A000058 -id:A000058 - cp's OEIS frontend

A225670 Slowest-growing sequence of odd primes p where 1/(p+1) sums to 1 without actually reaching it.

3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 53, 2539, 936599, 127852322431, 510819260848900502567, 1553192364608434843485965159509450536731, 52119893982548112392303882371161186032080710958633917215400463948724068502699

Offset: 1

Jonathan Sondow, May 11 2013

nonn

Is there a finite set of odd primes p where 1/(p+1) sums exactly to 1? (This would be an analog of 1/(2+1) + 1/(3+1) + 1/(5+1) + 1/(7+1) + 1/(11+1) + 1/(23+1) = 1 -- see A000058.)

Amiram Eldar, Table of n, a(n) for n = 1..23

Similar to A075442, A181503, A225669.
Cf. A000058.
See also A046689.

a[n_] := a[n] = Block[ {sm = Sum[ 1/(a[i] + 1), {i, n - 1}]}, NextPrime[ Max[ a[n - 1], 1/(1 - sm)]]]; a[0] = 2; Array[ a, 20]

A252730 a(n) = P_n(n) with P_0(z) = z+1 and P_n(z) = z + P_{n-1}(z)*(P_{n-1}(z)-z) for n>1.

Original entry on oeis.org

1, 3, 17, 871, 4870849, 483209576974811, 36956045653220845240164417232897, 8498748758632331927648392184620600167779995785955324343380396911247

Offset: 0

Alois P. Heinz, Dec 20 2014

nonn

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!)

Main diagonal of A177888.

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

p[n_] := p[n] = Function[z, z + If[n == 0, 1, p[n-1][z]*(p[n-1][z] - z)]];
a[n_] := p[n][n];
Table[a[n], {n, 0, 8}] (* Jean-François Alcover, Jun 12 2018, from Maple *)

A275664 a(n) is the sum of the LCM and GCD of all previous terms, with a(0) = 2.

Original entry on oeis.org

2, 4, 6, 14, 86, 3614, 6526886, 21300113901614, 226847426110843688722000886, 25729877366557343481074291996721923093306518970391614

Offset: 0

Andres Cicuttin, Aug 04 2016

nonn

Starting from 1, instead of from 2, it is generated A129871 (A variant of Sylvester's sequence A000058).

Andres Cicuttin, Table of n, a(n) for n = 0..13

Cf. A129871, A000058.

a = {2}; Do[AppendTo[a, LCM @@ a + GCD @@ a], {i, 1, 10}]; Column[a]

a(0) = 2, a(n+1) = lcm(a(0),a(1),..,a(n)) + gcd(a(0),a(1),..,a(n)).

A293390 Least m such that the exponents in expression for n as a sum of distinct powers of 2 are pairwise distinct mod m; a(0) = 0 by convention.

Original entry on oeis.org

0, 1, 1, 2, 1, 3, 2, 3, 1, 2, 3, 4, 2, 4, 3, 4, 1, 3, 2, 5, 3, 3, 4, 5, 2, 5, 4, 5, 3, 5, 4, 5, 1, 2, 3, 3, 2, 4, 5, 6, 3, 4, 3, 6, 4, 4, 5, 6, 2, 3, 5, 6, 4, 6, 5, 6, 3, 6, 5, 6, 4, 6, 5, 6, 1, 4, 2, 4, 3, 5, 3, 7, 2, 4, 4, 4, 5, 5, 6, 7, 3, 5, 4, 7, 3, 5, 6

Offset: 0

Rémy Sigrist, Oct 08 2017

The set of exponents in expression for n as a sum of distinct powers of 2 corresponds to the n-th row of A133457.
The sum of digits of n in base 2^a(n), say s, can be computed without carry in base 2; the Hamming weight of s equals the Hamming weight of n.
a(n) >= A000120(n) for any n > 0.
Apparently, a(n) = A000120(n) iff n = 0 or n belongs to A100290.
a(n) <= A070939(n) for any n >= 0.
For any sequence s of distinct nonnegative integers (s(n) being defined for n >= 0):
- let D_s be defined for any n > 0 by D_s(n) = a(Sum_{k=0..n-1} 2^s(k)),
- then D_s is the discriminator of s as introduced by Arnold, Benkoski, and McCabe in 1985,
- D_s(1) = 1,
- D_s(n) >= n for any n >= 1,
- D_s(n+1) >= D_s(n) for any n >= 1.

