A007018 -id:A007018 - cp's OEIS frontend

A003687 a(n+1) = a(n)-a(1)a(2)...a(n-1), if n>0. a(0)=1, a(1)=2.

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

Offset: 0

N. J. A. Sloane

a(n) = a(n-1)-a(n-2)^2+a(n-1)*a(n-2), if n>2. - Michael Somos, Mar 19 2004
Consider the mapping f(a/b) = (a - b)/(ab). Taking a = 2 b = 1 to start with and carrying out this mapping repeatedly on each new (reduced) rational number gives the following sequence 2/1,1/2,-1/2,-3/-2,-1/6,... Sequence contains the numerators. - Amarnath Murthy, Mar 24 2003
An infinite coprime sequence defined by recursion. - Michael Somos, Mar 19 2004

Seiichi Manyama, Table of n, a(n) for n = 0..27

Cf. A081478.

For n>1, a(2n-1) = -1, a(2n) = -A007018(n-1) - 1.

I:=[1,2,1]; [n le 3 select I[n] else Self(n-1)-Self(n-2)^2+Self(n-1)*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Dec 17 2015

{1}~Join~NestList[{(#1 - #2), #1 #2} & @@ # &, {2, 1}, 17] [[All, 1]] (* Michael De Vlieger, Sep 04 2016 *)

a(n)=local(an); if(n<1,(n==0),an=vector(max(2,n)); an[1]=2; an[2]=1; for(k=3,n,an[k]=an[k-1]-an[k-2]^2+an[k-1]*an[k-2]); an[n])

def A003687():
    x, y = 2, 1
    yield y
    while true:
       yield x
       x, y = x - y, x * y
a = A003687(); print([next(a) for i in range(20)])  # Peter Luschny, Dec 17 2015

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.

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

A182398 a(n) = (Sum_{k=1..2n} k^2n) mod 2n.

Original entry on oeis.org

1, 2, 1, 4, 5, 2, 7, 8, 3, 6, 11, 4, 13, 14, 5, 16, 17, 6, 19, 12, 1, 22, 23, 8, 25, 26, 9, 28, 29, 58, 31, 32, 11, 34, 35, 12, 37, 38, 13, 24, 41, 2, 43, 44, 15, 46, 47, 16, 49, 30, 17, 52, 53, 18, 45, 56, 19, 58, 59, 116, 61, 62, 3, 64, 65, 22, 67, 68, 23

Offset: 1

Michel Lagneau, Apr 27 2012

nonn

Sum_{k=1..n} k^n (mod n) = 0 if n odd.
Properties of this sequence:
a(n) = 1 for n = 1, 3, 21, 903, ...
a(n) = n if n not divisible by 3;
a(3*n) = n except for n = 7, 10, 14, 20, 21, 26, 28, 30, 35, ...
a(21*n) = n, except for n = 10, 20, 26, 30, 40, 43, 50, 52, ...
a(903*n) = n, except for n = 10, ....
It appears that a(A007018(n)/2) = 1 and conjecturally a(m*A007018(n)/2) = m for a majority of value m.
No, a(A007018(n)/2) <> 1 for n > 4. (For example, a(A007018(5)/2) = a(1631721) = 1807.) - Jonathan Sondow, Oct 18 2013
0 < a(n) < 10 for n: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 15, 18, 21, 24, 27, 42, 63, 84, 105, 126, 147, 168, 189, 903, 1806, 2709, 3612, 4515, 5418, 6321, 7224, 8127, .... Search limit was 25000. - Robert G. Wilson v, Jun 18 2015

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (terms 1..2499 from Michel Lagneau)
J. M. Grau, A. M. Oller-Marcen, and J. Sondow, On the congruence 1^m + 2^m + ... + m^m == n (mod m) with n|m, arXiv 1309.7941, 2013.
Mathematics Stack Exchange, 1^n + 2^n + ... + (p-1)^n mod p = ?

