A000058 -id:A000058 - cp's OEIS frontend

A177888 P_n(k) with P_0(z) = z+1 and P_n(z) = z + P_(n-1)(z)*(P_(n-1)(z)-z) for n>1; square array P_n(k), n>=0, k>=0, read by antidiagonals.

1, 2, 1, 3, 3, 1, 4, 5, 7, 1, 5, 7, 17, 43, 1, 6, 9, 31, 257, 1807, 1, 7, 11, 49, 871, 65537, 3263443, 1, 8, 13, 71, 2209, 756031, 4294967297, 10650056950807, 1, 9, 15, 97, 4691, 4870849, 571580604871, 18446744073709551617, 113423713055421844361000443, 1

Offset: 0

Alois P. Heinz, Dec 14 2010

			Square array P_n(k) begins:
  1,              2,          3,      4,       5,    6,    7,     8, ...
  1,              3,          5,      7,       9,   11,   13,    15, ...
  1,              7,         17,     31,      49,   71,   97,   127, ...
  1,             43,        257,    871,    2209, 4691, 8833, 15247, ...
  1,           1807,      65537, 756031, 4870849,  ...
  1,        3263443, 4294967297,    ...
  1, 10650056950807,        ...

Alois P. Heinz, Antidiagonals n = 0..13, flattened
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!)

Columns k=0-10 give: A000012, A000058(n+1), A000215, A000289(n+1), A000324(n+1), A001543(n+1), A001544(n+1), A067686, A110360(n+1), A110368(n+1), A110383(n+1).
Rows n=0-2 give: A000027(k+1), A005408, A056220(k+1).
Main diagonal gives A252730.
Coefficients of P_n(z) give: A177701.

p:= proc(n) option remember;
      z-> z+ `if`(n=0, 1, p(n-1)(z)*(p(n-1)(z)-z))
    end:
seq(seq(p(n)(d-n), n=0..d), d=0..8);

p[n_] := p[n] = Function[z, z + If [n == 0, 1, p[n-1][z]*(p[n-1][z]-z)] ]; Table [Table[p[n][d-n], {n, 0, d}], {d, 0, 8}] // Flatten (* Jean-François Alcover, Dec 13 2013, translated from Maple *)

A003096 a(n) = a(n-1)^2 - 1, a(0) = 2.

Original entry on oeis.org

2, 3, 8, 63, 3968, 15745023, 247905749270528, 61457260521381894004129398783, 3776994870793005510047522464634252677140721938309041881088

Offset: 0

N. J. A. Sloane

After a(0) = 2 and a(1) = 3, this can never be prime, since a(n) = (a(n-1) + 1) * (a(n-1) - 1). Each term is relatively prime to its successor. - Jonathan Vos Post, Jun 06 2008

Mensa (see Dutch link below) indicates high intelligence by offering a self test containing a number of problems, one of which is "Complete each series with the element that logically continues the series: 3968, 63, 8, 3". - David A. Corneth, May 19 2024

R. K. Guy, How to factor a number, Proc. 5th Manitoba Conf. Numerical Math., Congress. Num. 16 (1975), 49-89.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

G. C. Greubel, Table of n, a(n) for n = 0..12
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!)
Mensa, self-test indicative for high IQ
Index entries for sequences of form a(n+1)=a(n)^2 + ...

Cf. A003095, A077124, A139244.

[n le 1 select 2 else Self(n-1)^2 -1: n in [1..12]]; // G. C. Greubel, Oct 27 2022

a := proc(n) local k, v: v := 2: for k from 1 to n do v := v^2-1: od: v: end:
seq(a(n), n = 0 .. 8); # Lorenzo Sauras Altuzarra, Feb 01 2023

NestList[#^2-1&,2,10] (* Harvey P. Dale, Nov 06 2011 *)

PARI
```
a(n)=if(n<1,2*(n==0),a(n-1)^2-1)
```

def A003096(n): return 2 if (n==0) else A003096(n-1)^2 -1
[A003096(n) for n in range(12)] # G. C. Greubel, Oct 27 2022

a(n-1) = ceiling(c^(2^n)) where c = 1.295553... = A077124. - Benoit Cloitre, Nov 29 2002

A005267 a(n) = -1 + a(0)a(1)...*a(n-1) with a(0) = 3.

