A062481 -id:A062481 - cp's OEIS frontend

A160491 First differences of A062481.

7, 235, 16141, 48811741, 13011865891, 232617453293191, 143882557561868665, 8862794004464425240057, 99999991137061880347498904071, 191843424957750480504146841291811

Offset: 1

Cino Hilliard, May 15 2009

nonn

a(1), a(3) and a(6) are the only primes found for n up to 10000 using The C/Gmp program in the link which is 17 times faster than the PARI routine. We read large blocks of primes into memory and test in memory.
If another prime exists, it is larger than 418977 digits.
A concept of order can be introduced here by researching and analyzing the next-nextprime, next-next-nextprime etc.

			For n = 3, 4^7 - 3^3 = 16141, the 3rd entry in the sequence.

C.L. Hilliard,Primes less than 3 trillion

ppower(n) = { for(x=1,n, y=(x+1)^prime(x+1)-x^prime(x); print1(y",") ); }

a(n) = A062481(n+1) - A062481(n).

Edited by R. J. Mathar, May 17 2009

A123113 Main diagonal of prime power sum array.

Original entry on oeis.org

2, 13, 280, 17489, 48909526, 13423779037, 232729381165100, 146367546237420097, 8864305651125125485354, 100000100010100010100010101101, 193529735150413879906083607547512

Offset: 1

Jonathan Vos Post, Sep 28 2006

Main diagonal of the infinite array T(k,n) = 1 + Sum_{i=1..k} n^prime(i).
a(n) is prime for n = 1, 2, 4 -- what is the next prime in the sequence?
The next prime in the sequence is for n = 20. It has 93 digits. - Harvey P. Dale, Jan 18 2017

			a(1) = 2 = 1 + 1^2.
a(2) = 13 = 1 +2^2 +2^3.
a(3) = 280 = 1 +3^2 +3^3 +3^5.
a(4) = 17489 = 1 +4^2 +4^3 +4^5 +4^7.
a(5) = 48909526 = 1 +5^2 +5^3 +5^5 +5^7 +5^11.
a(6) = 13423779037 = 1 +6^2 +6^3 +6^5 +6^7 +6^11 +6^13.
a(7) = 232729381165100 = 1 +7^2 +7^3 +7^5 +7^7 +7^11 +7^13 +7^17.
a(8) = 146367546237420097 = 1 +8^2 +8^3 +8^5 +8^7 +8^11 +8^13 +8^17 +8^19.

G. C. Greubel, Table of n, a(n) for n = 1..75

Cf. A000040, A062481.

[1 + (&+[n^NthPrime(j): j in [1..n]]): n in [1..15]]; // G. C. Greubel, Jul 21 2021

Table[Total[n^Prime[Range[n]]]+1,{n,15}] (* Harvey P. Dale, Jan 18 2017 *)

[1 + sum(n^nth_prime(j) for j in (1..n)) for n in (1..15)] # G. C. Greubel, Jul 21 2021

a(n) = 1 + n^2 + n^3 + n^5 + ... + n^prime(n).

A135484 a(n) = Sum_{i=1..n} i^prime(i), where prime(i) denotes i-th prime number.

Original entry on oeis.org

1, 9, 252, 16636, 48844761, 13109538777, 232643623525984, 144347831699381856, 8863082467484200477785, 100000008863082467484200477785, 192043424966613562971631041769596

Offset: 1

Ctibor O. Zizka, Feb 07 2008

nonn

G. C. Greubel, Table of n, a(n) for n = 1..75

Cf. A000040, A087480.

Partial sums of A062481.

f[n_] := Sum[i^Prime@i, {i, n}]; Array[f, 12] (* Robert G. Wilson v, Feb 12 2008 *)
Accumulate[Table[n^Prime[n],{n,15}]] (* Harvey P. Dale, Nov 30 2023 *)

a(n) = sum(k=1, n, k^prime(k)); \\ Michel Marcus, Oct 15 2016

Edited and extended by Robert G. Wilson v, Feb 12 2008

A318199 a(n) is the largest integer m such that m^n <= n^prime(n).

