A000058 -id:A000058 - cp's OEIS frontend

A144746 a(n) = a(n-1)^2 - a(n-1) - 1, a(0)=6.

6, 29, 811, 656909, 431528777371, 186217085698878552894269, 34676803006183479266409218250231853558140150091, 1202480666729655584789949373132702064208272454072740050128160074167965751208292536045867158189

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, cf. A144743.

Cf. A000058, A082732, A144743, A144744, A144745, A144747, A144748.

NestList[#^2-#-1&,6,8]  (* Harvey P. Dale, Jan 22 2011 *)

a(n, s=6)={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 and a(0)=6.

a(n) ~ c^(2^n), where c = 5.33565954034691307256446890777476398311129407641143635105306409567572... . - Vaclav Kotesovec, May 06 2015

Corrected and edited by M. F. Hasler, Oct 06 2014

A144747 Recurrence sequence a(n)=a(n-1)^2-a(n-1)-1, a(0)=7.

Original entry on oeis.org

7, 41, 1639, 2684681, 7207509387079, 51948191564824694742765161, 2698614606855723567054656642857156538246857652590759, 7282520796335071470236496456671241855257664867148949932302276253455702665493855273950765616767079605321

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, cf. A144743.

Cf. A000058, A082732, A144743, A144744, A144745, A144746, A144748.

a = {}; k = 7; Do[k = k^2 - k - 1; AppendTo[a, k], {n, 1, 10}]; a

a(n, s=7)={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 and a(0)=7.

a(n) ~ c^(2^n), where c = 6.3622623884585267364822329679498420997632627444610172910703030892754... . - Vaclav Kotesovec, May 06 2015

Edited by M. F. Hasler, Oct 06 2014

A144748 Recurrence sequence a(n)=a(n-1)^2-a(n-1)-1, a(0)=8.

Original entry on oeis.org

8, 55, 2969, 8811991, 77651176572089, 6029705223029665929437251831, 36357345076631233348346773693633697407708655232275600729, 1321856541021241383115043586121503961331042183698683965174269952435581223368633124721267107619465028785549730711

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, cf. A144743.

Cf. A000058, A082732, A144743, A144744, A144745, A144746, A144747.

a = {}; k = 8; Do[k = k^2 - k - 1; AppendTo[a, k], {n, 1, 10}]; a
NestList[#^2-#-1&,8,10] (* Harvey P. Dale, Mar 14 2016 *)

a(n, s=8)={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 and a(0)=8.

a(n) ~ c^(2^n), where c = 7.3813237216360344087566795911708086794628396333350474334044779783264... . - Vaclav Kotesovec, May 06 2015

Edited by M. F. Hasler, Oct 06 2014

A164643 Semiprimes pq with pq - 1 divisible by p + q.

Original entry on oeis.org

6, 21, 301, 697, 1333, 1909, 2041, 3901, 24601, 26977, 96361, 130153, 163201, 250321, 275833, 296341, 389593, 486877, 495529, 542413, 808861, 1005421, 1005649, 1055833, 1063141, 1232053, 1284121, 1403221, 1618597, 1787917, 2287933, 2462881, 2488201, 2666437

Offset: 1

Mohamed Bouhamida, Aug 19 2009

nonn

The first three terms are Syl(0)*Syl(1), Syl(1)*Syl(2) and Syl(2)*Syl(3). Syl means Sylvester's sequence, see A000058.
Products of two consecutive numbers p and q in Sylvester's sequence with primes p and q are in the sequence.
Let p and q be consecutive prime Sylvester numbers. Then: pq - 1 = p*(p^2 - p + 1) - 1 = p^3 - p^2 + p - 1 = (p^2 + 1)*(p - 1) = (p + p^2 - p + 1)*(p - 1) = (p + q)*(p - 1) it means that: (pq - 1) is divisible by (p + q). - Mohamed Bouhamida, Aug 21 2009
(p-k)*(q-k) = k^2 + 1 for some integer k, providing a fast way for finding appropriate p,q. - Max Alekseyev, Aug 26 2009

Donovan Johnson, Table of n, a(n) for n = 1..1000

Cf. A001358, A000058.

isA001358 := proc(n) RETURN ( numtheory[bigomega](n) =2 ) ; end:
isA164643 := proc(n) if isA001358(n) then p := op(1,op(1,ifactors(n)[2]) ) ; q := n/p ; if (p*q-1) mod (p+q) =0 then true; else false; fi; else false; fi; end:
for n from 4 to 3000000 do if isA164643(n) then print(n) ; fi; od: # R. J. Mathar, Aug 24 2009

dsQ[n_]:=Module[{prs=Transpose[FactorInteger[n]][[1]]},Divisible[n-1, Total[prs]]]; Select[Select[Range[2000000], PrimeOmega[#] ==2&], dsQ] (* Harvey P. Dale, Jun 15 2011 *)

Extended by R. J. Mathar, Aug 24 2009

More terms from Max Alekseyev, Aug 26 2009

A174864 a(1) = 1, a(n) = square of the sum of previous terms.