			For n=42:
- 42 = 2^5 + 2^3 + 2^1,
- 5 mod 1 = 3 mod 1,
- 5 mod 2 = 3 mod 2,
- 5 mod 3, 3 mod 3 and 1 mod 3 are all distinct,
- hence a(42) = 3.

Robert Israel, Table of n, a(n) for n = 0..10000
Sajed Haque, Jeffrey Shallit, Discriminators and k-Regular Sequences, arXiv:1605.00092 [cs.DM], 2016.

Cf. A000041, A000045, A000058, A000069, A000108, A000110, A000120, A000215, A001147, A001566, A001969, A002808, A003095, A005823, A016726, A062383, A070939, A076793, A100290, A133457, A173195, A270097, A270151, A270176, A272633, A272881, A272882, A273037, A273041, A273043, A273044, A273056, A273062, A273064, A273068, A273237, A273377

f:= proc(n) local L,D,k;
  L:= convert(n,base,2);
  L:= select(t -> L[t+1]=1, [$0..nops(L)-1]);
  if nops(L) = 1 then return 1 fi;
  D:= {seq(seq(L[j]-L[i],i=1..j-1),j=2..nops(L))};
  D:= `union`(seq(numtheory:-divisors(i),i=D));
  min({$2..max(D)+1} minus D)
end proc:
0, seq(f(i),i=1..100); # Robert Israel, Oct 08 2017

{0}~Join~Table[Function[r, SelectFirst[Range@ 10, Length@ Union@ Mod[r, #] == Length@ r &]][Join @@ Position[#, 1] - 1 &@ Reverse@ IntegerDigits[n, 2]], {n, 86}] (* Michael De Vlieger, Oct 08 2017 *)

a(n) = if (n, my (d=Vecrev(binary(n)), x = []); for (i=1, #d, if (d[i], x = concat(x, i-1))); for (m=1, oo, if (#Set(vector(#x, i, x[i]%m))==#x, return (m))), return (0))

a(2*n) = a(n) for any n >= 0.
a(2^k-1) = k for any k >= 0.
a(n) = 1 iff n = 2^k for some k >= 0.
a(n) = 2 iff n belongs to A173195.
a(Sum_{k=1..n} 2^(k^2)) = A016726(n) for any n >= 1.
a(Sum_{k=1..n} 2^A000069(k)) = A062383(n) for any n >= 1.
a(Sum_{k=0..n} 2^(2^k)) = A270097(n) for any n >= 0.
a(Sum_{k=1..n} 2^A000045(k+1)) = A270151(n) for any n >= 1.
a(Sum_{k=1..n} 2^A000041(k)) = A270176(n) for any n >= 1.
a(A076793(n)) = A272633(n) for any n >= 0.
a(Sum_{k=1..n} 2^A001969(k)) = A272881(n) for any n >= 1.
a(Sum_{k=1..n} 2^A005823(k)) = A272882(n) for any n >= 1.
a(Sum_{k=1..n} 2^A000215(k-1)) = A273037(n) for any n >= 1.
a(Sum_{k=1..n} 2^A000108(k)) = A273041(n) for any n >= 1.
a(Sum_{k=1..n} 2^A001566(k)) = A273043(n) for any n >= 1.
a(Sum_{k=1..n} 2^A003095(k)) = A273044(n) for any n >= 1.
a(Sum_{k=1..n} 2^A000058(k-1)) = A273056(n) for any n >= 1.
a(Sum_{k=1..n} 2^A002808(k)) = A273062(n) for any n >= 1.
a(Sum_{k=1..n} 2^(k!)) = A273064(n) for any n >= 1.
a(Sum_{k=1..n} 2^(k^k)) = A273068(n) for any n >= 1.
a(Sum_{k=1..n} 2^A000110(k)) = A273237(n) for any n >= 1.
a(Sum_{k=1..n} 2^A001147(k)) = A273377(n) for any n >= 1.

A348640 Denominators of the remainders in the greedy Egyptian fraction representation of 1 with square denominators (A348626).

