A277341 a(n) is the nearest integer to prime(n)^prime(n+1)/prime(n+1)^prime(n).

1, 2, 5, 101, 19, 873, 44, 3455, 716066, 122, 3682385, 42002, 239, 74612, 38038256, 75356321, 487, 168475200, 414001, 701, 473945370, 786827, 996734911, 1854156102515, 1757001, 1408, 2223586, 1578, 2777435, 102598699146418244788937, 5067957, 14314401926, 2577, 172311367283303079, 3045

Offset: 1

Author

Ahmad J. Masad, Oct 09 2016

Keywords

• nonn

Comments

Conjecture 1: For each positive integer m, there exists a minimum sufficiently large positive integer r that depends on m, such that taking any two distinct positive integers r1, r2 >= r, we have abs(a(r1) - a(r2)) >= m. For the special case of m=1 it is conjectured that r=1, which would imply (if the conjecture were true) that all terms of this sequence are distinct. - Ahmad J. Masad, Jun 28 2018

A complementary conjecture to Conjecture 1: For each nonnegative integer q, there are infinitely many possible positive integers k, t, w, s such that k < t <= w < s and (t-k) > (s-w) and abs((the nearest integer to (k^t/t^k)) - (the nearest integer to (w^s/s^w))) = q. These two conjectures together describe partially the significance of the set of primes among the set of natural numbers. - Ahmad J. Masad, Mar 29 2018

Conjecture 3: The Riemann hypothesis is true if and only if all terms of this sequence are distinct. This conjecture idea comes from the visual representation of the logarithmic scatterplot of the first 10000 terms of this sequence. - Ahmad J. Masad, Jan 09 2019

Conjecture 4: For each value of n, a(n+1) > a(n) if and only if A058077(n+1) > A058077(n), checked for n <= 10000. Note that the logarithmic scatterplot of A058077 seems to be similar to the logarithmic scatterplot of this sequence. - Ahmad J. Masad, Jun 28 2019

Notification: the conjecture that says that all terms of this sequence are distinct has been checked for the first 10000 terms; that is, the first 10000 terms of this sequence are distinct. - Ahmad J. Masad, Aug 25 2019

Conjecture 5: For each value of n > 1, if a(n) has the same number of digits as a(n+1) and a(n+1) > a(n), then prime(n+2) - prime(n+1) = prime(n+1) - prime(n). This conjecture has been verified for all n < 10000. - Ahmad J. Masad, Oct 08 2019

Examples

			For n = 4, we have ((prime(4)^prime(5))/(prime(5)^prime(4))) = (7^11)/(11^7) = 1977326743/19487171 = 101.4681271..., and 101 is the nearest integer to 101.4681271..., so a(4) = 101.
		

Links

• Daniel Suteu, Table of n, a(n) for n = 1..10000 (first 1000 terms from Vincenzo Librandi)
• Rémy Sigrist, Colored logarithmic scatterplot of (n, a(n)) for n = 1..10000 (where the color is function of prime(n+1)-prime(n))
• Rémy Sigrist, Colored logarithmic scatterplot of (n, a(n)) for n = 1..100000 (where the color is function of prime(n+1)-prime(n))
• Daniel Suteu, Perl program

Crossrefs

Cf. A053089, A058077, A078422.

Programs

Magma

[Round((NthPrime(n)^NthPrime(n+1))/(NthPrime(n+1)^NthPrime(n))): n in [1..40]]; // Vincenzo Librandi Oct 18 2016

Mathematica

Table[Round[((Prime[n]^Prime[n + 1])/(Prime[n + 1]^Prime[n]))], {n, 35}] (* Michael De Vlieger, Oct 14 2016 *)
Round[(#[[1]]^#[[2]])/#[[2]]^#[[1]]]&/@Partition[Prime[Range[40]],2,1] (* Harvey P. Dale, Jun 29 2022 *)

PARI

a(n) = round(prime(n)^prime(n+1)/prime(n+1)^prime(n)); \\ Michel Marcus, Jan 13 2018