Original entry on oeis.org

1, 2, 6, 11, 34, 48, 112, 139, 274, 794, 860, 2125, 3259, 3313, 4842, 9741, 18637, 17946, 32306, 41558, 39471, 66148, 82046, 131305, 265464, 313781, 288660, 339008, 313761, 366288, 1287573, 1451134, 2014343, 1824089, 3743848, 3371509, 4510880, 5976406

Offset: 1

Stefano Spezia, Aug 21 2018

nonn

The sequence is not monotonic, for example a(18) < a(17).

Conjecture: there is no run of consecutive increasing terms with more than 17 terms.

Stefano Spezia, Table of n, a(n) for n = 1..10000

Cf. A000040, A062481, A333138.

Digits:= 2000:
a:= n-> floor(n^(ithprime(n)/n)):
seq(a(n),n=1..40); # Muniru A Asiru, Sep 17 2018

a[n_]:=Floor[n^(Prime[n]/n)]; Array[a,40]

a(n) = sqrtnint(n^prime(n), n); \\ Michel Marcus, Mar 12 2020
vector(40, n, a(n))

a(n) = floor(n^(prime(n)/n)).

a(n) = floor(A062481(n)^(1/n)).

A076955 Product_{i=1..n} i^prime(i).

Original entry on oeis.org

1, 8, 1944, 31850496, 1555200000000000, 20311991333683200000000000, 4725188984058417024032990822400000000000, 680971499131541103151460312870118086801149132800000000000

Offset: 1

Amarnath Murthy, Oct 20 2002

nonn

n-th partial product of A062481.

Cf. A076954.

seq(product(i^ithprime(i),i=1..n),n=1..13);

a(1) =1, a(n+1) = a(n)*(n+1)^prime(n+1).

More terms from Sascha Kurz, Jan 22 2003

A096250 Decimal expansion of Sum_{n>=1} 1/n^prime(n).

Original entry on oeis.org

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

Offset: 1

Cino Hilliard, Jul 31 2004

Cf. A062481 (n^prime(n)).

xtothepx(n) = { local(x,s,a); default(realprecision,200); s=0; print1(1","); for(x=1,n,s+=1./x^prime(x)); a=Vec(Str(s)); for(x=3,n,print1(eval(a[x]),",")); print(); print(s) }

suminf(n=1, 1/n^prime(n)) \\ Michel Marcus, Nov 18 2020

A160501 (n+1)^prime(n+1) + n^prime(n).

Original entry on oeis.org

9, 251, 16627, 48844509, 13109522141, 232643574681223, 144347818589843079, 8863082234840576951801, 100000008862938119652501095929, 192043424957750480504146841291811

Offset: 1

Cino Hilliard, May 15 2009

nonn

a(2)=251 is the only prime found for n up to 10000 using The C/Gmp program in the link which is 17 times faster than the PARI routine.
Here there are divisibility rules: If prime(n) and prime(n+1) do not differ by 6, then n^2+n+1 is a divisor. So finding primes in this case will be difficult since 5/6 of the numbers are composite at the onset.
If another prime exists, it is larger than 418977 digits.

			For n = 3, 4^7 + 3^5 = 16627, the 3rd entry in the sequence.

Harvey P. Dale, Table of n, a(n) for n = 1..95
C. L. Hilliard,Sum of prime indices to respective primes

Cf. A160491.

Table[n^Prime[n]+(n+1)^Prime[n+1],{n,10}] (* Harvey P. Dale, Sep 10 2016 *)
Total/@Partition[Table[n^Prime[n] ,{n,15}],2,1] (* Harvey P. Dale, Sep 22 2020 *)

ppower(n) = { for(x=1,n, y=(x+1)^prime(x+1) + x^prime(x); print1(y", ") ); }

a(n) = (n+1)^prime(n+1) + n^prime(n) = A062481(n)+A062481(n+1).

