A098882 -id:A098882 - cp's OEIS frontend

A048656 a(n) is the number of unitary (and also of squarefree) divisors of n!.

1, 2, 4, 4, 8, 8, 16, 16, 16, 16, 32, 32, 64, 64, 64, 64, 128, 128, 256, 256, 256, 256, 512, 512, 512, 512, 512, 512, 1024, 1024, 2048, 2048, 2048, 2048, 2048, 2048, 4096, 4096, 4096, 4096, 8192, 8192, 16384, 16384, 16384, 16384, 32768, 32768, 32768, 32768

Offset: 1

Labos Elemer

nonn

Let K(n) be the field that is generated over the rationals Q by adjoining the square roots of the numbers 1,2,3,...,n, i.e., K(n) = Q(sqrt(1),sqrt(2),...,sqrt(n)); a(n) is the degree of this field over Q. - Avi Peretz (njk(AT)netvision.net.il), Mar 20 2001
For n>1, a(n) is the number of ways n! can be expressed as the product of two coprime integers p and q such that 0 < p/q < 1, if negative integers are considered as well. This is the answer to the 2nd problem of the International Mathematical Olympiad 2001. Example, for n = 3, the a(3) = 4 products are 3! = (-2)*(-3) = (-1)*(-6) = 1*6 = 2*3. - Bernard Schott, Jan 21 2021
a(n) = number of subsets S of {1,2,...,n} such that every number in S is a prime. - Clark Kimberling, Sep 17 2022

			For n = 7, n! = 5040 = 16*9*5*7 with 4 distinct prime factors, so a(7) = A034444(7!) = 16.
The subsets S of {1, 2, 3, 4} such that every number in S is a prime are these: {}, {2}, {3}, {2, 3}; thus, a(4) = 4. - _Clark Kimberling_, Sep 17 2022

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
International Mathematical Olympiad 2001, Hong Kong Preliminary Selection Contest, Problem 2.
Index to divisibility sequences.
Index to sequences related to Olympiads.

Cf. A000720, A001221, A034444, A357214, A357215.

Cf. A098882, A098990.

Table[2^PrimePi[n], {n, 1, 70}] (* Clark Kimberling, Sep 17 2022 *)

a(n)=2^primepi(n) \\ Charles R Greathouse IV, Apr 07 2012

A001221(n!) = A000720(n) so a(n) = A034444(n!) = 2^A000720(n).

Sum_{n>=1} 1/a(n) = A098882 + 1 = A098990 - 1. - Amiram Eldar, Mar 13 2025

A098990 Decimal expansion of Sum_{n>=1} prime(n)/(2^n).

Original entry on oeis.org

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

Offset: 1

Joseph Biberstine (jrbibers(AT)indiana.edu), Nov 07 2004

Relates the growth of the n-th prime function A000040(n) to the base-2 exponential of n.

			3.6746439660113287789956763090840294116777975887794373283122052201763...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 2.2.1, p. 96.

Thomas Bloom, Is Sum p_n/2^n irrational? (Here p_n is the nth prime), Erdős Problems.
Peter J. Cho and Henry H. Kim, The average of the smallest prime in a conjugacy class, International Mathematics Research Notices, Vol. 2020, No. 6 (2020), pp. 1718-1747, arXiv preprint, arXiv:1601.03012 [math.NT], 2016.
Paul Erdős, Remarks on number theory. I., Mat. Lapok, Vol. 12 (1961), pp. 10-17; Math. Rev. 26 #2410.
Steven R. Finch, Average least nonresidues, December 4, 2013. [Cached copy, with permission of the author]
Paul Pollack, The average least quadratic nonresidue modulo m and other variations on a theme of Erdős, J. Number Theory, Vol. 132, No. 6 (2012), pp. 1185-1202, alternative link.
Terence Tao, Erdős problem database, see no. 251.

Cf. A000040, A053760, A098882.

f:=N->sum(ithprime(n)/2^n,n=1..N); evalf[106](f(500)); evalf[106](f(1000));

RealDigits[Sum[Prime[i]/2^i,{i,1000}],10,120][[1]] (* Harvey P. Dale, Apr 10 2012 *)

suminf(k=1, prime(k)/2^k) \\ Michel Marcus, Jan 13 2016

Equals Sum_{n>=1} prime(n)/2^n.
Equals 2 plus the constant in A098882. - R. J. Mathar, Sep 02 2008
Equals lim_{n->oo} (1/n) * Sum_{k=1..n} A053760(k). - Amiram Eldar, Oct 29 2020

cp's OEIS Frontend

A048656 a(n) is the number of unitary (and also of squarefree) divisors of n!.

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