A078422 -id:A078422 - cp's OEIS frontend

A104127 (1+prime(n))^prime(n).

9, 64, 7776, 2097152, 743008370688, 793714773254144, 2185911559738696531968, 5242880000000000000000000, 55572324035428505185378394701824, 6863037736488300000000000000000000000000000

Offset: 1

Cino Hilliard, Mar 06 2005

Sum of reciprocals rapidly converges to 0.1268651887726617214821302614..

(1+#)^#&/@Prime[Range[20]] (* Harvey P. Dale, Nov 19 2015 *)

A104127(n) := block(
        return((1+A000040(n))^A000040(n))
  )$
  for n : 1 thru 20 do printf(true,"~d,",A104127(n)) ; /* R. J. Mathar, Feb 27 2012 */

ptopp1(n) = { local(x,z,sr=0); forprime(x=1,n, z=(x+1)^x; sr+=1./z; print1(z","); ); print(); print(sr) }

Definition (which equaled A078422) corrected and dummy variables in PARI program removed by R. J. Mathar, Oct 23 2009

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

A373495 a(1) = 2; thereafter, a(n) = prime(n)^prime(n-1) (mod 10).

Original entry on oeis.org

2, 9, 5, 7, 1, 7, 7, 9, 7, 9, 1, 3, 1, 3, 3, 7, 9, 1, 7, 1, 7, 9, 7, 9, 7, 1, 3, 3, 9, 3, 7, 1, 3, 9, 9, 1, 3, 3, 3, 7, 9, 1, 1, 7, 7, 9, 1, 7, 3, 9, 3, 9, 1, 1, 3, 3, 9, 1, 3, 1, 3, 7, 7, 1, 7, 7, 1, 3, 7, 9, 3, 9, 3, 7, 9, 7, 9, 7, 1, 9, 9, 1, 1, 7, 9, 7, 9, 7, 1, 3, 3, 9, 3, 1, 9, 7, 9, 1, 3, 1, 7, 3, 3, 9, 1

Offset: 1

Robert G. Wilson v, Jun 06 2024

nonn

This sequence is not periodic.

			a(2) = 3^2 (mod 10) = 9.
a(3) = 5^3 (mod 10) = 5.

Ilan Vardi, "Computational Recreations in Mathematica," Addison-Wesley Publishing, Redwood City, CA, 1991, p. 226-229.

Paolo Xausa, Table of n, a(n) for n = 1..10000
Robert P. Munafo, Hypercalc - The Calculator That Doesn't Overflow.
Robert P. Munafo, Sequence A092188, Smallest Positive Integer M such that 2^3^4^5^...^N = M mod N.
Robert G. Wilson v, Mathematica coding for "SuperPowerMod" from Vardi.
Wolfram cloud Function Repository, PowerTowerMod.

Cf. A000040, A092188, A133612, A171881, A171882, A213013, A336111, A342176.

a[n_] := Switch[ Mod[ Prime[n], 10], 1, 1, 3, If[ Mod[ Prime[n -1], 4] == 1, 3, 7], 5, 5, 7, If[ Mod[ Prime[n -1], 4] == 1, 7, 3], 9, 9]; a[1] = 2; a[2] = 9; Array[a, 105]
Join[{2}, Map[PowerMod[#[[2]], #[[1]], 10] &, Partition[Prime[Range[100]], 2, 1]]] (* Paolo Xausa, Jul 14 2025 *)

a(n) = if(n<2, 2, lift(Mod(prime(n),10)^prime(n-1))) \\ Hugo Pfoertner, Jul 07 2024

a(n) = A078422(n-1) mod 10. - R. J. Mathar, Jul 14 2025

A097499 Numbers of the form p^q + q^p where p and q are consecutive primes.

Original entry on oeis.org

17, 368, 94932, 1996813914, 36314872537968, 8660320497414243870, 244552822542936127033092, 257904243416235317958787975746, 3091062959814255272215316579358416079052

Offset: 1

Cino Hilliard, Aug 24 2004

nonn

The first term is the only prime in the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..75

Cf. A053089, A078422.

lst={}; Do[p=Prime[n]; q=Prime[n+1]; a=p^q+q^p; AppendTo[lst,a],{n,4!}]; lst (* Vladimir Joseph Stephan Orlovsky, Aug 16 2009 *)
First[#]^Last[#]+Last[#]^First[#]&/@Partition[Prime[Range[10]],2,1] (* Harvey P. Dale, Sep 20 2011 *)

f(n) = for(x=1,n,p=prime(x);q=prime(x+1);v=p^q+q^p;print1(v","))

a(n) = A053089(n) + A078422(n). - Amiram Eldar, Jul 07 2024

Offset corrected by Amiram Eldar, Jul 07 2024

A280609 Odd prime powers with prime exponents.

