A048649 -id:A048649 - cp's OEIS frontend

A051179 a(n) = 2^(2^n) - 1.

1, 3, 15, 255, 65535, 4294967295, 18446744073709551615, 340282366920938463463374607431768211455, 115792089237316195423570985008687907853269984665640564039457584007913129639935

Offset: 0

Alan DeKok (aland(AT)ox.org)

In a tree with binary nodes (0, 1 children only), the maximum number of unique child nodes at level n.
Number of binary trees (each vertex has 0, or 1 left, or 1 right, or 2 children) such that all leaves are at level n. Example: a(1) = 3 because we have (i) root with a left child, (ii) root with a right child and (iii) root with two children. a(n) = A000215(n) - 2. - Emeric Deutsch, Jan 20 2004
Similarly, this is also the number of full balanced binary trees of height n. (There is an obvious 1-to-1 correspondence between the two sets of trees.) - David Hobby (hobbyd(AT)newpaltz.edu), May 02 2010
Partial products of A000215.
The first 5 terms n (only) have the property that phi(n)=(n+1)/2, where phi(n) = A000010(n) is Euler's totient function. - Lekraj Beedassy, Feb 12 2007
If A003558(n) is of the form 2^n and A179480(n+1) is even, then (2^(A003558(n)) - 1) is in A051179. Example: A003558(25) = 8 with A179480(25) = 4, even. Then (2^8 - 1) = 255. - Gary W. Adamson, Aug 20 2012
For any odd positive a(0), the sequence defined by a(n) = a(n-1) * (a(n-1) + 2) gives a constructive proof that there exist integers with at least n distinct prime factors, e.g., a(n), since omega(a(n)) >= n. As a corollary, this gives a constructive proof of Euclid's theorem stating that there are infinitely many primes. - Daniel Forgues, Mar 07 2017
From Sergey Pavlov, Apr 24 2017: (Start)
I conjecture that, for n > 7, omega(a(n)) > omega(a(n-1)) > n.
It seems that the largest prime divisor p(n+1) of a(n+1) is always bigger than the largest prime divisor of a(n): p(n+1) > p(n). For 3 < n < 8, p(n+1) > 100 * p(n).
(End)
It appears that a(n) is the integer whose bits indicate the possible subset sums of the first n powers of two. For another example, see the calculation for primes at A368491 - Yigit Oktar, Mar 20 2025

			15 = 3*5; 255 = 3*5*17; 65535 = 3*5*17*257; ... - _Daniel Forgues_, Mar 07 2017

M. Aigner and G. M. Ziegler, Proofs from The Book, Springer-Verlag, Berlin, 1999; see p. 4.

Vincenzo Librandi, Table of n, a(n) for n = 0..11
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!)
J. H. Conway, Integral lexicographic codes, Discrete Mathematics 83.2-3 (1990): 219-235. See p. 235.
Ben Delo and Filip Saidak, Euclid's theorem redux, Fib. Q., 57:4 (2019), 331-336.
Index entries for sequences of form a(n+1)=a(n)^2 + ....

Cf. A001146, A007018, A048649.
Cf. A003558, A179480, A000215.
Cf. A368491

[2^(2^n)-1: n in [0..8]]; // Vincenzo Librandi, Jun 20 2011

A051179:=n->2^(2^n)-1; seq(A051179(n), n=0..8); # Wesley Ivan Hurt, Feb 08 2014

Table[2^(2^n)-1,{n,0,9}] (* Vladimir Joseph Stephan Orlovsky, Mar 16 2010 *)

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

def A051179(n): return (1<<(1<Chai Wah Wu, May 03 2023

a(n) = A000215(n) - 2.
a(n) = (a(n-1) + 1)^2 - 1, a(0) = 1. [ or a(n) = a(n-1)(a(n-1) + 2) ].
1 = 2/3 + 4/15 + 16/255 + 256/65535 + ... = Sum_{n>=0} A001146(n)/a(n+1) with partial sums: 2/3, 14/15, 254/255, 65534/65535, ... - Gary W. Adamson, Jun 15 2003
a(n) = b(n-1) where b(1)=1, b(n) = Product_{k=1..n-1} (b(k) + 2). - Benoit Cloitre, Sep 13 2003
A136308(n) = A007088(a(n)). - Jason Kimberley, Dec 19 2012
A000215(n) = a(n+1) / a(n). - Daniel Forgues, Mar 07 2017
Sum_{n>=0} 1/a(n) = A048649. - Amiram Eldar, Oct 27 2020

A051158 Decimal expansion of Sum_{n >= 0} 1/(2^2^n+1).

