A152441 -id:A152441 - cp's OEIS frontend

A380840 Decimal expansion of Sum_{p prime} 1/(p-1)^3.

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

Offset: 1

Artur Jasinski, Mar 19 2025

nonn

			1.1475290977585800469332838..,

Artur Jasinski, Complete list of the 117 convergent cases of Sum_{primes p} 1/((p-1)^m p^n (p+1)^r) for exponents -3 <= n,m,r <= 3
Index to constants which are prime zeta sums {0,3,0}

Cf. A085541, A086242, A085548, A136141, A152441, A154945, A179119, A369632, A324833.

PARI
```
sumeulerrat(1/(p-1)^3)
```

A246549 Prime powers p^e where p is a prime and e >= 3 (prime powers without 1, the primes, or the squares of primes).

Original entry on oeis.org

8, 16, 27, 32, 64, 81, 125, 128, 243, 256, 343, 512, 625, 729, 1024, 1331, 2048, 2187, 2197, 2401, 3125, 4096, 4913, 6561, 6859, 8192, 12167, 14641, 15625, 16384, 16807, 19683, 24389, 28561, 29791, 32768, 50653, 59049, 65536, 68921, 78125, 79507, 83521, 103823, 117649, 130321, 131072, 148877, 161051, 177147, 205379

Offset: 1

Joerg Arndt, Aug 29 2014

nonn

Consists of 8 and the terms of A088247. - R. J. Mathar, Sep 01 2014

Jens Kruse Andersen, Table of n, a(n) for n = 1..10000

Cf. A000961, A152441, A246547, A246550.

With[{nn=60},Take[Union[Flatten[Table[p^Range[3,nn/3],{p,Prime[ Range[ nn]]}]]],nn]] (* Harvey P. Dale, Dec 10 2015 *)

for(n=1, 10^6, if(isprimepower(n)>=3, print1(n, ", ")));

m=10^6; v=[]; forprime(p=2, m^(1/3), e=3; while(p^e<=m, v=concat(v, p^e); e++)); v=vecsort(v) \\ Faster program. Jens Kruse Andersen, Aug 29 2014

from math import isqrt
from sympy import primerange, integer_nthroot, primepi
def A246549(n):
    def g(x,a,b,c,m): yield from (((d,) for d in enumerate(primerange(b+1,isqrt(x//c)+1),a+1)) if m==2 else (((a2,b2),)+d for a2,b2 in enumerate(primerange(b+1,integer_nthroot(x//c,m)[0]+1),a+1) for d in g(x,a2,b2,c*b2,m-1)))
    def f(x): return int(n+x-sum(primepi(integer_nthroot(x, k)[0]) for k in range(3, 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 11 2024

Sum_{n>=1} 1/a(n) = Sum_{p prime} 1/(p^2*(p-1)) = A152441. - Amiram Eldar, Oct 24 2020

A135177 a(n) = p^2*(p-1), where p = prime(n).

Original entry on oeis.org

4, 18, 100, 294, 1210, 2028, 4624, 6498, 11638, 23548, 28830, 49284, 67240, 77658, 101614, 146068, 201898, 223260, 296274, 352870, 383688, 486798, 564898, 697048, 903264, 1020100, 1082118, 1213594, 1283148, 1430128, 2032254, 2230930

Offset: 1

Omar E. Pol, Nov 25 2007

			a(4) = 294 because the 4th prime number is 7, 7^2 = 49, 7-1 = 6 and 49 * 6 = 294.

Antti Karttunen, Table of n, a(n) for n = 1..10000 (first 1000 terms from Vincenzo Librandi)
Index to sequences related to prime powers.

Cf. A001248 (p^2), A030078 (p^3), A045991 (n^2 * (n-1)), A065414, A065483, A138416 (terms halved), A152441.

Column 4 of A379010.

[(p^3-p^2): p in PrimesUpTo(200)]; // Vincenzo Librandi, Dec 15 2010

Table[Prime[n]^3-Prime[n]^2, {n, 1, 12}] (* Vladimir Joseph Stephan Orlovsky, Apr 29 2008 *)
Table[p^3-p^2,{p,Prime[Range[40]]}] (* Harvey P. Dale, Jan 15 2015 *)

forprime(p=2,1e3,print1(p^2*(p-1)", ")) \\ Charles R Greathouse IV, Jun 16 2011

A135177(n) = eulerphi(prime(n)^3); \\ Antti Karttunen, Dec 14 2024

A135177(n) = ((p->(p-1)*p*p)(prime(n))); \\ Antti Karttunen, Dec 14 2024

a(n) = p^3 - p^2 = A030078(n) - A001248(n).
a(n) = A000010(prime(n)^3). - R. J. Mathar, Oct 15 2017
Sum_{n>=1} 1/a(n) = A152441. - Amiram Eldar, Nov 09 2020
From Amiram Eldar, Nov 22 2022: (Start)
Product_{n>=1} (1 + 1/a(n)) = A065483.
Product_{n>=1} (1 - 1/a(n)) = A065414. (End)
a(n) = 2*A138416(n). - Antti Karttunen, Dec 14 2024

A372695 Cubefull numbers that are not prime powers.