Original entry on oeis.org

3, 2, 5, 29, 869, 756029, 571580604869, 326704387862983487112029, 106735757048926752040856495274871386126283608869, 11392521832807516835658052968328096177131218666695418950023483907701862019030266123104859068029

Offset: 0

N. J. A. Sloane

The next term is too large to include.
An infinite coprime sequence defined by recursion. - Michael Somos, Mar 14 2004
Let u(k), v(k) be defined by u(1)=1, v(1)=3, u(k+1)=v(k)-u(k), v(k+1)=u(k)v(k); then a(n)=v(2n). - Benoit Cloitre, Apr 02 2002
For positive n, a(n) has digital root 2 or 5 depending on whether n is odd or even. (T. Koshy) - Lekraj Beedassy, Apr 11 2005

R. K. Guy and R. Nowakowski, "Discovering primes with Euclid," Delta (Waukesha), Vol. 5, pp. 49-63, 1975.
T. Koshy, "Intriguing Properties Of Three Related Number Sequences", in Journal of Recreational Mathematics, Vol. 32(3) pp. 210-213, 2003-2004 Baywood NY.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Indranil Ghosh, Table of n, a(n) for n = 0..12
R. K. Guy and R. Nowakowski, Discovering primes with Euclid, Research Paper No. 260 (Nov 1974), The University of Calgary Department of Mathematics, Statistics and Computing Science.
Index entries for sequences of form a(n+1)=a(n)^2 + ...

Cf. A000058, A000289, A117805.

a(n)=if(n<2,3*(n>=0)-(n>0),a(n-1)^2+a(n-1)-1)

def a(n):
    if n == 0: return 2
    t = a(n-1)
    l = t+1
    u = t
    return l * u - 1
print([a(n) for n in range(0, 8)]) # Darío Clavijo, Aug 24 2024

a(n) = -1 + a(0)a(1)...a(n-1).
a(n) = -1 + Product_{iHenry Bottomley, Jul 31 2000
a(n+1) = a(n)^2 + a(n) - 1 if n>1. a(0)=3, a(1)=2.
An induction shows that a(n+1) = A117805(n) - 1. - R. J. Mathar, Apr 22 2007; M. F. Hasler, May 04 2007
For n>0, a(n) = a(0)^2 + a(1)^2 + ... + a(n-1)^2 - n - 6. - Max Alekseyev, Jun 19 2008

A004168 a(n+1) = a(n)*(a(n)+1).

Original entry on oeis.org

3, 12, 156, 24492, 599882556, 359859081592975692, 129498558604939936868397356895854556, 16769876680757063368089314196389622249367851612542961252860614401811692

Offset: 0

N. J. A. Sloane

The next term (a(8)) has 141 digits. - Harvey P. Dale, Jul 02 2021

Cf. A000058, A007018.

[n eq 1 select 3 else  Self(n-1)*(Self(n-1)+1): n in [1..10]]; // Vincenzo Librandi, Feb 23 2016

A004168 := proc(n) option remember; if n=0 then 3 else A004168(n-1)*(A004168(n-1)+1); fi; end;

a = {3}; Do[AppendTo[a, a[[n - 1]] (a[[n - 1]] + 1)], {n, 2, 8}]; a (* Michael De Vlieger, Feb 23 2016 *)
NestList[#(#+1)&,3,7] (* Harvey P. Dale, Jul 02 2021 *)

a(n) = A082732(n+3) - 1. - Max Alekseyev, Aug 09 2019

a(7) from Vincenzo Librandi, Feb 23 2016

A051070 a(n) is the n-th term in sequence A_n, respecting the offset, or a(n) = -1 if A_n has fewer than n terms.

Original entry on oeis.org

1, 2, 1, 0, 2, 3, 0, 7, 8, 4, 63, 1, 316, 78, 16, 2048, 7652, 26627, 8, 24000, 232919, 1145406, 3498690007594650042368, 2058537, 58, 26, 27, 59, 9272780, 3, 69273668, 4870847, 2387010102192469724605148123694256128, 1, 1, -53, 43, 0, -4696, 173, 44583, 111111111111111111111111111111111111111111, 30402457, 668803781, 1134903170, 382443020332

