A048298 -id:A048298 - cp's OEIS frontend

A059864 a(n) = Product_{i=4..n} (prime(i)-5), where prime(i) is i-th prime.

1, 1, 1, 2, 12, 96, 1152, 16128, 290304, 6967296, 181149696, 5796790272, 208684449792, 7930009092096, 333060381868032, 15986898329665536, 863292509801938944, 48344380548908580864, 2997351594032332013568

Offset: 1

Labos Elemer, Feb 28 2001

nonn

Such products arise in Hardy-Littlewood prime k-tuplet conjectural formulas.

Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 84-94.
R. K. Guy, Unsolved Problems in Number Theory, A8, A1
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979.
G. Polya, Mathematics and Plausible Reasoning, Vol. II, Appendix Princeton UP, 1954

G. C. Greubel, Table of n, a(n) for n = 1..350
C. K. Caldwell, Prime k-tuple Conjecture
Steven R. Finch, Hardy-Littlewood Constants [Broken link]
Steven R. Finch, Hardy-Littlewood Constants [From the Wayback machine]
G. H. Hardy and J. E. Littlewood, Some problems of 'partitio numerorum'; III: on the expression of a number as a sum of primes, Acta Mathematica, Vol. 44, pp. 1-70, 1923.
G. Niklasch, Some number theoretical constants: 1000-digit values [Cached copy]
G. Polya, Heuristic reasoning in the theory of numbers, Am. Math. Monthly,66 (1959), 375-384.

Cf. A000847, A002110, A005867, A022008, A048298, A049296, A051160, A051161, A051162.
Cf. A051163, A051164, A051165, A051166, A051167, A051168, A059861, A059862, A059863.
Cf. A059864, A059865.

[n le 3 select 1 else (&*[NthPrime(j) -5: j in [4..n]]): n in [1..30]]; // G. C. Greubel, Feb 02 2023

Join[{1,1,1},FoldList[Times,Prime[Range[4,20]]-5]] (* Harvey P. Dale, Dec 29 2018 *)

a(n) = prod(k=4, n, prime(k)-5); \\ Michel Marcus, Dec 12 2017

def A059864(n): return product(nth_prime(j) -5 for j in range(4,n+1))
[A059864(n) for n in range(1,31)] # G. C. Greubel, Feb 02 2023

A060147 Nim-binomial transform of the Nim-squares sequence {0,1,3,2,6,7,5,4,13,12,14,...}.

Original entry on oeis.org

0, 1, 3, 0, 6, 0, 0, 0, 13, 0, 0, 0, 0, 0, 0, 0, 24, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 52, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 103, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

John W. Layman, Mar 06 2001

The Nim-binomial transform of the Nim-squares consists of the Nim-squares of the terms of the Nim-binomial transform of the integers (given in A048298).

Multiplicative with a(2^e) = A006017(e), a(p^e) = 0 otherwise. - David W. Wilson, Jun 12 2005

See A048298.

a(n) = n X n, where Nim-multiplication is used, if n=2^k, else a(n)=0.

A368540 The smallest unitary divisor d of n such that n/d is a term of A138302.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 27, 1, 1, 1, 1, 32, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 27, 1, 8, 1, 1, 1, 1, 1, 1, 1, 64, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 32, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 27, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 125, 1, 1

Offset: 1

Amiram Eldar, Dec 29 2023

First differs from A368167 at n = 64 and from A367513 at n = 128.

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

Cf. A048298, A077610, A138302, A367168, A367513, A368167.

f[p_, e_] := If[e == 2^IntegerExponent[e, 2], 1, p^e]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i, 2] == 1 << valuation(f[i, 2], 2), 1, f[i, 1]^f[i, 2]));}

from math import prod
from sympy import factorint
def A368540(n): return prod(p**e for p, e in factorint(n).items() if not e or (e&-e)^e) # Chai Wah Wu, Dec 30 2023

a(n) = n / A367168(n).
Multiplicative with a(p^e) = p^(e-A048298(e)).
a(n) >= 1, with equality if and only if n is in A138302.

A372504 Multiplicative with a(p^e) = e if e is a power of 2, and 0 otherwise.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, May 04 2024

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

Cf. A005361, A048298, A138302, A209229, A355823, A368473, A372470.

f[p_, e_] := If[e == 2^IntegerExponent[e, 2], e, 0]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = vecprod(apply(x -> if(x == 1 << valuation(x, 2), x, 0), factor(n)[, 2]));

Multiplicative with a(p^e) = A048298(e) = A209229(e) * e.
a(n) = A355823(n) * A005361(n).
a(A138302(n)) = A005361(A138302(n)) = A368473(n).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} f(1/p) = 1.31285540951965780409..., where f(x) = (1-x) * (1 + Sum_{k>=0} 2^k * x^(2^k)).

A060146 Take the first 2n integers and using each integer once and only once as either a numerator or a denominator, construct n fractions whose sum is an integer; a(n) = number of distinct solutions for n.

Original entry on oeis.org

1, 1, 7, 21, 190, 1007, 6972, 111554, 1040635

Offset: 1

Jack Brennen, May 13 2008

nonn

The old entry with this sequence number was a duplicate of A048298.

{ npairs(n) = loca(r,q,z); r=0;
forvec(p=vector(n,i,[1,2*n]),
q = eval( setminus( Set(vector(2*n,i,i)), Set(p) ) );
for(j=1,n!,
z=numtoperm(n,j);
if(type( sum(j=1,#p,p[j]/q[z[j]]) )=="t_INT",r++); );, 2); r }
/* Max Alekseyev, May 14 2008 */

a(6)-a(8) from Max Alekseyev, May 14 2008
Edited by Charles R Greathouse IV, Oct 28 2009
a(9) from Sean A. Irvine, Oct 29 2022

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A059864 a(n) = Product_{i=4..n} (prime(i)-5), where prime(i) is i-th prime.

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A060147 Nim-binomial transform of the Nim-squares sequence {0,1,3,2,6,7,5,4,13,12,14,...}.

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A368540 The smallest unitary divisor d of n such that n/d is a term of A138302.

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A372504 Multiplicative with a(p^e) = e if e is a power of 2, and 0 otherwise.

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A060146 Take the first 2n integers and using each integer once and only once as either a numerator or a denominator, construct n fractions whose sum is an integer; a(n) = number of distinct solutions for n.

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