reviewed

approved

proposed

reviewed

editing

proposed

Numbers n k such that A086793(nk) is 1.

Primes Prime numbers in the sequence are also primes with digit sum = 14 (A106756). - Zak Seidov, May 21 2006

ss={8, 14}; Do[If[15==Total@Flatten[IntegerDigits/@Divisors[n]], AppendTo[ss, n]], {n, 20, 2000}]; ss - _(* _Zak Seidov_, May 21 2006 *)

_Zak Seidov, _, May 16 2006

approved

editing

Primes numbers in the sequence are also primes with digit sum = 14 (A106756). - _Zak Seidov (zakseidov(AT)yahoo.com), _, May 21 2006

ss={8, 14}; Do[If[15==Total@Flatten[IntegerDigits/@Divisors[n]], AppendTo[ss, n]], {n, 20, 2000}]; ss - _Zak Seidov (zakseidov(AT)yahoo.com), _, May 21 2006

proposed

approved

nonn,base

approved

proposed

Decimal expansion of Pi csch PiNumbers n such that A086793(n) is 1.

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

8, 14, 20, 26, 59, 62, 122, 123, 143, 149, 167, 206, 239, 257, 293, 302, 341, 347, 383, 419, 422, 491, 509, 563, 617, 653, 743, 761, 941, 1049, 1133, 1193, 1202, 1203, 1229, 1283, 1313, 1319, 1331, 1373, 1409, 1427, 1481, 1553, 1571, 1607

pi csch pi where where csch(z) = hyberbolic cosecant, csch(z) = 1/sinh(z) = 2/(e^z - e^-z) and directly implemented in Mathematica as Csch[z]. pi csch pi = PRODUCT[from n = 2 to infinity] (n^2 - 1)/(n^2 + 1). This is one of an infinite set of infinite products, the next being PRODUCT[from n = 2 to infinity] (n^3 - 1)/(n^3 + 1) = 2/3. This is related to Ramanujan's surprising formula: PRODUCT[from n = 1 to infinity] (prime(n)^2 - 1)/(prime(n)^2 + 1) = 5/2 and we use it in finding A112407 the semiprime analog.

Primes numbers in the sequence are also primes with digit sum = 14 (A106756). - Zak Seidov (zakseidov(AT)yahoo.com), May 21 2006

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.

Eric W. Weisstein, <a href="http://mathworld.wolfram.com/InfiniteProduct.html">Infinite Product</a>.

Eric W. Weisstein, <a href="http://mathworld.wolfram.com/HyperbolicCosecant.html">Hyperbolic Cosecant</a>

0.272029...

ss={8, 14}; Do[If[15==Total@Flatten[IntegerDigits/@Divisors[n]], AppendTo[ss, n]], {n, 20, 2000}]; ss - Zak Seidov (zakseidov(AT)yahoo.com), May 21 2006

Cf. A114528-A114536A034690, A086793, A106756.

cons,easy,nonn,new

nonn

Jonathan Vos Post (jvospost2(AT)yahoo.com), Dec 07 2005

Zak Seidov, May 16 2006

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.

cons,easy,nonn,new

Decimal expansion of Pi csch Pi.

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

1,1

pi csch pi where where csch(z) = hyberbolic cosecant, csch(z) = 1/sinh(z) = 2/(e^z - e^-z) and directly implemented in Mathematica as Csch[z]. pi csch pi = PRODUCT[from n = 2 to infinity] (n^2 - 1)/(n^2 + 1). This is one of an infinite set of infinite products, the next being PRODUCT[from n = 2 to infinity] (n^3 - 1)/(n^3 + 1) = 2/3. This is related to Ramanujan's surprising formula: PRODUCT[from n = 1 to infinity] (prime(n)^2 - 1)/(prime(n)^2 + 1) = 5/2 and we use it in finding A112407 the semiprime analog.

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

Eric W. Weisstein, <a href="http://mathworld.wolfram.com/InfiniteProduct.html">Infinite Product</a>.

Eric W. Weisstein, <a href="http://mathworld.wolfram.com/HyperbolicCosecant.html">Hyperbolic Cosecant</a>

0.272029...

Cf. A114528-A114536.

cons,easy,nonn

Jonathan Vos Post (jvospost2(AT)yahoo.com), Dec 07 2005

approved