Edited by R. J. Mathar, May 30 2009

A098137 a(n) = n^n + n^prime(n).

Original entry on oeis.org

2, 12, 270, 16640, 48831250, 13060740672, 232630514810750, 144115188092633088, 8862938119652888516418, 100000000000000000010000000000, 191943424957750480504432152962422, 8505622499821102144576131693030930382848, 4695452425098908797088971409337724910182721066

Offset: 1

Parthasarathy Nambi, Sep 27 2004

nonn

			a(1) = 1^1 + 1^2 = 2.
a(2) = 2^2 + 2^3 = 12.
a(3) = 3^3 + 3^5 = 270.

Cf. A000312, A062481, A089622.

[n^n + n^(NthPrime(n)): n in [1..20]]; // Vincenzo Librandi, Aug 27 2015

Table[n^Prime[n] + n^n, {n, 1, 15}] (* Stefan Steinerberger, Feb 28 2006 *)

first(m)=vector(m,i,i^i + i^prime(i)) \\ Anders Hellström, Aug 26 2015

a(n) = A000312(n) + A062481(n). - Michel Marcus, Aug 27 2015

More terms from Stefan Steinerberger, Feb 28 2006

a(12)-a(13) from Vincenzo Librandi, Aug 27 2015

A215633 Decimal expansion of Sum_{n>=1} 1/n^(n^prime(n)).

Original entry on oeis.org

1, 0, 0, 3, 9, 0, 6, 2, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1

Offset: 1

Balarka Sen, Aug 18 2012

Is this a finite sequence? (or equivalently, is this constant rational?)
Unlikely: the "apparent" rationality seems to be caused by the rapid growth of the denominators; one may instead speculate that the value is transcendental. - Joerg Arndt, Feb 05 2013
The constant is approximately 257/256 (see A021260).

			1.00390625000000000000000000... = 1 + 1/2^8 + 1/3^243 + 1/4^16384+...

Cf. A062481.

PARI
```
suminf(n=1,1/n^(n^prime(n)))
```

A259654 a(n) = prime(n)^prime(n) - prime(n)^n.

Original entry on oeis.org

2, 18, 3000, 821142, 285311509560, 302875101765444, 827240261885926425504, 1978419655660296605560938, 20880467999847912032553880249104, 2567686153161211134561828214310308893183268, 17069174130723235958610643029033906279148329600

Offset: 1

Ilya Gutkovskiy, Jul 02 2015

Cf. A062481, A051674.

[NthPrime(n)^NthPrime(n)-NthPrime(n)^n: n in [1..12]]; // Vincenzo Librandi, Jul 03 2015

Table[Prime[n]^Prime[n] - Prime[n]^n, {n, 1, 10}]
#[[1]]^#[[1]]-#[[1]]^#[[2]]&/@Table[{Prime[n],n},{n,15}] (* Harvey P. Dale, Nov 10 2016 *)

a(n,p=prime(n))=p^p - p^n \\ Charles R Greathouse IV, Jul 22 2016

a(n) = A051674(n) - A062457(n).

cp's OEIS Frontend

A160491 First differences of A062481.

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A123113 Main diagonal of prime power sum array.

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A135484 a(n) = Sum_{i=1..n} i^prime(i), where prime(i) denotes i-th prime number.

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A318199 a(n) is the largest integer m such that m^n <= n^prime(n).

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Maple

Mathematica

PARI

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A076955 Product_{i=1..n} i^prime(i).

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Maple

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A096250 Decimal expansion of Sum_{n>=1} 1/n^prime(n).

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PARI

PARI

A160501 (n+1)^prime(n+1) + n^prime(n).

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Mathematica

PARI

Formula

Extensions

A098137 a(n) = n^n + n^prime(n).

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