Original entry on oeis.org

1, 1, 4, 36, 1764, 3261636, 10650053687364, 113423713055411194304049636, 12864938683278671740537145884937248491231415124195364

Offset: 1

Giovanni Teofilatto, Mar 31 2010

a(n) divides a(n+1) with result a square.
Except for first two terms, partial sum k of a(n) is divisible by 6.
These numbers are divisible by their digital roots, which makes the sequence a subsequence of A064807. - Ivan N. Ianakiev, Oct 09 2013
a(n) is the number of binary trees in which the nodes are labeled by nonnegative integer heights, the left and right children of each node (if present) must have smaller height, and the root has height n-2. For instance, there are four trees with root height 1: the left and right children of the root may or may not be present, and must each be at height 0 if present. - David Eppstein, Oct 25 2018

Vincenzo Librandi, Table of n, a(n) for n = 1..13
Index entries for sequences of form a(n+1)=a(n)^2 + ...

Cf. A000058, A007018.

t = {1}; Do[AppendTo[t, Total[t]^2], {n, 9}]; t (* Vladimir Joseph Stephan Orlovsky, Feb 24 2012 *)
Join[{1},FoldList[(#+Sqrt[#])^2&,1,Range[7]]] (* Ivan N. Ianakiev, May 08 2015 *)

a=vector(10);a[1]=a[2]=1;for(n=3,#a,a[n]=a[n-1]*(sqrtint(a[n-1])+1)^2);a

a(n+1) = (a(n) + sqrt(a(n)))^2 = a(n) * (sqrt(a(n)) + 1)^2 for n > 1. - Charles R Greathouse IV, Jun 30 2011
a(n) = A000058(n-1) - A000058(n-2), n>=2. - Ivan N. Ianakiev, Oct 09 2013
a(n+2) + 1 = ( A000058(n+1)^2+1 ) / ( A000058(n)^2+1 ). - Bill Gosper, Hugo Pfoertner, May 09 2021

A231830 a(0) = 1; for n > 0, a(n) = 1 + 4*Product_{i=1..n-1} a(i)^2.

Original entry on oeis.org

1, 5, 101, 1020101, 1061522231810040101, 1196154511175776540960913502483611007728163340227060101

Offset: 0

Michel Marcus, Nov 14 2013

nonn

Sequence designed to show that there are an infinity of primes congruent to 1 modulo 4 (A002144). Terms are not necessarily prime. Their smallest prime factors from A002144 are: 5, 101, 1020101, 53, 686743037.
Next term is too large to include.
From Max Alekseyev, Apr 21 2023: (Start)
Similarly to Sylvester's sequence (A000058), it is unknown if all terms are squarefree.
Primes dividing terms of this sequence are listed in A362252. Since terms are pairwise coprime, for each n prime A362252(n) divides exactly one term, whose index is A362253(n). That is, A362252(n) divides a(A362253(n)). (End)

S. A. Shirali, A family portrait of primes-a case study in discrimination, Math. Mag. Vol. 70, No. 4 (Oct., 1997), pp. 263-272.

Cf. A000058, A002144, A007018, A231831, A362252, A362253.

lista(nn) = {a = vector(nn); a[1] = 5; for (n=2, nn, a[n] = 4*prod(i=1, n-1, a[i]^2) + 1;); a;}

For n > 1, a(n) = (a(n-1) - 1) * a(n-1)^2 + 1. - Max Alekseyev, Mar 25 2023

a(0)=1 prepended by Max Alekseyev, Mar 25 2023

A231831 a(0) = 1; for n > 0, a(n) = -1 + 4*Product_{i=0..n-1} a(i)^2.

Original entry on oeis.org

1, 3, 35, 44099, 85762231424099, 630794963141019085083178800095033630804099

Offset: 0

Michel Marcus, Nov 14 2013

nonn

Sequence designed to show that there are an infinity of primes congruent to 3 modulo 4 (A002145). Terms are not necessarily prime. Their smallest prime factor from A002145 are: 3, 7, 11, 23, 4111, 2809343.
Next term is too large to include.
Similarly to Sylvester's sequence (A000058), it is unknown if all terms are squarefree (see also MathOverflow link). - Max Alekseyev, Mar 26 2023
Primes dividing terms of this sequence are listed in A362250. Since terms are pairwise coprime, for each n prime A362250(n) divides exactly one term, whose index is A362251(n). That is, A362250(n) divides a(A362251(n)). - Max Alekseyev, Apr 16 2023

S. A. Shirali, A family portrait of primes-a case study in discrimination, Math. Mag. Vol. 70, No. 4 (Oct., 1997), pp. 263-272.
fredrickmnelson et al., Does a(0)=6, a(n+1)=a(n)^3-a(n), define a square-free sequence?, MathOverflow, 2023.