Offset: 1

Robert G. Wilson v, Aug 23 2000

a(58) = A000058(58) = 192523...920807 (58669977298272603 digits) is too large to include in the b-file. - Pontus von Brömssen, May 19 2022
Comment from N. J. A. Sloane, Dec 26 2022 (Start)
Note that a(n) = -1 can arise in two ways: either A_n has fewer than n terms, or A_n has at least n terms, but its n-th term is -1.
Here is a summary of the terms with n <= 80.
a(n) = -1 occurs just twice, for n = 53 and 54, in both cases because the relevant New York subway lines do not have enough stops.
a(1) though a(65) are known, although a(58) = = 192523...920807 has 58669977298272603 digits.
a(66) is the first unknown value.
Also unknown for n <= 80 are a(67), a(72), a(74), a(75), a(76), and a(77) (counts of numbers <= 2^n represented by various quadratic forms; some of these do not even have b-files), and a(80), which like a(66) is a graph-theory question. (End)

			a(19) = 8 because A000019(19) = 8.
a(20) = 24000 because A000020(20) = 24000.

Seth A. Troisi, Table of n, a(n) for n = 1..57 (terms 1..48 from Pontus von Brömssen).
N. J. A. Sloane, My favorite integer sequences, in Sequences and their Applications (Proceedings of SETA '98).
N. J. A. Sloane, "A Handbook of Integer Sequences" Fifty Years Later, arXiv:2301.03149 [math.NT], 2023, p. 21.
Index entries for sequences whose definition involves A_n (or An).

See A091967, A107357, A102288 for other versions. See also A031214, A031135.

Cf. A000019, A000020, A000058.

for m from 1 do
  url:= sprintf("https://oeis.org/A%06d/b%06d.txt",m,m);
  S:= URL:-Get(url);
  L:= StringTools[Split](S,"\n");
  for t in L do
    g:= sscanf(t, "%d %d");
    if nops(g) = 2 and g[1] = m then
      a[m]:= g[2];
      break
    fi;
  od;
  if not assigned(a[m]) then break fi;
od:
seq(a[i],i=1..m-1); # Robert Israel, May 31 2015

Rechecked and 4 more terms added by N. J. A. Sloane, May 25 2005
a(36) and a(42) corrected and a(43) to a(46) added by Robert Israel, May 31 2015
Definition revised by N. J. A. Sloane, Nov 27 2016

A075442 Slowest-growing sequence of primes whose reciprocals sum to 1.

Original entry on oeis.org

2, 3, 7, 43, 1811, 654149, 27082315109, 153694141992520880899, 337110658273917297268061074384231117039, 8424197597064114319193772925959967322398440121059128471513803869133407474043

Offset: 1

T. D. Noe, Sep 16 2002

This sequence was mentioned by K. S. Brown. The sequence is generated by a greedy algorithm given by the Mathematica program. The sum converges quadratically.
It is easily shown that this sequence is infinite. For suppose there was a finite representation of unity as a sum of unit fractions with distinct prime denominators. Multiply the equation by the product of all denominators to obtain this product of prime numbers on one side of the equation and a sum of products consisting of this product with always exactly one of the prime numbers removed on the other side. Then each of the prime numbers divides one side of the equation but not the other, since it divides all the products added except exactly one. Contradiction. - Peter C. Heinig (algorithms(AT)gmx.de), Sep 22 2006
{a(n)} = 2, 3, 7, ..., so A225671(1) = 3. - Jonathan Sondow, May 13 2013

R. K. Guy, Unsolved Problems in Number Theory, D11.

Robert G. Wilson v, Table of n, a(n) for n = 1..14
K. S. Brown, Odd, Greedy and Stubborn (Unit Fractions)
Eric Weisstein's World of Mathematics, Egyptian Fraction

Cf. A000058, A225669, A225671.

x=1; lst={}; Do[n=Ceiling[1/x]; If[PrimeQ[n], n++ ]; While[ !PrimeQ[n], n++ ]; x=x-1/n; AppendTo[lst, n], {10}]; lst
a[n_] := a[n] = Block[{sm = Sum[1/(a[i]), {i, n - 1}]}, NextPrime[ Max[ a[n - 1], 1/(1 - sm)]]]; a[0] = 1; Array[a, 10] (* Robert G. Wilson v, Oct 28 2010 *)

a(n)=if(n<3, return(prime(n))); my(x=1.); for(i=1,n-1,x-=1/a(i)); nextprime(1/x) \\ Charles R Greathouse IV, Apr 29 2015

a_vector(N=10)= my(r=1, v=vector(N)); for(i=1, N, v[i]= nextprime(1+1/r); r-= 1/v[i]); v; \\ Ruud H.G. van Tol, Jul 29 2023

A092666 a(n) = number of Egyptian fractions 1 = 1/x_1 + ... + 1/x_k (for any k), with 0 < x_1 <= ... <= x_k = n.