Original entry on oeis.org

1, 4, 2, 4, 36, 36, 1764, 2352, 115248, 416333400, 107225418169800, 562904175532925098845000, 1857180475556752726157213892231405000, 424594887903818740281781489141947299544299873193026842805000, 27616236678198713245845367246922973802897093015095664467139174240964043973815461112656369429045000

Offset: 0

Max Alekseyev, Oct 26 2021

a(n) divides LCM( A348626(1), ..., A348626(n) )^2.

			The first few remainders are 1, 3/4, 1/2, 1/4, 5/36, 1/36, 13/1764, 1/2352, 1/115248, 11/416333400, ... - _N. J. A. Sloane_, Apr 21 2025

Cf. A000058, A348626, A348641 (numerators), A382719.

s=1; for(n=1, 20, print1(denominator(s), ", "); t=sqrtint(floor(1/s))+1; s-=1/t^2);

a(n) = denominator of 1 - Sum_{k=1..n} 1/A348626(k)^2.

A348641 Numerators of the remainders in the greedy Egyptian fraction representation of 1 with square denominators (A348626).

Original entry on oeis.org

1, 3, 1, 1, 5, 1, 13, 1, 1, 11, 817, 10252633, 100287877217, 6528073355352461938177, 62417959978427831731164878741347502689913, 70288410375198910851231147751405037331087262102769745506188780420713, 1637848790982120651632223869737258212156187623721099799629950249330321081907360495884020503587938103781073751577

Offset: 0

Max Alekseyev, Oct 26 2021

			The first few remainders are 1, 3/4, 1/2, 1/4, 5/36, 1/36, 13/1764, 1/2352, 1/115248, 11/416333400, ... - _N. J. A. Sloane_, Apr 21 2025

Cf. A000058, A348626, A348640 (denominators), A382719.

s=1; for(n=1, 20, print1(numerator(s), ", "); t=sqrtint(floor(1/s))+1; s-=1/t^2);

a(n) = numerator of 1 - Sum_{k=1..n} 1/A348626(k)^2.

A369607 Greedy solution a(1) < a(2) < ... to 1/a(1) + 1/a(2) + ... = (1 - 1/a(1)) * (1 - 1/a(2)) * ....

Original entry on oeis.org

3, 6, 29, 803, 643727, 414383582243, 171713753231982206218247, 29485613049014079571725771288849499850026859243, 869401376876189366008603664962520703088459987798626788985159595026678611496977754082506135887

Offset: 1

Max Alekseyev, Jan 27 2024

nonn

For any n, (x1, x2, ..., xn) = (a(1), a(2), ..., a(n-1), a(n)-1) forms a solution to 1/x1 + ... + 1/xn = (1 - 1/x1) * ... * (1 - 1/xn), proving that A369470(n) >= A369469(n) >= 1.

Max Muller et al., The diophantine equation Sum_{n=1..N} 1/x_n = Product_{k=1..N} (1-1/x_k), MathOverflow, 2024.

Cf. A000058, A348625, A348626, A369469, A369470.

a(n+2) = a(n+1)^2 + (a(n) - 2)*a(n+1) - a(n)^3 + 2*a(n)^2 - 2*a(n) + 2.

A065035 a(n+1) = a(n)^2 + 3*a(n) + 1.

Original entry on oeis.org

0, 1, 5, 41, 1805, 3263441, 10650056950805, 113423713055421844361000441, 12864938683278671740537145998360961546653259485195805, 165506647324519964198468195444439180017513152706377497841851388766535868639572406808911988131737645185441

Offset: 0

Henry Bottomley, Nov 03 2001

a(k) is the optimal competitive ratio of any memoryless algorithm for the weighted k-server problem (Chiplunkar and Vishwanathan). - David Eppstein, Dec 31 2013

This is a divisibility sequence, that is if n divides m then a(n) divides a(m). Cf. A002065. - Peter Bala, Mar 26 2018

			a(3) = a(2)^2 + 3*a(2) + 1 = 25 + 15 + 1 = 41.

Harry J. Smith, Table of n, a(n) for n = 0..12
Ashish Chiplunkar and Sundar Vishwanathan, On Randomized Memoryless Algorithms for the Weighted k-server Problem, 54th IEEE Symp. Foundations of Computer Science (FOCS 2013), pp. 11-19; arXiv:1301.0123
Index entries for sequences of form a(n+1)=a(n)^2 + ...

