A053089 -id:A053089 - cp's OEIS frontend

A078422 a(n) = prime(n+1)^prime(n).

9, 125, 16807, 19487171, 1792160394037, 9904578032905937, 5480386857784802185939, 74615470927590710561908487, 4316720717749415770740818372739989

Offset: 1

Lior Manor, Dec 29 2002

nonn

			a(7) = 17^13 = 5480386857784802185939.

Cf. A000040, A053089, A373495 (sequence of last digits).

Table[Prime[n+1]^Prime[n], {n, 1, 10}] (* Vaclav Kotesovec, Jan 25 2016 *)
#[[2]]^#[[1]]&/@Partition[Prime[Range[10]],2,1] (* Harvey P. Dale, Oct 16 2023 *)

a(n) = A000040(n+1)^A000040(n) - Omar E. Pol, Feb 27 2020

A138323 a(n) = Sum_{k = 1..n} prime(k)^prime(k + 1).

Original entry on oeis.org

8, 251, 78376, 1977405119, 34524689549050, 8650450444070886983, 239081086135595395734136, 257829867026393862843621801395

Offset: 1

Vladimir Joseph Stephan Orlovsky, May 07 2008

nonn

			2^3=8
2^3+3^5=8+243=251
2^3+3^5+5^7=8+243+78125=78376

Harvey P. Dale, Table of n, a(n) for n = 1..75

Cf. A051674, A053089, A074745.

P3[n_] := Sum[Prime[i]^Prime[i + 1], {i, 1, n}]; Table[P3[n], {n, 1, 8}]
Accumulate[#[[1]]^#[[2]]&/@Partition[Prime[Range[10]],2,1]] (* Harvey P. Dale, Jan 18 2025 *)

a(n) = sum(k=1, n, prime(k)^prime(k+1)); \\ Michel Marcus, Jan 25 2016

A140893 a(n) = prime(n)^prime(n+1) - prime(n+1)^prime(n).

Original entry on oeis.org

-1, 118, 61318, 1957839572, 32730551749894, 8640511341348431996, 233592048827366522661214, 257755012474380136537664158772, 3091054326372819773383775097721670599074, 2141662167055484666186673758527328459608763158

Offset: 1

Juri-Stepan Gerasimov, Jul 07 2008

sign

a(n) > 0 for n>=2. - Robert Israel, Nov 02 2014

a(n) = A053089(n) - A078422(n). - Michel Marcus, Oct 10 2016

			n=1: a(1) = prime(1)^prime(1+1) - prime(1+1)^prime(1) = 2^3 - 3^2 = 8 - 9 = -1.
n=3: a(3) = prime(3)^prime(4) - prime(4)^prime(3) = 5^7 - 7^5 = 78125 - 16807 = 61318.

Robert Israel, Table of n, a(n) for n = 1..70

Cf. A000040, A007965, A143776.

[NthPrime(n)^NthPrime(n+1)-NthPrime(n+1)^NthPrime(n): n in [1..10]]; // Vincenzo Librandi, Nov 02 2014

seq(ithprime(i)^ithprime(i+1)-ithprime(i+1)^ithprime(i), i=1..20); # Robert Israel, Nov 02 2014

Array[Prime[ # ]^Prime[ #+1]-Prime[ #+1]^Prime[ # ]&,16] (* Vladimir Joseph Stephan Orlovsky, Oct 11 2009 *)

Corrected and extended by Vladimir Joseph Stephan Orlovsky, Oct 11 2009

a(10) from Vincenzo Librandi, Nov 02 2014

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

Original entry on oeis.org

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

Ahmad J. Masad, Oct 09 2016

nonn

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

			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.

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

Cf. A053089, A058077, A078422.

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

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 *)

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

A358895 Numbers k such that p(k)^p(k + 1) < p(k + 2)^p(k), where p(k) = prime(k).

Original entry on oeis.org

1, 2, 3, 10, 33, 41, 45, 52, 60, 98, 113, 120, 262, 294, 296, 318, 343, 377, 408, 428, 444, 475, 477, 486, 572, 601, 673, 700, 774, 837, 870, 913, 934, 936, 944, 984, 1050, 1115, 1169, 1182, 1230, 1232, 1287, 1391, 1445, 1456, 1550, 1584, 1647, 1653, 1674

Offset: 1

Clark Kimberling, Dec 06 2022

nonn

			For k = 3, we have 78125 = p(3)^p(4) < p(5)^p(3) = 161051.

Cf. A000040, A053089, A358896.

p[n_] := Prime[n];
u = Select[Range[3000], p[#]^p[# + 1] < p[# + 2]^p[#] &]  (* A358895 *)
Prime[u]  (* A358896 *)

A358896 Primes p(k) such that p(k)^p(k + 1) < p(k + 2)^p(k).

Original entry on oeis.org

2, 3, 5, 29, 137, 179, 197, 239, 281, 521, 617, 659, 1667, 1931, 1949, 2111, 2309, 2591, 2801, 2969, 3119, 3371, 3389, 3467, 4157, 4421, 5021, 5279, 5879, 6449, 6761, 7127, 7331, 7349, 7457, 7757, 8387, 8969, 9437, 9547, 10007, 10037, 10529, 11549, 12071

Offset: 1

Clark Kimberling, Dec 06 2022

nonn

			For k = 3, we have 5^7 = p(3)^p(4) < p(5)^p(3) = 11^5.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A000040, A053089, A358895.

p[n_] := Prime[n];
u = Select[Range[3000], p[#]^p[# + 1] < p[# + 2]^p[#] &]  (* A358895 *)
Prime[u]  (* A358896 *)
Select[Partition[Prime[Range[1500]],3,1],#[[1]]^#[[2]]<#[[3]]^#[[1]]&][[All,1]] (* Harvey P. Dale, Dec 17 2022 *)

A358897 Numbers k such that p(k)^p(k) < p(k+1)^p(k-1), where p(k) = prime(k).

