n) = a(n-1)*a(n-2) + 1, a(0) = 1, a(1) = 3.

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A005831 a(n+1) = a(n) * (a(n-1) + 1).

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A067686 a(n) = a(n-1) * a(n-1) - B * a(n-1) + B, a(0) = 1 + B for B = 7.

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A081461 Consider the mapping f(a/b) = (a^2+b^3)/(a^3+b^2) from rationals to rationals. Starting with 1/2 (a=1, b=2) and applying the mapping to each new (reduced) rational number gives 1/2, 9/5, 103/377, ... . Sequence gives values of the numerators.