Original entry on oeis.org

1, 1, 1, 2, 1, 7, 1, 10, 10, 26, 1, 107, 1, 83, 375, 384, 1, 1418, 1, 4781, 7812, 1529, 1, 33665, 9789, 4276, 27787, 168107, 1, 584667, 1, 586340, 1177696, 52334, 5285597, 14746041, 1, 218959, 13092673, 84854683, 1, 279357910, 1, 491060793, 2001103921

Offset: 1

Max Alekseyev, Mar 02 2004

nonn

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

Toshitaka Suzuki, Table of n, a(n) for n = 1..390
Index entries for sequences related to Egyptian fractions

Cf. A020473, A092667, A092669, A002966, A000058, A259633.

a(n) = A020473(n) - A020473(n-1).

a(n) = 1 if n is prime.

Edited by Max Alekseyev, May 05 2010

A129871 A variant of Sylvester's sequence: a(0)=1 and for n>0, a(n) = (a(0)a(1)...*a(n-1)) + 1.

Original entry on oeis.org

1, 2, 3, 7, 43, 1807, 3263443, 10650056950807, 113423713055421844361000443, 12864938683278671740537145998360961546653259485195807

Offset: 0

Ben Branman, Sep 16 2011

nonn

A variant of A000058, starting with an extra 1.

Jean-Marie Monier, Analyse, Exercices corrigés, 2ème année, MP, Dunod, 1997, Exercice 3.3.4 page 284.

Mohammad K. Azarian, Sylvester's Sequence and the Infinite Egyptian Fraction Decomposition of 1, Problem 958, College Mathematics Journal, Vol. 42, No. 4, September 2011, p. 330.
Mohammad K. Azarian, Sylvester's Sequence and the Infinite Egyptian Fraction Decomposition of 1, Solution College Mathematics Journal, Vol. 43, No. 4, September 2012, pp. 340-342.
Junnosuke Koizumi, Irrationality of the reciprocal sum of doubly exponential sequences, arXiv:2504.05933 [math.NT], 2025.
Vjekoslav Kovač, On simultaneous rationality of two Ahmes series, arXiv:2406.17593 [math.NT], 2024.

Cf. A000058 which is the main entry for this sequence.

Cf. A118227.

a129871 n = a129871_list !! n
a129871_list = 1 : a000058_list  -- Reinhard Zumkeller, Dec 18 2013

a[0] = 1; a[n_] := a[n] = Product[a[k], {k, 0, n - 1}] + 1

For n>0, a(n) = A000058(n-1).
a(1) = 2, a(n+1) = a(n)^2 - a(n) + 1. a(n) = round(c^(2^n)), where c = 1.264... is the Vardi constant, A076393. - Thomas Ordowski, Jun 11 2013
From Bernard Schott, Apr 06 2021: (Start)
Sum_{n>=0} 1/a(n) = 2.
Sum_{n>=0} (-1)^(n+1)/a(n) = 2 * (A118227 - 1). (End)

Corrected and rewritten by Ben Branman, Sep 16 2011

Edited by Max Alekseyev, Oct 11 2012

A091967 a(n) is the n-th term of sequence A_n, ignoring the offset, or -1 if A_n has fewer than n terms.

Original entry on oeis.org

0, 2, 1, 0, 2, 3, 0, 6, 6, 4, 44, 1, 180, 42, 16, 1096, 7652, 13781, 8, 24000, 119779, 458561, 152116956851941670912, 1054535, -53, 26, 27, 59, 4806078, 2, 35792568, 3010349, 2387010102192469724605148123694256128, 2, 0, -53, 43, 0, -4097, 173, 37338, 111111111111111111111111111111111111111111, 30402457, 413927966

