A114529 -id:A114529 - cp's OEIS frontend

A112407 Decimal expansion of a semiprime analog of a Ramanujan formula.

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

Offset: 0

Jonathan Vos Post, Dec 21 2005

This is related to Ramanujan's surprising formula: Prod[from n = 1 to infinity] (prime(n)^2 - 1)/ (prime(n)^2 + 1) = 2/5 and we use it in finding A112407 as the semiprime analog. We also use: A090986 = Decimal expansion of Pi csch Pi = Prod[from n = 2 to infinity] (n^2 - 1)/(n^2 + 1).
Since every integer above 1 is a k-almost prime for some k, we factor the (n^2 - 1)/(n^2 + 1) infinite product and use Ramanujan's formula, to have: Prod[from n = 1 to infinity] (prime(n)^2-1)/(prime(n)^2+1) * Prod[from n = 1 to infinity] (semiprime(n)^2 - 1)/(semiprime(n)^2 + 1) * Prod[from n = 1 to infinity] (3-almostprime(n)^2 - 1)/ (3-almostprime(n)^2 + 1) * ... * Prod[from n = 1 to infinity] (k-almostprime(n)^2 - 1)/ (k-almostprime(n)^2 + 1) * ... = Prod[from n = 2 to infinity] (n^2 - 1)/(n^2 + 1) = pi csch pi as each integer appear once and only once in numerator and once and only once in denominator.
2/5 is the first (Ramanujan, prime) term in this infinite product of infinite products. This here is the second (semiprime) term. A155799 is the third (3-almost prime) term. All of these have slow convergence.

			0.75449970170951407835571816895054...

Borwein, J.; Bailey, D.; and Girgensohn, R. "Two Products." Section 1.2 in Experimentation in Mathematics: Computational Paths to Discovery. Natick, MA: A. K. Peters, pp. 4-7, 2004.

R. J. Mathar, Hardy-Littlewood constants embedded into infinite products over all positive integers, arXiv:0903.2514, Table 1, k=s=2.
Eric Weisstein's World of Mathematics, Infinite Product.
Eric Weisstein's World of Mathematics, Hyperbolic Cosecant

Cf. A001358, A090986, A114529-A114536.

spQ[n_] := Plus @@ Last /@ FactorInteger@n == 2; p = 1; Do[If[spQ[n], p = N[p*(n^2 - 1)/(n^2 + 1), 64]], {n, 10^6}]; p (* Robert G. Wilson v *) (* This program converges slowly. For {n, 10^6}, only the first 6 digits are correct. - Jason Yuen, Aug 10 2025 *)

A(lim)=my(x=1.);forprime(p=2,lim\2,forprime(q=2,min(p,lim\p),x*=1-2/((p*q)^2+1)));x \\ Charles R Greathouse IV, Aug 15 2011

Decimal expansion of a = prod[from n = 1 to infinity] (semiprime(n)^2 - 1)/(semiprime(n)^2 + 1) = prod[from n = 1 to infinity] (A001358(n)^2 - 1)/(A001358(n)^2 + 1).

log a = -2*sum_{l=1..infinity} P_2(2*(2l-1))/(2l-1), where P_k(s) are the k-almost prime zeta functions of arXiv:0803.0900 [math.NT]. - R. J. Mathar, Jan 27 2009

Edited and extended by R. J. Mathar, Jan 27 2009

A113285 Let S(n)=sigma(|n|)-2*n; sequence gives numbers n such that S(S(S(S(n))))=n. May be called {2,1}-Sociable number of orders 1 or 2 or 4.

Original entry on oeis.org

51, 72, 120, 132, 672, 2602, 4756, 10054, 14884, 45840, 51168, 116252, 523776, 906202, 3003698, 5271836, 65071776, 77260656, 82842816, 89761152, 138357404, 139626548, 459818240, 985948800, 1381340160, 1476304896, 1489384608, 2183133696, 3835877062

Offset: 1

Yasutoshi Kohmoto, Jan 27 2006

nonn

Orders of cycles are 4,4,1,4,1,4,4,4,4,2,2,2,1,2,4,4,4,4,4,4,4,4,...

			{51,-30,132,72} is a {2,1}-Aliquot cycle.

Eric Weisstein's World of Mathematics, Sociable Numbers

Cf. A113791, A114528, A114529.

fQ[n_] := Nest[ DivisorSigma[1, # ] - 2# &, n, 4] == n; t = {}; Do[ If[ fQ[n], AppendTo[t, n]], {n, 3*10^7}]; t (* Robert G. Wilson v *)

a(12)-a(22) from Robert G. Wilson v, Jan 30 2006

a(23)-a(29) from Charles R Greathouse IV, Nov 13 2010

A114528 Let S(n)=sigma(|n|)-3*n; sequence gives numbers n such that S(S(S(S(n))))=n. May be called {3,1}-Sociable number of orders 1 or 2 or 4.

Original entry on oeis.org

1248, 1596, 28272, 30240, 32760, 463296, 678032, 1906128, 2178540, 4694328, 4697616, 4698072, 11110976, 12865770, 23569920, 30998250, 31235904, 37501072, 45532800, 63723600, 76980288, 95801088, 142990848, 146078592, 163032720, 614533696, 1044244800, 1379454720

Offset: 1

Yasutoshi Kohmoto, Feb 15 2006

nonn

Eric Weisstein's World of Mathematics, Sociable Numbers

Cf. A113285, A114529. Contains A069146 as a subsequence.

S(n)=sigma(abs(n))-3*n
is(n)=S(S(S(S(n))))==n \\ Charles R Greathouse IV, Jan 16 2012

a(5)-a(12) from Charles R Greathouse IV, Jan 20 2010

a(13)-a(28) from Donovan Johnson, Jan 16 2012

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A112407 Decimal expansion of a semiprime analog of a Ramanujan formula.

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A113285 Let S(n)=sigma(|n|)-2*n; sequence gives numbers n such that S(S(S(S(n))))=n. May be called {2,1}-Sociable number of orders 1 or 2 or 4.

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A114528 Let S(n)=sigma(|n|)-3*n; sequence gives numbers n such that S(S(S(S(n))))=n. May be called {3,1}-Sociable number of orders 1 or 2 or 4.

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