Leo Depuydt - cp's OEIS frontend

User: Leo Depuydt

Leo Depuydt has authored 3 sequences.

A246447 The odd primes squared plus 1 and the composites squared minus 1.

10, 15, 26, 35, 50, 63, 80, 99, 122, 143, 170, 195, 224, 255, 290, 323, 362, 399, 440, 483, 530, 575, 624, 675, 728, 783, 842, 899, 962, 1023, 1088, 1155, 1224, 1295, 1370, 1443, 1520, 1599, 1682, 1763, 1850, 1935, 2024, 2115, 2210, 2303, 2400, 2499, 2600

Offset: 1

Leo Depuydt, Aug 26 2014

The odd primes squared plus 1 and the nonprimes squared minus 1 are the numerators or denominators of an infinite product converging to 1 whose denominators or numerators, conversely, are the squares of said numbers, that is, (p^2+1/p^2)*(q^2-1)/q^2)..., where p is an odd prime and q is a nonprime.

Union of A066872 and (A062312 - 1) with 0 and 5 removed. - Robert Israel, Aug 26 2014

Jens Kruse Andersen, Table of n, a(n) for n = 1..10000

Cf. A066872, A062312.

lista(nn) = {for (n=3, nn, if (isprime(n), print1(n^2+1, ", "), print1(n^2-1, ", ")););} \\ Michel Marcus, Aug 26 2014

for n in range(3,10**3):
  if isprime(n):
    print(n**2+1,end=', ')
  else:
    print(n**2-1,end=', ') # Derek Orr, Sep 19 2014

More terms from Michel Marcus, Aug 26 2014

A231327 Denominator of rational component of zeta(4n)/zeta(2n).

Original entry on oeis.org

1, 15, 105, 675675, 34459425, 16368226875, 218517792968475, 30951416768146875, 694097901592400930625, 23383376494609715287281703125, 2289686345687357378035370971875, 219012470258383844016431785453125, 4791965046290912124048163518904807546875

Offset: 0

Leo Depuydt, Nov 07 2013

Denominator of a close variant of Euler's infinite prime product zeta(2n) = Product_{k>=1} (prime(k)^(2n))/(prime(k)^(2n)-1), namely with all minus signs changed into plus signs, as follows: zeta(4n)/zeta(2n) = Product_{k>=1} prime(k)^(2n)/(prime(k)^(2n)+1).
For a detailed account of the results in question, including proof and relation to the zeta function, see the PDF file submitted as supporting material in A231273.
The reference to Apostol below is a discussion of the equivalence of 1) zeta(2s)/zeta(s) and 2) a related infinite prime product, that is, Product_{sigma>1} prime(n)^s/(prime(n)^s + 1), with s being a complex variable such that s = sigma + i*t where sigma and t are real (following Riemann), using a type of proof different from the one posted below involving zeta(4n)/zeta(2n). - Leo Depuydt, Nov 22 2013
Denominator of B(4*n)*4^n*(2*n)!/(B(2*n)*(4*n)!) where B(n) are the Bernoulli numbers (see A027641 and A027642). - Robert Israel, Aug 22 2014

T. M. Apostol, Introduction to Analytic Number Theory, Springer, 1976, p. 231.

Cf. A231273 (the corresponding numerator).
Cf. A114362 and A114363 (closely related results).
Cf. A001067, A046968, A046988, A098087, A141590, and A156036 (same number sequence as found in numerator, though in various transformations (alternation of sign, intervening numbers, and so on)).
Cf. A027641 and A027642.

seq(denom(bernoulli(4*n)*4^n*(2*n)!/(bernoulli(2*n)*(4*n)!)),n=0..100); # Robert Israel, Aug 22 2014

Denominator[Table[Zeta[4 n]/Zeta[2 n], {n, 0, 15}]] (* T. D. Noe, Nov 15 2013 *)

A231273 Numerator of zeta(4n)/(zeta(2n) * Pi^(2n)).

Original entry on oeis.org

1, 1, 1, 691, 3617, 174611, 236364091, 3392780147, 7709321041217, 26315271553053477373, 261082718496449122051, 2530297234481911294093, 5609403368997817686249127547, 61628132164268458257532691681, 354198989901889536240773677094747

Offset: 0

Leo Depuydt, Nov 07 2013

Integer component of the numerator of a close variant of Euler's infinite prime product zeta(2n) = Product_{k>=1} (prime(k)^(2n))/(prime(k)^(2n)-1), namely with all minus signs changed into plus signs, as follows: zeta(4n)/zeta(2n) = Product_{k>=1} (prime(k)^(2n))/(prime(k)^(2n)+1). The transcendental component is Pi^(2n).
For a detailed account of the results, including proof and relation to the zeta function, see Links for the PDF file submitted as supporting material.
The reference to Apostol is to a discussion of the equivalence of 1) zeta(2s)/zeta(s) and 2) a related infinite prime product, that is, Product_{sigma>1} prime(n)^s/(prime(n)^s + 1), with s being a complex variable such that s = sigma + i*t where sigma and t are real (following Riemann), using a type of proof different from the one posted below involving zeta(4n)/zeta(2n). On this, see also Hardy and Wright cited below. - Leo Depuydt, Nov 22 2013, Nov 27 2013
The background of the sequence is now described in the link below to L. Depuydt, The Prime Sequence ... . - Leo Depuydt, Aug 22 2014
From Robert Israel, Aug 22 2014: (Start)
Numerator of (-1)^n*B(4*n)*4^n*(2*n)!/(B(2*n)*(4*n)!), where B(n) are the Bernoulli numbers (see A027641 and A027642).
Not the same as abs(A001067(2*n)): they differ first at n=17.
(End)

T. M. Apostol, Introduction to Analytic Number Theory, Springer, 1976, p. 231.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Fourth Edition, Clarendon Press, 1960, p. 255.

Leo Depuydt, Details, including proof and relation to zeta function

Cf. A231327 (corresponding denominator).
Cf. A114362 and A114363 (closely related results).
Cf. A001067, A046968, A046988, A098087, A141590, A156036 (same number sequence, though in various transformations (alternation of signs, intervening numbers, and so on)).
Cf. A027641 and A027642

seq(numer((-1)^n*bernoulli(4*n)*4^n*(2*n)!/(bernoulli(2*n)*(4*n)!)),n=0..100); # Robert Israel, Aug 22 2014

Numerator[Table[Zeta[4n]/(Zeta[2n] * Pi^(2n)), {n, 0, 15}]] (* T. D. Noe, Nov 18 2013 *)

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User: Leo Depuydt

A246447 The odd primes squared plus 1 and the composites squared minus 1.

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A231327 Denominator of rational component of zeta(4n)/zeta(2n).

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A231273 Numerator of zeta(4n)/(zeta(2n) * Pi^(2n)).

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Maple

Mathematica