Offset: 1

Proposed by several people, including Jeff Burch and Michael Joseph Halm

sign

This version ignores the offset of A_n and just counts from the beginning of the terms shown in the OEIS entry.
Thus a(8) = 6 because A_8 begins 1,1,2,2,3,4,5,6,... [even though A_8(8) is really 7].
The value a(n) = -1 could arise in two different ways, but it will be easy to decide which. - N. J. A. Sloane, Nov 27 2016
From M. F. Hasler, Sep 22 2013: (Start)
The value of a(91967) can be chosen at will.
Note that this sequence may change if the initial terms in A_n are altered, which does happen from time to time, usually because of the addition of an initial term.
After a(47), currently unknown, the sequence continues with a(48) = A48(47) = 1497207322929, a(49) = A49(48) = unknown, a(50) = A50(49) = unknown, a(51) = A51(50) = 1125899906842625, a(52)=97, a(53) = -1 (since A000053 has only 29 terms). (End)
a(58) = A000058(57) = 138752...985443 (29334988649136302 digits) is too large to include in the b-file. - Pontus von Brömssen, May 21 2022

			a(1) = 0 since A000001 has offset 0, and begins with A000001(0) = 0.
a(26) = 26 because the 26th term of A000026 = 26.

Pontus von Brömssen, Table of n, a(n) for n = 1..57 (terms n = 1..46 from M. F. Hasler)
OEIS, List of finite sequences, sorted by A-number.
N. J. A. Sloane, My favorite integer sequences, in Sequences and their Applications (Proceedings of SETA '98).
Index entries for sequences whose definition involves A_n (or An).

See A051070, A107357, A102288 for other versions.

Cf. A000001, A000002, A000003, A000004, A000005, A000006, A000007, A000008, A000009, A000010, A000011, A000012, A000013, A000014, A000015, etc.

Corrected and extended by Jud McCranie; further extended by N. J. A. Sloane and E. M. Rains, Dec 08 1998
Corrected and extended by N. J. A. Sloane, May 25 2005
a(26), a(36) and a(42) corrected by M. F. Hasler, Jan 30 2009
a(43) and a(44) added by Daniel Sterman, Nov 27 2016
a(1) corrected by N. J. A. Sloane, Nov 27 2016 at the suggestion of Daniel Sterman
Definition and comments changed by N. J. A. Sloane, Nov 27 2016

A144743 Recurrence sequence a(n)=a(n-1)^2-a(n-1)-1, a(0)=3.

Original entry on oeis.org

3, 5, 19, 341, 115939, 13441735781, 180680260792773944179, 32645356640144805339284259388335434039861, 1065719310162246533488642668727242229836148490441005113524301742665845135502859459

Offset: 0

Artur Jasinski, Sep 20 2008

nonn

a(0)=3 is the smallest integer generating an increasing sequence of the form a(n)=a(n-1)^2-a(n-1)-1.

Conjecture: A130282 and this sequence are disjoint. If this is true, for n >= 1, a(n+1) is the smallest m such that (m^2-1) / (a(n)^2-1) + 1 is a square. - Jianing Song, Mar 19 2022

Cf. A000058, A082732, A144744, A144745, A144746, A144747, A144748.

a = {3}; k = 3; Do[k = k^2 - k - 1; AppendTo[a, k], {n, 1, 10}]; a

a(n,s=3)=for(i=1,n,s=s^2-s-1);s \\ M. F. Hasler, Oct 06 2014

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

a(n) ~ c^(2^n), where c = 2.07259396780115004655284076205241023281287049774423620992171834046728756... . - Vaclav Kotesovec, May 06 2015

Edited by M. F. Hasler, Oct 06 2014

cp's OEIS Frontend

A177888 P_n(k) with P_0(z) = z+1 and P_n(z) = z + P_(n-1)(z)*(P_(n-1)(z)-z) for n>1; square array P_n(k), n>=0, k>=0, read by antidiagonals.

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A003096 a(n) = a(n-1)^2 - 1, a(0) = 2.

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A005267 a(n) = -1 + a(0)*a(1)*...*a(n-1) with a(0) = 3.

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A004168 a(n+1) = a(n)*(a(n)+1).

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A051070 a(n) is the n-th term in sequence A_n, respecting the offset, or a(n) = -1 if A_n has fewer than n terms.

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Maple

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A075442 Slowest-growing sequence of primes whose reciprocals sum to 1.

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Mathematica

PARI

PARI

A092666 a(n) = number of Egyptian fractions 1 = 1/x_1 + ... + 1/x_k (for any k), with 0 < x_1 <= ... <= x_k = n.

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A005267 a(n) = -1 + a(0)a(1)...*a(n-1) with a(0) = 3.

A129871 A variant of Sylvester's sequence: a(0)=1 and for n>0, a(n) = (a(0)a(1)...*a(n-1)) + 1.