Cf. A000058, A002145, A007018, A231830, A362250, A362251.

lista(nn) = {a = vector(nn); a[1] = 3; for (n=2, nn, a[n] = 4*prod(i=1, n-1, a[i]^2) - 1;); a;}

For n > 1, a(n) = (a(n-1) + 1) * a(n-1)^2 - 1. - Max Alekseyev, Mar 26 2023

a(0) = 1 prepended by Max Alekseyev, Mar 26 2023

A362250 Primes dividing terms of A231831.

Original entry on oeis.org

3, 5, 7, 11, 19, 23, 89, 101, 137, 157, 211, 373, 659, 877, 881, 1399, 1597, 1627, 1663, 1811, 2029, 2069, 2087, 2153, 2381, 2677, 2939, 3433, 3491, 3511, 3617, 3673, 4111, 4127, 4547, 4721, 5059, 5483, 6529, 6793, 6827, 7757, 8209, 8297, 8677, 9203, 9463, 9811, 10139, 10159, 11321

Offset: 1

Max Alekseyev, Apr 13 2023

nonn

Since the terms of A231831 are pairwise coprime, each prime divides at most one term of A231831. Indices of the corresponding terms are listed in A362251, and so a(n) divides A231831(A362251(n)).

Max Alekseyev, Table of n, a(n) for n = 1..16944
fredrickmnelson et al., Does a(0)=6, a(n+1)=a(n)^3-a(n), define a square-free sequence?, MathOverflow, 2023.

Cf. A000058, A007996, A180871, A231830, A231831, A362251, A362252, A362253.

A362251 a(n) is the unique index such that prime A362250(n) divides A231831(a(n)).

Original entry on oeis.org

1, 2, 2, 3, 3, 4, 9, 4, 4, 8, 3, 31, 12, 7, 7, 9, 44, 8, 22, 29, 36, 37, 8, 21, 5, 26, 4, 20, 24, 12, 76, 30, 5, 47, 5, 13, 9, 25, 6, 41, 51, 9, 53, 6, 27, 39, 5, 12, 78, 64, 10, 185, 113, 205, 91, 85, 43, 195, 32, 117, 20, 133, 142, 119, 64, 70, 199, 41, 125, 79, 243, 70, 35, 105, 67, 156

Offset: 1

Max Alekseyev, Apr 13 2023

nonn

Max Alekseyev, Table of n, a(n) for n = 1..16944

Cf. A000058, A007996, A180871, A231830, A231831, A362250, A362252, A362253.

A362252 Primes dividing terms of A231830.

Original entry on oeis.org

5, 53, 89, 101, 373, 877, 1109, 1181, 1597, 1613, 2029, 2069, 2153, 2213, 2381, 2741, 3617, 4273, 6529, 6737, 7417, 7717, 11321, 12653, 13009, 13309, 16829, 17729, 23581, 23993, 25373, 32569, 33353, 33857, 34841, 35053, 36097, 37201, 38609, 41513, 42461, 48661, 55829, 58369, 59093, 63281

Offset: 1

Max Alekseyev, Apr 21 2023

nonn

Since the terms of A231830 are pairwise coprime, each prime divides at most one term of A231830. Indices of the corresponding terms are listed in A362253, and so a(n) divides A231830(A362253(n)).

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

Cf. A000058, A007996, A180871, A231830, A231831, A362250, A362251, A362253.

cp's OEIS Frontend

A144746 a(n) = a(n-1)^2 - a(n-1) - 1, a(0)=6.

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A144747 Recurrence sequence a(n)=a(n-1)^2-a(n-1)-1, a(0)=7.

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A144748 Recurrence sequence a(n)=a(n-1)^2-a(n-1)-1, a(0)=8.

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A164643 Semiprimes pq with pq - 1 divisible by p + q.

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A174864 a(1) = 1, a(n) = square of the sum of previous terms.

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A231830 a(0) = 1; for n > 0, a(n) = 1 + 4*Product_{i=1..n-1} a(i)^2.

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A231831 a(0) = 1; for n > 0, a(n) = -1 + 4*Product_{i=0..n-1} a(i)^2.

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A362250 Primes dividing terms of A231831.

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