A007404 -id:A007404 - cp's OEIS frontend

A356242 a(n) is the number of Fermat numbers dividing n, counted with multiplicity.

0, 0, 1, 0, 1, 1, 0, 0, 2, 1, 0, 1, 0, 0, 2, 0, 1, 2, 0, 1, 1, 0, 0, 1, 2, 0, 3, 0, 0, 2, 0, 0, 1, 1, 1, 2, 0, 0, 1, 1, 0, 1, 0, 0, 3, 0, 0, 1, 0, 2, 2, 0, 0, 3, 1, 0, 1, 0, 0, 2, 0, 0, 2, 0, 1, 1, 0, 1, 1, 1, 0, 2, 0, 0, 3, 0, 0, 1, 0, 1, 4, 0, 0, 1, 2, 0, 1

Offset: 1

Amiram Eldar, Jul 30 2022

nonn

The multiplicity of a divisor d (not necessarily a prime) of n is defined in A169594 (see also the first formula).
A000244(n) is the least number k such that a(k) = n.
The asymptotic density of occurrences of 0 is 1/2.
The asymptotic density of occurrences of 1 is (1/2) * Sum_{k>=0} 1/(2^(2^k)+1) = (1/2) * A051158 = 0.2980315860... .

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Fermat Number.
Wikipedia, Fermat number.

Cf. A000244, A000215, A007404, A051158, A051179, A169594, A356241.

Cf. A080307 (positions of nonzeros), A080308 (positions of 0's).

f = Table[(2^(2^n) + 1), {n, 0, 5}]; a[n_] := Total[IntegerExponent[n, f]]; Array[a, 100]

a(n) = Sum_{k>=1} v(A000215(k), n), where v(m, n) is the exponent of the largest power of m that divides n.
a(A000215(n)) = 1.
a(A000244(n)) = a(A000351(n)) = a(A001026(n)) = n.
a(A003593(n)) = A112754(n).
a(n) >= A356241(n).
a(A051179(n)) = n.
a(A080307(n)) > 0 and a(A080308(n)) = 0.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{k>=0} 1/(2^(2^k)) = 0.8164215090... (A007404).

A386537 Exponent of the highest power of 2 dividing the n-th number whose prime factorization exponents are all powers of 2 (A138302).

Original entry on oeis.org

0, 1, 0, 2, 0, 1, 0, 0, 1, 0, 2, 0, 1, 0, 4, 0, 1, 0, 2, 0, 1, 0, 0, 1, 2, 0, 1, 0, 0, 1, 0, 2, 0, 1, 0, 0, 1, 0, 2, 0, 1, 0, 4, 0, 1, 0, 2, 0, 0, 0, 1, 0, 2, 0, 1, 0, 0, 1, 0, 2, 0, 1, 0, 0, 1, 0, 2, 0, 1, 0, 4, 0, 1, 0, 2, 0, 1, 0, 0, 1, 0, 2, 0, 1, 0, 0, 1

Offset: 1

Amiram Eldar, Jul 25 2025

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

Cf. A007404, A007814, A138302.

Similar sequences: A383004, A383005, A383006, A383007, A383029, A383032, A386536.

exp2nQ[n_] := AllTrue[FactorInteger[n][[;; , 2]], # == 2^IntegerExponent[#, 2] &];
IntegerExponent[Select[Range[200], exp2nQ], 2]

isexp2n(n) = {my(f = factor(n)); for(i=1, #f~, if(f[i, 2] >> valuation(f[i, 2], 2) > 1, return (0))); 1;}
list(lim) = for(k = 1, lim, if(isexp2n(k), print1(valuation(k, 2), ", ")));

a(n) = A007814(A138302(n)).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = (1 + Sum_{k>=0} (2^k + 1)/2^(2^k)) / (1 + Sum_{k>=0} 1/2^(2^k)) - 1 = 0.70550483007968767769... .