Cf. A007018, A014117, A031971, A229307, A229311.

for n from 1 to 100 do: s:=sum('k^(2*n)', 'k'=1..2*n)
: x:=irem(s,2*n): printf(`%d, `,x):od:
# second Maple program:
a:= n-> add(k&^(2*n) mod (2*n), k=1..2*n) mod (2*n):
seq(a(n), n=1..100);

Table[Mod[Total[PowerMod[Range[2*n], 2*n, 2*n]], 2*n], {n, 100}] (* T. D. Noe, Apr 28 2012 *)

a(n) = A031971(2n) mod 2n. - Jonathan Sondow, Oct 18 2013

A368190 Number of (undirected) cycles in the n-Dorogovtsev-Goltsev-Mendes graph.

Original entry on oeis.org

0, 1, 11, 249, 74835, 5890739121, 34755832523270764251, 1207969003612007237832573159205646499369, 1459189113687796591938380205390010178829792070192521048490799792728844237848995

Offset: 0

Eric W. Weisstein, Dec 16 2023

nonn

Using the indexing convention that DGM(0) = P_2.

For n > 0, DGM(n) contains a unique longest cycle of length 3*2^(n-1).

Andrew Howroyd, Table of n, a(n) for n = 0..11
Eric Weisstein's World of Mathematics, Dorogovtsev-Goltsev-Mendes Graph.
Eric Weisstein's World of Mathematics, Graph Cycle.

Cf. A007018, A368456, A367967 (5-cycles), A367968 (6-cycles).

a(n) = {my(t=0,b=1); for(k=1, n, t = 3*t + b^3; b += b^2); t} \\ Andrew Howroyd, Dec 30 2023

a(n) = 3*a(n-1) + A007018(n-1)^3 for n > 0. - Andrew Howroyd, Dec 30 2023

Offset corrected and a(5) from Eric W. Weisstein, Dec 29 2023

Terms a(6) and beyond from Andrew Howroyd, Dec 30 2023

A081478 Consider the mapping f(a/b) = (a - b)/(ab). Taking a = 2 and b = 1 to start with and carrying out this mapping repeatedly on each new (reduced) rational number gives the following sequence 2/1,1/2,-1/2,-3/-2,-1/6,... Sequence contains the denominators.

Original entry on oeis.org

1, 2, 2, -2, 6, -6, 42, -42, 1806, -1806, 3263442, -3263442, 10650056950806, -10650056950806, 113423713055421844361000442, -113423713055421844361000442, 12864938683278671740537145998360961546653259485195806

Offset: 1

Amarnath Murthy, Mar 24 2003

The mapping f(a/b) = (a + b)/(a - b). Taking a = 2 and b = 1 to start with and carrying out this mapping repeatedly on each new (reduced) rational number gives the periodic sequence 2/1,3/1,2/1,3/1,...

Seiichi Manyama, Table of n, a(n) for n = 1..26

A003687 gives the numerators.

Cf. A007018.

Last /@ NestList[{(#1 - #2), #1 #2} & @@ # &, {2, 1}, 16] (* Michael De Vlieger, Sep 04 2016 *)

# Variant with first four terms slightly different. Absolute values.
def A081478_abs():
    x, y = 1, 2
    yield x
    while True:
       yield x
       x, y = x * y, x//y + 1
a = A081478_abs(); print([next(a) for i in range(17)])  # Peter Luschny, Dec 17 2015

a(2n-1) = A007018(n-1), a(2n) = -A007018(n-1) for n >= 2. - Jianing Song, Oct 10 2021

More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 06 2003

A100016 a(0) = 1; a(n+1) = a(n) * (next prime larger than a(n)).

Original entry on oeis.org

1, 2, 6, 42, 1806, 3270666, 10697259354222, 114431357691543566765996394, 13094535623129987017538646614449662873664453962869814, 171466863185420237392391564368967506501628543653753176958938044126997508808439363294403869833497610468982

Offset: 0

Parthasarathy Nambi, Nov 18 2004

nonn

Terms are very close to A007018(n), with equality for the first 5 terms. Indeed, sequence A007018 is defined like the present sequence, with nextprime(a(n)) replaced by a(n)+1. - M. F. Hasler, May 20 2019

			If n=1, then the prime immediately greater than n is 2. Hence the next number is n*p = 1*2 = 2.
If n=2, then the next prime is 3, so the next number in the sequence is 2*3=6.
If n=6, then the next prime is 7, so the next number in the sequence is 6*7=42.