Original entry on oeis.org

9, 25, 27, 49, 121, 125, 169, 243, 289, 343, 361, 529, 841, 961, 1331, 1369, 1681, 1849, 2187, 2197, 2209, 2809, 3125, 3481, 3721, 4489, 4913, 5041, 5329, 6241, 6859, 6889, 7921, 9409, 10201, 10609, 11449, 11881, 12167, 12769, 16129, 16807, 17161, 18769, 19321, 22201, 22801, 24389, 24649, 26569, 27889, 29791, 29929

Offset: 1

Ilya Gutkovskiy, Jan 06 2017

Intersection of A053810 and A061345.

			9 is in the sequence because 9 = 3^2;
25 is in the sequence because 25 = 5^2;
27 is in the sequence because 27 = 3^3, etc.

Eric Weisstein's World of Mathematics, Prime Power.

Cf. A000961, A034785, A051006, A053810, A061345, A078422, A246547, A246551, A246655.

Select[Range[30000], PrimePowerQ[#1] && PrimeQ[PrimeOmega[#1]] && Mod[#1, 2] == 1 & ]

from sympy import primepi, integer_nthroot, primerange
def A280609(n):
    def f(x): return int(n+x-sum(primepi(integer_nthroot(x, p)[0])-1 for p in primerange(x.bit_length())))
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    return bisection(f,n,n) # Chai Wah Wu, Sep 12 2024

a(n) = p^q, where p, q are primes and p > 2.

Sum_{n>=1} 1/a(n) = Sum_{p prime} P(p) - A051006 = 0.25699271237062131298..., where P(s) is the prime zeta function. - Amiram Eldar, Sep 13 2024

A180203 Differences between prime powers of primes, offsetting the prime and the power by only one. (For purposes of this sequence, 0 and 1 are treated as primes; see Formula.)

Original entry on oeis.org

1, 1, 7, 116, 16682, 19470364, 1792140906866, 9902785872511900, 5480376953206769280002, 74609990540732925759722548, 4316720643133944843150107810831502

Offset: 1

Rick Stephens (rickstphns68(AT)hotmail.com), Aug 15 2010

nonn

Except for the first few terms, first differences of A078422. - Franklin T. Adams-Watters, Mar 08 2014

			n=7 13^11 - 11^7 = 1 792 160 394 037 - 19 487 171 = 1 792 140 906 866.

Cf. A078422, A088385.

a(n)= {EP(n+2)^EP(n+1)}-{EP(n+1)^EP(n)}, where EP(1) = 0, EP(2) = 1, and EP(n) = Prime(n-2) for n > 2.

a(8)-a(11) from Robert G. Wilson v, Aug 23 2010

A268062 a(n) = Sum_{k=1..n} prime(k+1)^prime(k).

Original entry on oeis.org

9, 134, 16941, 19504112, 1792179898149, 9906370212804086, 5480396764155014990025, 74620951324354865576898512, 4316720792370367095095683949638501, 17761887757410618772194137156551786713472772, 4113915065494528452775640793448453170290434881585

Offset: 1

Emre APARI, Jan 25 2016

Partial sums of A078422. - Michel Marcus, Jan 26 2016

			a(3) = 3^2+5^3+7^5 = 16941.

Cf. A051674, A074745, A078422, A138323.

[&+[NthPrime(k+1)^NthPrime(k): k in [1..n]]: n in [1..12]]; // Vincenzo Librandi, Jan 26 2016

Table[Sum[Prime[k+1]^Prime[k], {k, 1, n}], {n, 1, 12}] (* Vaclav Kotesovec, Jan 25 2016 *)

a(n) = sum(k=1, n, prime(k+1)^prime(k)); \\ Michel Marcus, Jan 26 2016

[sum(nth_prime(i+1)^nth_prime(i) for i in [1..n]) for n in [1..15]] # Tom Edgar, Jan 25 2016

cp's OEIS Frontend

A104127 (1+prime(n))^prime(n).

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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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Magma

Mathematica

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A373495 a(1) = 2; thereafter, a(n) = prime(n)^prime(n-1) (mod 10).

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Programs

Mathematica

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Formula

A097499 Numbers of the form p^q + q^p where p and q are consecutive primes.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A280609 Odd prime powers with prime exponents.

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Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Python

Formula

A180203 Differences between prime powers of primes, offsetting the prime and the power by only one. (For purposes of this sequence, 0 and 1 are treated as primes; see Formula.)

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Author

Keywords

Comments

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