Original entry on oeis.org

46, 99, 263, 295, 297, 319, 344, 378, 409, 429, 487, 573, 602, 838, 914, 937, 945, 985, 1051, 1116, 1170, 1231, 1233, 1288, 1392, 1446, 1457, 1551, 1585, 1648, 1675, 1708, 1710, 1831, 1879, 1908, 1983, 2032, 2064, 2154, 2176, 2250, 2310, 2327, 2344, 2524

Offset: 1

Clark Kimberling, Dec 06 2022

nonn

			For k=46, let p = prime(45) = 197, q = prime(46) = 199, and r = prime(47) = 211. Then q^q < r^p, where (r^p) = (2.5815...)*q^q.

Cf. A000040, A053089, A358898.

p[n_] := Prime[n];
u = Select[1 + Range[3000], p[#]^p[#] < p[# + 1]^p[# - 1] &] (* A358897 *)
Prime[u]  (* A358898 *)

A358898 Primes p(k) such that p(k)^p(k) < p(k+1)^p(k-1).

Original entry on oeis.org

199, 523, 1669, 1933, 1951, 2113, 2311, 2593, 2803, 2971, 3469, 4159, 4423, 6451, 7129, 7351, 7459, 7759, 8389, 8971, 9439, 10009, 10039, 10531, 11551, 12073, 12163, 13009, 13339, 13933, 14251, 14563, 14593, 15683, 16141, 16453, 17209, 17683, 17989, 18919

Offset: 1

Clark Kimberling, Dec 06 2022

nonn

			For k=46, let p = prime(45) = 197, q = prime(46) = 199, and r = prime(47) = 211.  Then q^q < r^p, where (r^p) = (2.5815...)*q^q.

Cf. A000040, A053089, A358897.

p[n_] := Prime[n];
u = Select[1 + Range[3000], p[#]^p[#] < p[# + 1]^p[# - 1] &] (* A358897 *)
Prime[u]  (* A358898 *)

A358899 Numbers k such that p(k)^p(k) > p(k-1)^p(k+1), where p(k) = prime(k).

Original entry on oeis.org

3, 5, 10, 35, 190, 206, 294, 296, 320, 332, 336, 430, 458, 463, 530, 558, 608, 616, 636, 726, 805, 837, 870, 891, 1117, 1171, 1198, 1230, 1232, 1275, 1328, 1371, 1391, 1410, 1445, 1571, 1634, 1651, 1709, 1832, 1880, 1987, 2028, 2066, 2075, 2244, 2249, 2311

Offset: 1

Clark Kimberling, Dec 06 2022

nonn

			For k = 3, we have prime(2) = 2, prime(3) = 5, prime(4) = 7, and 3125 = 5^5 > 3^7 = 2187.

Cf. A000040, A053089, A358900.

p[n_] := Prime[n];
u = Select[1 + Range[3000], p[#]^p[#] > p[# - 1]^p[# + 1] &]  (* A358899 *)
Prime[u]  (* A358900 *)

A358900 Primes prime(k) such that prime(k)^prime(k) > prime(k-1)^prime(k+1).

Original entry on oeis.org

5, 11, 29, 149, 1151, 1277, 1931, 1949, 2129, 2237, 2267, 2999, 3251, 3299, 3821, 4049, 4481, 4547, 4721, 5501, 6197, 6449, 6761, 6947, 8999, 9461, 9719, 10007, 10037, 10427, 10937, 11351, 11549, 11777, 12071, 13217, 13829, 13997, 14591, 15727, 16183, 17291

Offset: 1

Clark Kimberling, Dec 06 2022

nonn

			For k = 3, we have prime(2) = 2, prime(3) = 5, prime(4) = 7, and 3125 = 5^5 > 3^7 = 2187.

Alois P. Heinz, Table of n, a(n) for n = 1..2000 (first 600 terms from Harvey P. Dale)

Cf. A000040, A053089, A358899.

p[n_] := Prime[n];
u = Select[1 + Range[3000], p[#]^p[#] > p[# - 1]^p[# + 1] &]  (* A358899 *)
Prime[u]  (* A358900 *)
Select[Partition[Prime[Range[2000]],3,1],#[[2]]^#[[2]]>#[[1]]^#[[3]]&][[;;,2]] (* Harvey P. Dale, Oct 12 2023 *)

cp's OEIS Frontend

A078422 a(n) = prime(n+1)^prime(n).

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A138323 a(n) = Sum_{k = 1..n} prime(k)^prime(k + 1).

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A140893 a(n) = prime(n)^prime(n+1) - prime(n+1)^prime(n).

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

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A358895 Numbers k such that p(k)^p(k + 1) < p(k + 2)^p(k), where p(k) = prime(k).

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A358896 Primes p(k) such that p(k)^p(k + 1) < p(k + 2)^p(k).

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A358897 Numbers k such that p(k)^p(k) < p(k+1)^p(k-1), where p(k) = prime(k).

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A358898 Primes p(k) such that p(k)^p(k) < p(k+1)^p(k-1).

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A358899 Numbers k such that p(k)^p(k) > p(k-1)^p(k+1), where p(k) = prime(k).

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