Cf. A000058, A002065, A007018, A028387, A060136.

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

{ for (n=0, 12, a=if(n, a^2 + 3*a + 1, 0); write("b065035.txt", n, " ", a) ) } \\ Harry J. Smith, Oct 03 2009

a(n) = A007018(n)-1 = A000058(n)-2 = A060136(3, n) = A028387(a(n-1)). - Michael Somos, Feb 10 2002

A084594 a(n) = Sum_{r=0..2^(n-1)} Binomial(2^n,2r)*3^r.

Original entry on oeis.org

1, 4, 28, 1552, 4817152, 46409906716672, 4307758882900393634270543872, 37113573186414494550922197215584520229965687291643953152

Offset: 0

Mario Catalani (mario.catalani(AT)unito.it), May 31 2003

a(n)/A084595(n) converges to sqrt(3). Related to Newton's iteration.

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!)
Eric Weisstein's World of Mathematics, Newton's Iteration.

Cf. A001146, A026150, A084595,

Table[Sum[Binomial[2^n, 2 r]3^r, {r, 0, 2^(n - 1)}], {n, 0, 8}]
Table[Simplify[Expand[(1/2) ((1 + Sqrt[3])^(2^n) + (1 - Sqrt[3])^(2^n))]], {n, 0, 7}] (* Artur Jasinski, Oct 11 2008 *)

a(n) = ( (1+sqrt(3))^(2^n) + (1-sqrt(3))^(2^n) )/2.
a(n) = A026150(2^n).
a(n) = 2*a(n-1)^2 - A001146(n-1), n>1.
a(n) = a(n-1)^2 + 3*A084595(n-1)^2.

A084595 For n > 0: a(n) = Sum_{r=0..2^(n-1)-1} binomial(2^n, 2r+1)*3^r.

Original entry on oeis.org

1, 2, 16, 896, 2781184, 26794772135936, 2487085750646543836443049984, 21427531469765285263614058238314319540132878612321796096

Offset: 0

Mario Catalani (mario.catalani(AT)unito.it), May 31 2003

A084594(n)/a(n) converges to sqrt(3). Related to Newton's iteration.

a(n) is divisible by 2^n.

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!)
Eric Weisstein's World of Mathematics, Newton's Iteration.

Cf. A001146, A002605, A084594.

For n>0: Table[Sum[Binomial[2^n, 2 r + 1]3^r, {r, 0, 2^(n - 1) - 1}], {n, 1, 8}]

a(n) = if (n==0, 1, sum(r=0, 2^(n-1)-1, binomial(2^n, 2*r+1)*3^r)); \\ Michel Marcus, Sep 09 2019; corrected Jun 13 2022

a(n) = ((1+sqrt(3))^(2^n) - (1-sqrt(3))^(2^n))/(2*sqrt(3)).
For n > 1:
a(n) = 2*a(n-1)*sqrt(3*a(n-1)^2 + A001146(n-1)).
a(n) = 2*a(n-1)*A084594(n-1).
a(n) = A002605(2^n).

cp's OEIS Frontend

A225670 Slowest-growing sequence of odd primes p where 1/(p+1) sums to 1 without actually reaching it.

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A252730 a(n) = P_n(n) with P_0(z) = z+1 and P_n(z) = z + P_{n-1}(z)*(P_{n-1}(z)-z) for n>1.

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A275664 a(n) is the sum of the LCM and GCD of all previous terms, with a(0) = 2.

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A293390 Least m such that the exponents in expression for n as a sum of distinct powers of 2 are pairwise distinct mod m; a(0) = 0 by convention.

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A348640 Denominators of the remainders in the greedy Egyptian fraction representation of 1 with square denominators (A348626).

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A348641 Numerators of the remainders in the greedy Egyptian fraction representation of 1 with square denominators (A348626).

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PARI

Formula

A369607 Greedy solution a(1) < a(2) < ... to 1/a(1) + 1/a(2) + ... = (1 - 1/a(1)) * (1 - 1/a(2)) * ....

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A065035 a(n+1) = a(n)^2 + 3*a(n) + 1.

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