Similar to A074839.

Even more similar to A007018.

a[0] = 1; a[n_] := a[n - 1]*NextPrime[a[n - 1]]; Table[ a[n], {n, 0, 9}] (* Robert G. Wilson v, Nov 23 2004 *)
NestList[# NextPrime[#]&,1,10] (* Harvey P. Dale, Oct 15 2016 *)

(nxt_A100016(n)=n*nextprime(n+1)); A100016_vec=vector(10,i,t=if(i>1,nxt_A100016(t),1)) \\ M. F. Hasler, May 20 2019

from itertools import islice
from sympy import nextprime
def A100016_gen(): # generator of terms
    yield (a:=1)
    while (a:=a*nextprime(a)): yield a
A100016_list = list(islice(A100016_gen(),10)) # Chai Wah Wu, Mar 19 2024

More terms from Robert G. Wilson v, Nov 23 2004

A139145 a(1) = 1, a(2n) = a(n)^2, a(2n+1) = a(n)*(a(n)+1).

Original entry on oeis.org

1, 1, 2, 1, 2, 4, 6, 1, 2, 4, 6, 16, 20, 36, 42, 1, 2, 4, 6, 16, 20, 36, 42, 256, 272, 400, 420, 1296, 1332, 1764, 1806, 1, 2, 4, 6, 16, 20, 36, 42, 256, 272, 400, 420, 1296, 1332, 1764, 1806, 65536, 65792, 73984, 74256, 160000, 160400, 176400, 176820, 1679616

Offset: 1

Reinhard Zumkeller, Apr 10 2008

nonn

Cf. A007018.

from functools import lru_cache
@lru_cache(maxsize=None)
def A139145(n): return (m:=A139145(n>>1))*(m+(n&1)) if n>1 else 1 # Chai Wah Wu, Mar 19 2024

a(2^n - 1) = A007018(n-1).

A343511 a(n) = 1 + Sum_{d|n, d < n} a(d)^2.

Original entry on oeis.org

1, 2, 2, 6, 2, 10, 2, 42, 6, 10, 2, 146, 2, 10, 10, 1806, 2, 146, 2, 146, 10, 10, 2, 23226, 6, 10, 42, 146, 2, 314, 2, 3263442, 10, 10, 10, 42814, 2, 10, 10, 23226, 2, 314, 2, 146, 146, 10, 2, 542731938, 6, 146, 10, 146, 2, 23226, 10, 23226, 10, 10, 2, 141578, 2, 10, 146, 10650056950806, 10

Offset: 1

Ilya Gutkovskiy, Apr 17 2021

nonn

a(n) depends only on the prime signature of n (see formulas). - Bernard Schott, Apr 24 2021

Alois P. Heinz, Table of n, a(n) for n = 1..3000
Index entries for sequences related to prime signature

Cf. A025487, A006881, A007018, A007304, A067824, A082588, A333120.

a:= proc(n) option remember;
      1+add(a(d)^2, d=numtheory[divisors](n) minus {n})
    end:
seq(a(n), n=1..65);  # Alois P. Heinz, Apr 17 2021

a[n_] := a[n] = 1 + Sum[If[d < n, a[d]^2, 0], {d, Divisors[n]}]; Table[a[n], {n, 65}]

lista(nn) = {my(va = vector(nn)); for (n=1, nn, va[n] = 1 + sumdiv(n, d, if (dMichel Marcus, Apr 18 2021

from functools import lru_cache
from sympy import divisors
@lru_cache(maxsize=None)
def A343511(n): return 1+sum(A343511(d)**2 for d in divisors(n) if d < n) # Chai Wah Wu, Apr 17 2021

G.f.: x / (1 - x) + Sum_{n>=1} a(n)^2 * x^(2*n) / (1 - x^n).
a(p^k) = A007018(k) for p prime.
From Bernard Schott, Apr 24 2021: (Start)
a(A006881(n)) = 10 for signature [1, 1].
a(A054753(n)) = 146 for signature [2, 1].
a(A007304(n)) = 314 for signature [1, 1, 1].
a(A065036(n)) = 23226 for signature [3, 1].
a(A085986(n)) = 42814 for signature [2, 2].
a(A085987(n)) = 141578 for signature [2, 1, 1]. (End)

A361085 Least prime p > prime(n) such that at least one of p * prime(n)# +- 1 is not squarefree, where prime(n)# is the n-th primorial A002110(n).