A232734 Decimal expansion of Integral {x=0..infinity} 1/2^(2^x) dx.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Nov 29 2013

			0.546306835952482741736098769624101388937635539081659135416783399176163689841...

I. S. Gradsteyn, I. M. Ryzhik, Table of integrals, series and products, (1980) 8.212.

Cf. A007400, A007404 (sum instead of integral).

RealDigits[-ExpIntegralEi[-Log[2]]/Log[2], 10, 100] // First

eint1(log(2))/log(2) \\ Charles R Greathouse IV, Dec 02 2013

-Ei(-log(2))/log(2), where Ei is the exponential integral function.
Also equals (2*Integral_{x = 0..1/2} log(log(1/x)) dx - log(log(2)))/(2*log(2)).
From Peter Bala, Feb 05 2024: (Start)
Equals 1/log(2) * Integral_{x >= 1} 1/(x * 2^x) dx.
Equals 1/log(4) * Integral_{x = 0..1} 1/(log(2) - log(x)) dx.
Equals Integral_{x >= 1} log(x)/2^x dx = (log(2))^2 * Integral_{x >= 0} x*(2^x) /(2^(2^x)). See Gradsteyn and Ryzhik, Section 8.212, formulas (4) and (16). (End)

A085010 a(n)=2^(2^n)*sum(k=0,n,1/2^(2^k)).

Original entry on oeis.org

1, 3, 13, 209, 53505, 3506503681, 15060318633198616577, 277813843495134114797235287762174738433, 94535152227927400227782074307303551040545228366095741656402842333161034088449

Offset: 0

Benoit Cloitre, Jun 19 2003

nonn

Cf. A007404.

a(n)=A074854(2^n)=; a(n)=floor(c*2^(2^n)) where c=sum(k>=0, 1/2^(2^k))=0.81642150902...

a(n + 1) = 1 + a(n)*2^(2^n), a(0) = 1 [From Peter Moxey (pmoxey(AT)live.com), Mar 14 2010]

A339253 Decimal expansion of the unique real nontrivial zero of the Fredholm series, i.e., the complex equation Sum_{k>=0} z^(2^k) = 0 (negated).

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, Nov 28 2020

The trivial zero is z = 0.
This constant was found by Mahler (1980), who also found 3 pairs of conjugate complex zeros, and later (1982) 5 more pairs.
Zannier and Veneziano (2020) proved that there are infinitely many complex zeros in the complex unit disk.

			-0.65862675430016392241347283057950164594093279622043...

David Masser, Auxiliary Polynomials in Number Theory, Cambridge University Press, 2016. See pp. 27-29.

Kurt Mahler, On a special function, Journal of Number Theory, Vol. 12, No. 1 (1980), pp. 20-26; alternative link.
Kurt Mahler, On the zeros of a special sequence of polynomials, Mathematics of Computation, Vol. 39, No. 159 (1982), pp. 207-212; alternative link.
Umberto Zannier and Francesco Veneziano, A note on the zeroes of the Fredholm series, arXiv:2006.11922 [math.CV], 2020.

Cf. A007404, A036987.

m = 10; RealDigits[x /. FindRoot[Sum[x^(2^k), {k, 0, m}] == 0, {x, -0.65}, WorkingPrecision -> 120], 10, 100][[1]]

cp's OEIS Frontend

A356242 a(n) is the number of Fermat numbers dividing n, counted with multiplicity.

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A386537 Exponent of the highest power of 2 dividing the n-th number whose prime factorization exponents are all powers of 2 (A138302).

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A232734 Decimal expansion of Integral {x=0..infinity} 1/2^(2^x) dx.

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A085010 a(n)=2^(2^n)*sum(k=0,n,1/2^(2^k)).

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A339253 Decimal expansion of the unique real nontrivial zero of the Fredholm series, i.e., the complex equation Sum_{k>=0} z^(2^k) = 0 (negated).

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