Original entry on oeis.org

5, 9, 6, 0, 6, 3, 1, 7, 2, 1, 1, 7, 8, 2, 1, 6, 7, 9, 4, 2, 3, 7, 9, 3, 9, 2, 5, 8, 6, 2, 7, 9, 0, 6, 4, 5, 4, 6, 2, 3, 6, 1, 2, 3, 8, 4, 7, 8, 1, 0, 9, 9, 3, 2, 6, 2, 1, 4, 4, 2, 4, 5, 9, 9, 6, 0, 9, 1, 0, 8, 9, 9, 7, 7, 4, 8, 8, 6, 0, 8, 8, 8, 9, 9, 3, 6, 1, 9, 1, 8, 4, 6, 4, 6, 4, 4, 0, 7, 4

Offset: 0

Robert Lozyniak (11(AT)onna.com)

			0.59606317211782167942...

Joerg Arndt, Matters Computational (The Fxtbook), section 38.7, p.740 (gives method for divisionless computation corresponding to PARI/GP code below).
S. Audinarayana Moorthy, Problem E2455, The American Mathematical Monthly, Vol. 81, No. 1 (1974), p. 85, solution, ibid., Vol. 82, No. 2 (1975), pp. 173-174.
Michael Coons, On the rational approximation of the sum of the reciprocals of the Fermat numbers, Raman. J., Vol. 28 (2013), pp. 39-65.
Michael Coons, Addendum to: On the rational approximation of the sum of the reciprocals of the Fermat numbers, arXiv:1511.08147 [math.NT], 2015.
Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 247.
Solomon W. Golomb, On the sum of the reciprocals of the Fermat numbers and related irrationalities, Canad. J. Math., Vol. 15 (1963), pp. 475-478.

A048649 + A051158 = 2.
Terms in continued fraction: A159243. - Enrique Pérez Herrero, Nov 17 2009
Cf. A000215, A000120, A007814.

RealDigits[Sum[1/(2^2^n + 1), {n, 0, 10}], 10, 111][[1]] (* Robert G. Wilson v, Jul 03 2014 *)

/* divisionless routine from fxtbook */
s2(y, N=7)=
{ local(in, y2, A); /* as powerseries correct to order = 2^N-1 */
    in = 1; /* 1+y+y^2+y^3+...+y^(2^k-1) */
    A = y; for(k=2, N, in *= (1+y); y *= y; A += y*(in + A); );
    return( A ); }
a=0.5*s2(0.5) /* computation of the constant 0.596063172117821... */
/* Joerg Arndt, Apr 15 2010 */

suminf(n=0, 1/(2^2^n+1)) \\ Michel Marcus, May 15 2020

Equals (1/2) * Sum_{k>=1} A000120(k)/2^k (S. Audinarayana Moorthy, 1974). - Amiram Eldar, May 15 2020

Equals 1 - Sum_{n>=1} A007814(n)/2^n = 2/3 - Sum_{n>=1} A007814(n)/4^n = 3/5 - Sum_{n>=1} A007814(n)/16^n. - Amiram Eldar, Nov 06 2020

A384251 The number of integers k from 1 to n such that the greatest divisor of k that is an infinitary divisor of n is odd.

Original entry on oeis.org

1, 1, 3, 3, 5, 3, 7, 4, 9, 5, 11, 9, 13, 7, 15, 15, 17, 9, 19, 15, 21, 11, 23, 12, 25, 13, 27, 21, 29, 15, 31, 16, 33, 17, 35, 27, 37, 19, 39, 20, 41, 21, 43, 33, 45, 23, 47, 45, 49, 25, 51, 39, 53, 27, 55, 28, 57, 29, 59, 45, 61, 31, 63, 48, 65, 33, 67, 51, 69

Offset: 1

Amiram Eldar, May 23 2025

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A006519, A048649, A077609.
Analogous sequences: A026741, A384055.
The number of integers k from 1 to n such that the greatest divisor of k that is an infinitary divisor of n is: A384247(1), A384249 (squarefree), A384250 (powerful), this sequence (odd), A384252 (power of 2).

a[n_] := n * If[OddQ[n], 1, 1 - 1/2^(2^(IntegerExponent[IntegerExponent[n, 2], 2]))]; Array[a, 100]

a(n) = n * if(n%2, 1, (1 - 1/(1 << (1 << valuation(valuation(n, 2), 2)))));

Multiplicative with a(2^e) = 2^e * (1 - 1/2^A006519(e)), and a(p^e) = p^e if p is an odd prime.
a(n) = A384247(n)/A384252(n).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = (3 - A048649)/2 = 0.79803158605891083971... .

