A120458 -id:A120458 - cp's OEIS frontend

A120459 Row sums of A120458.

1, 3, 14, 161, 3124, 181259, 6732438, 493478345, 24995572328, 2255433009731, 470444892889498, 38714638073629151, 7749166585021832892, 1203906832960860262109, 121893712541593098356318, 17161342484454585041813495, 4656941131185104848296141136, 1513056629126772227843475996471

Offset: 0

Roger L. Bagula, Jun 24 2006

nonn

Cf. A120458, A291140.

a[n_] := 1 + Sum[Prime[k]^n, {k, 1, n}]; Array[a, 18, 0] (* Amiram Eldar, Jun 08 2025 *)

a(n) = 1 + sum(k = 1, n, prime(k)^n); \\ Amiram Eldar, Jun 08 2025

From Amiram Eldar, Jun 08 2025: (Start)
a(n) = 1 + Sum_{k=1..n} prime(k)^n.
a(n) = A291140(n) + 1 for n >= 1. (End)

More terms from Amiram Eldar, Jun 08 2025

A030514 a(n) = prime(n)^4.

Original entry on oeis.org

16, 81, 625, 2401, 14641, 28561, 83521, 130321, 279841, 707281, 923521, 1874161, 2825761, 3418801, 4879681, 7890481, 12117361, 13845841, 20151121, 25411681, 28398241, 38950081, 47458321, 62742241, 88529281, 104060401, 112550881, 131079601, 141158161

Offset: 1

Jeff Burch

Numbers with 5 divisors (1, p, p^2, p^3, p^4, where p is the n-th prime). - Alexandre Wajnberg, Jan 15 2006
Subsequence of A036967. - Reinhard Zumkeller, Feb 05 2008
The n-th number with p divisors is equal to the n-th prime raised to power p-1, where p is prime. - Omar E. Pol, May 06 2008
The general product formula for even s is: product_{p = A000040} (p^s-1)/(p^s+1) = 2*Bernoulli(2s)/( binomial(2s, s)*Bernoulli^2(s)), where the infinite product is over all primes. Here, with s = 4, product_{n = 1, 2, ...} (a(n)-1)/(a(n)+1) = 6/7. In A030516, where s = 6, the product of the ratios is 691/715. For s = 8, the 8th row in A120458, the corresponding product of ratios is 7234/7293. - R. J. Mathar, Feb 01 2009
Except for the first three terms, all others are congruent to 1 mod 240. - Robert Israel, Aug 29 2014

R. J. Mathar, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Prime Power.
OEIS Wiki, Index entries for number of divisors
Index to sequences related to prime signature

Cf. A030078, A085964, A131991, A131992, A000005, A000040, A001248, A157290, A215267.

Cf. A258601.

a030514 = (^ 4) . a000040
a030514_list = map (^ 4) a000040_list
-- Reinhard Zumkeller, Jun 03 2015

[NthPrime(n)^4: n in [1..100] ]; // Vincenzo Librandi, Apr 22 2011

map(p -> p^4, select(isprime,[2,seq(2*i+1,i=1..100)])); # Robert Israel, Aug 29 2014

Array[Prime[#]^4 &, 5!] (* Vladimir Joseph Stephan Orlovsky, Sep 01 2008 *)

a(n)=prime(n)^4 \\ Charles R Greathouse IV, Mar 21 2013

[p**4 for p in prime_range(100)]
# Zerinvary Lajos, May 15 2007

a(n) = A000040(n)^(5-1) = A000040(n)^4, where 5 is the number of divisors of a(n). - Omar E. Pol, May 06 2008
A000005(a(n)) = 5. - Alexandre Wajnberg, Jan 15 2006
A056595(a(n)) = 2. - Reinhard Zumkeller, Aug 15 2011
Sum_{n>=1} 1/a(n) = P(4) = 0.0769931397... (A085964). - Amiram Eldar, Jul 27 2020
From Amiram Eldar, Jan 23 2021: (Start)
Product_{n>=1} (1 + 1/a(n)) = zeta(4)/zeta(8) = 105/Pi^4 (A157290).
Product_{n>=1} (1 - 1/a(n)) = 1/zeta(4) = 90/Pi^4 (A215267). (End)

Description corrected by Eric W. Weisstein

A030627 Numbers with 9 divisors.

Original entry on oeis.org

36, 100, 196, 225, 256, 441, 484, 676, 1089, 1156, 1225, 1444, 1521, 2116, 2601, 3025, 3249, 3364, 3844, 4225, 4761, 5476, 5929, 6561, 6724, 7225, 7396, 7569, 8281, 8649, 8836, 9025, 11236, 12321, 13225, 13924, 14161, 14884, 15129

Offset: 1

Jeff Burch

nonn

Numbers of the form p^8 (8th row of A120458) or p^2*r^2 (A085986), where p and r are distinct primes. - R. J. Mathar, Mar 01 2010

R. J. Mathar, Table of n, a(n) for n = 1..1000

Cf. A000005, A030515, A030516, A030626, A085986, A120458, A179645.

Select[Range[90000],DivisorSigma[0,#]==9&] (* Vladimir Joseph Stephan Orlovsky, May 05 2011 *)

is(n)=numdiv(n)==9 \\ Charles R Greathouse IV, Jun 19 2016

from math import isqrt
from sympy import primepi, primerange
def A030627(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return int(n+x+(t:=primepi(s:=isqrt(y:=isqrt(x))))+(t*(t-1)>>1)-sum(primepi(y//k) for k in primerange(1, s+1))-primepi(isqrt(s)))
    return bisection(f,n,n) # Chai Wah Wu, Feb 21 2025

A000005(a(n)) = 9. - Juri-Stepan Gerasimov, Oct 10 2009

Sum_{n>=1} 1/a(n) = (P(2)^2 - P(4))/2 + P(8) = 0.0678286..., where P is the prime zeta function. - Amiram Eldar, Jul 03 2022

A197987 a(n) = prime(n)^(n+1).

Original entry on oeis.org

4, 27, 625, 16807, 1771561, 62748517, 6975757441, 322687697779, 41426511213649, 12200509765705829, 787662783788549761, 243569224216081305397, 37929227194915558802161, 3177070365797955661914307, 566977372488557307219621121, 205442259656281392806087233013

Offset: 1

Bruno Berselli, Oct 20 2011

Subsequence of A000961, A120458.
First five elements are also consecutive members of A133018. - Omar E. Pol, Oct 20 2011
Third diagonal of A319075. - Omar E. Pol, Sep 13 2018

			The fourth prime number is 7, so a(4) = 7^(4+1) = 7^5 = 16807. - _Omar E. Pol_, Oct 20 2011

Bruno Berselli, Table of n, a(n) for n = 1..50

Cf. A000961, A062457, A093360, A120458, A133018, A319075.

Magma
```
[NthPrime(n)^(n+1): n in [1..16]];
```

Table[Prime[n]^(n+1),{n,20}] (* Harvey P. Dale, Dec 16 2012 *)

for(n=1, 16, print1(prime(n)^(n+1)", "));

a(n) = A000040(n)^(n+1). - Omar E. Pol, Oct 20 2011

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A120459 Row sums of A120458.

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A030514 a(n) = prime(n)^4.

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A030627 Numbers with 9 divisors.

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A197987 a(n) = prime(n)^(n+1).

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