Original entry on oeis.org

3, 5, 29, 31, 139, 167, 43, 127, 211, 41, 607, 1223, 71, 769, 1549, 947, 269, 1129, 163, 577, 673, 107, 4057, 1979, 433, 3833, 4177, 383, 1723, 409, 2399, 4517, 3803, 3061, 3299, 457, 3779, 971, 5749, 2843, 13709

Offset: 0

M. F. Hasler, Mar 28 2023

It appears that a product P of distinct primes rarely has the property that P +- 1 has a square factor, and this is even more rare when P has all of the first n primes as factor. This sequence is one possible way to quantify this observation. (One could also display the gap between a(n) and prime(n), or consider b(n) the least product of distinct primes > prime(n) that yields a product with the desired property.)

See also Zumkeller's contested comment in A007018 and the discussion in the linked MathOverflow page.

			a(0) = 3 because for P = (product of the first 0 primes) = 1, p = 3 is the least prime such that p*P + 1 = 4 = 2^2 is a square; for p = 2 neither p*P - 1 = 1 nor p*P + 1 = 3 has a nontrivial square factor.
a(1) = 5 because for P = (product of the first prime) = 2, p = 5 is the least prime such that p*P - 1 = 9 = 3^2 is a square; for p = 3 none of p*P - 1 = 5 nor p*P + 1 = 7 has a nontrivial square factor.
a(2) = 29 because for P = (product of the first two primes) = 6, p = 29 is the least prime such that p*P + 1 = 5^2*7 has a square factor; for all primes 3 < p < 29 both of p*P +- 1 are squarefree.

Dan Asimov, Interesting sequence on MathOverflow, math-fun mailing list, Mar 28 2023.
Fredrick M. Nelson, Does a(0)=6, a(n+1)=a(n)^3-a(n), define a square-free sequence?, MathOverflow, Mar 24 2023.

Cf. A002110 (factorials), A013929 (numbers that are not squarefree), A007018, A228649 (both related).

Map[(k = 1; While[AllTrue[Prime[k] # + {-1, 1}, SquareFreeQ], k++]; Prime[k]) &, FoldList[Times, 1, Prime@ Range[24] ] ] (* Michael De Vlieger, Mar 28 2023 *)

A361085(n, P=vecprod(primes(n)))=forprime(p=prime(n)+1,,(issquarefree(p*P-1)&&issquarefree(p*P+1))||return(p))

a(30) from Michael S. Branicky, Mar 29 2023

a(31)-a(40) from Jinyuan Wang, Mar 30 2023

cp's OEIS Frontend

A003687 a(n+1) = a(n)-a(1)a(2)...a(n-1), if n>0. a(0)=1, a(1)=2.

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

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

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A182398 a(n) = (Sum_{k=1..2n} k^2n) mod 2n.

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A368190 Number of (undirected) cycles in the n-Dorogovtsev-Goltsev-Mendes graph.

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A081478 Consider the mapping f(a/b) = (a - b)/(ab). Taking a = 2 and b = 1 to start with and carrying out this mapping repeatedly on each new (reduced) rational number gives the following sequence 2/1,1/2,-1/2,-3/-2,-1/6,... Sequence contains the denominators.

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Programs

Mathematica

Sage

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A100016 a(0) = 1; a(n+1) = a(n) * (next prime larger than a(n)).

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Mathematica

A139145 a(1) = 1, a(2n) = a(n)^2, a(2n+1) = a(n)*(a(n)+1).