A048164 a(0)=1, a(n+1)=1+(2^(2^n)+1)*a(n).

Original entry on oeis.org

1, 4, 21, 358, 92007, 6029862760, 25898063359598159721, 477734946799221833229035410333259818858, 162564778457687820218065957445498785826947155451688293007128627114802460256107

Offset: 0

N. J. A. Sloane, E. M. Rains

a(n) = height of lattice of orthogonal arrays with 2^2^n runs.

E. M. Rains, N. J. A. Sloane and J. Stufken, The Lattice of N-Run Orthogonal Arrays, J. Stat. Planning Inference, 102 (2002), 477-500 (Abstract, pdf, ps)

Cf. A039930, A048638.

RecurrenceTable[{a[0]==1,a[n]==1+(2^(2^(n-1))+1)a[n-1]},a,{n,10}] (* Harvey P. Dale, Dec 13 2013 *)

a(n) converges to nearest integer to c*(2^(2^n)-1), where c = 1.403936827882178... (see A048649).

A346192 Decimal expansion of Sum_{k>=0} 1/(2^(2^k)*(2^(2^k) - 1)).

Original entry on oeis.org

5, 8, 7, 5, 1, 5, 3, 1, 8, 8, 6, 0, 2, 8, 5, 1, 7, 6, 8, 6, 8, 1, 2, 6, 3, 3, 6, 6, 0, 6, 6, 8, 4, 1, 3, 2, 8, 3, 4, 3, 2, 7, 3, 8, 5, 6, 0, 1, 3, 3, 7, 2, 6, 3, 0, 3, 7, 3, 4, 6, 6, 3, 6, 1, 9, 6, 0, 3, 3, 4, 8, 6, 0, 3, 0, 5, 5, 8, 6, 1, 1, 0, 4, 0, 1, 5, 2, 7, 6, 8, 2, 8, 6, 9, 5, 8, 6, 6, 0, 0

Offset: 0

Amiram Eldar, Jul 09 2021

This constant is transcendental (Schwarz, 1967).

			0.58751531886028517686812633660668413283432738560133...

Wolfgang Schwarz, Remarks on the irrationality and transcendence of certain series, Mathematica Scandinavica, Vol. 20, No. 2 (1967), pp. 269-274; alternative link.
Index entries for transcendental numbers

Cf. A048649, A087046, , A346190, A346191.

RealDigits[Sum[1/((2^(2^n) - 1)*2^(2^n)), {n, 0, 10}], 10, 100][[1]]

Equals Sum_{k>=2} 1/A087046(k).

A048650 Continued fraction for Sum_{m>=0} 1/(2^2^m - 1).

Original entry on oeis.org

1, 2, 2, 9, 1, 3, 5, 1, 2, 1, 1, 1, 1, 8, 2, 1, 1, 2, 1, 12, 19, 24, 1, 18, 12, 1, 2, 1, 1, 2, 2, 1, 1, 1, 1, 2, 1, 8, 1, 1, 5, 2, 5, 8, 1, 4, 2, 5, 1, 1, 8, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 2, 1, 12, 18, 1, 7, 2, 1, 1, 2, 4, 1, 5, 4, 2, 1, 1, 1, 1, 1, 4, 64, 14, 1, 6, 3, 1, 6

Offset: 0

N. J. A. Sloane

			1.40393682788217832057620607413720935453...
1.403936827882178320576206074... = 1 + 1/(2 + 1/(2 + 1/(9 + 1/(1 + ...)))). - _Harry J. Smith_, May 03 2009

Harry J. Smith, Table of n, a(n) for n = 0..19999
G. Xiao, Contfrac
Index entries for continued fractions for constants

Cf. A048164, A048649 (decimal expansion).

{ allocatemem(932245000); default(realprecision, 21000); x=suminf(m=0, 1/(2^2^m - 1)); x=contfrac(x); for (n=1, 20000, write("b048650.txt", n-1, " ", x[n])); } \\ Harry J. Smith, May 07 2009

Deleted old PARI program. - Harry J. Smith, May 20 2009

Offset changed by Andrew Howroyd, Aug 03 2024

A346190 Decimal expansion of Sum_{k>=0} 1/(2^(2^(2*k+1)) - 1).

Original entry on oeis.org

5, 0, 7, 8, 1, 2, 5, 0, 0, 4, 6, 5, 6, 6, 1, 2, 8, 7, 3, 0, 7, 7, 3, 9, 2, 5, 7, 8, 1, 2, 5, 0, 0, 0, 0, 0, 0, 0, 5, 8, 7, 7, 4, 7, 1, 7, 5, 4, 1, 1, 1, 4, 3, 7, 5, 3, 9, 8, 4, 3, 6, 8, 2, 6, 8, 6, 1, 1, 1, 2, 2, 8, 3, 8, 9, 0, 9, 3, 3, 2, 7, 7, 8, 3, 8, 6, 0, 4, 3, 7, 6, 0, 7, 5, 4, 3, 7, 5, 8, 5

Offset: 0

Amiram Eldar, Jul 09 2021

This constant is transcendental (Schwarz, 1967).

			0.50781250046566128730773925781250000000587747175411...

Wolfgang Schwarz, Remarks on the irrationality and transcendence of certain series, Mathematica Scandinavica, Vol. 20, No. 2 (1967), pp. 269-274; alternative link.
Index entries for transcendental numbers

Cf. A048649, A076214, A346191, A346192.

RealDigits[Sum[1/(2^(2^(2*n + 1) - 1)), {n, 0, 10}], 10, 100][[1]]

Equals A076214 - A346191.

A346191 Decimal expansion of Sum_{k>=0} 1/(2^(2^(2*k)) - 1).

Original entry on oeis.org

1, 1, 2, 5, 0, 3, 0, 5, 1, 7, 5, 7, 8, 1, 2, 5, 0, 0, 0, 1, 0, 8, 4, 2, 0, 2, 1, 7, 2, 4, 8, 5, 5, 0, 4, 4, 3, 4, 0, 0, 7, 4, 5, 2, 8, 0, 0, 8, 6, 9, 9, 4, 1, 7, 1, 1, 4, 2, 5, 7, 8, 1, 2, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 7, 2, 7, 2, 3, 3, 7, 1, 1, 0, 1, 8, 8, 8, 8, 9, 2, 5, 0, 7, 7, 2

Offset: 1

Amiram Eldar, Jul 09 2021

This constant is transcendental (Schwarz, 1967).

			1.12503051757812500010842021724855044340074528008699...

Wolfgang Schwarz, Remarks on the irrationality and transcendence of certain series, Mathematica Scandinavica, Vol. 20, No. 2 (1967), pp. 269-274; alternative link.
Index entries for transcendental numbers

Cf. A048649, A076214, A346190, A346192.

RealDigits[Sum[1/(2^(2^(2*n) - 1)), {n, 0, 10}], 10, 100][[1]]

Equals A076214 - A346190.

A386261 a(n) = A001511(A001511(n)), where A001511 is the ruler function.

Original entry on oeis.org

1, 2, 1, 1, 1, 2, 1, 3, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 3, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1

Offset: 1

Amiram Eldar, Jul 17 2025

The first occurrence of k = 1, 2, ... is at n = 2^(2^(k-1) - 1) = A058891(k).

The asymptotic density of the occurrences of k = 1, 2, ... in this sequence is 2^(2^(k-1))/(2^(2^k)-1) = 2/3, 4/15, 16/255, 256/65535, 65536/4294967295, ...

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A001511, A003159, A048649, A058891.

Similar sequences: A010553, A010554, A051027, A055033, A058699, A068346, A132090, A178601, A181776, A284908, A386262.

f[n_] := IntegerExponent[n, 2] + 1; a[n_] := f[f[n]]; Array[a, 100]

a(n) = valuation(valuation(n, 2) + 1, 2) + 1;

a(n) >= 1, with equality if and only if n is in A003159.

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{m>=0} 1/(2^(2^m) - 1) = 1.4039368... (A048649).

cp's OEIS Frontend

A051179 a(n) = 2^(2^n) - 1.

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A051158 Decimal expansion of Sum_{n >= 0} 1/(2^2^n+1).

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A384251 The number of integers k from 1 to n such that the greatest divisor of k that is an infinitary divisor of n is odd.

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A048164 a(0)=1, a(n+1)=1+(2^(2^n)+1)*a(n).

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A346192 Decimal expansion of Sum_{k>=0} 1/(2^(2^k)*(2^(2^k) - 1)).

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A048650 Continued fraction for Sum_{m>=0} 1/(2^2^m - 1).

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A346190 Decimal expansion of Sum_{k>=0} 1/(2^(2^(2*k+1)) - 1).

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