Original entry on oeis.org

216, 432, 648, 864, 1000, 1296, 1728, 1944, 2000, 2592, 2744, 3375, 3456, 3888, 4000, 5000, 5184, 5488, 5832, 6912, 7776, 8000, 9261, 10000, 10125, 10368, 10648, 10976, 11664, 13824, 15552, 16000, 16875, 17496, 17576, 19208, 20000, 20736, 21296, 21952, 23328, 25000

Offset: 1

Michael De Vlieger, May 14 2024

nonn

Numbers k such that rad(k)^3 | k and omega(k) > 1. In other words, numbers with at least 2 distinct prime factors whose prime power factors have exponents that exceed 2.
Proper subset of the following sequences: A001694, A036966, A126706, A286708.
Superset of A372841.
Smallest term k with omega(k) = m is k = A002110(m)^3 = A115964(m).

			Table of smallest 12 terms and instances of omega(a(n)) = m for m = 2..4
    n      a(n)
  ------------------------
    1      216 = 2^3 * 3^3
    2      432 = 2^4 * 3^3
    3      648 = 2^3 * 3^4
    4      864 = 2^5 * 3^3
    5     1000 = 2^3 * 5^3
    6     1296 = 2^4 * 3^4
    7     1728 = 2^6 * 3^3
    8     1944 = 2^3 * 3^5
    9     2000 = 2^4 * 5^3
   10     2592 = 2^5 * 3^4
   11     2744 = 2^3 * 7^3
   12     3375 = 3^3 * 5^3
  ...
   43    27000 = 2^3 * 3^3 * 5^3
  ...
  587  9261000 = 2^3 * 3^3 * 5^3 * 7^3

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A001694, A007947, A024619, A036966, A115964, A126706, A286708, A372841.

Cf. A065483, A152441.

nn = 25000; Rest@ Select[Union@ Flatten@ Table[a^5 * b^4 * c^3, {c, Surd[nn, 3]}, {b, Surd[nn/(c^3), 4]}, {a, Surd[nn/(b^4 * c^3), 5]}], Not@*PrimePowerQ]

from math import gcd
from sympy import primepi, integer_nthroot, factorint
def A372695(n):
    def f(x):
        c = n+1+x+sum(primepi(integer_nthroot(x, k)[0]) for k in range(3, x.bit_length()))
        for w in range(1,integer_nthroot(x,5)[0]+1):
            if all(d<=1 for d in factorint(w).values()):
                for y in range(1,integer_nthroot(z:=x//w**5,4)[0]+1):
                    if gcd(w,y)==1 and all(d<=1 for d in factorint(y).values()):
                        c -= integer_nthroot(z//y**4,3)[0]
        return c
    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

Intersection of A036966 and A024619.

Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + 1/(p^2*(p-1))) - Sum_{p prime} 1/(p^2*(p-1)) - 1 = A065483 - A152441 - 1 = 0.0188749045... . - Amiram Eldar, May 17 2024

A366076 The number of prime factors of the largest divisor of n that is a cubefull number (A036966), counted with multiplicity.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 3, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 3, 0, 3, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 4, 4, 0, 0, 0, 0, 0, 0

Offset: 1

Amiram Eldar, Sep 28 2023

The sum of exponents larger than 2 in the prime factorization of n.

The number of distinct prime factors of the largest divisor of n that is a cubefull number is A295659(n).

Amiram Eldar, Table of n, a(n) for n = 1..10000
Rafael Jakimczuk and Matilde Lalín, Sums of omega(n) and Omega(n) over the k-free parts and k-full parts of some particular sequences, Integers, Vol. 22 (2022), Article #A113.

Cf. A001222, A004709, A036966, A046100, A085541, A152441, A295659, A360540, A366077.

Similar sequence: A275812 (number of prime factors of the powerful part).

f[p_, e_] := If[e < 3, 0, e]; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = vecsum(apply(x -> if(x < 3, 0, x), factor(n)[, 2]));

a(n) = A001222(A360540(n)).
a(n) = A001222(n) - A366077(n).
Additive with a(p^e) = 0 if e <= 2, and a(p^e) = e for e >= 3.
a(n) >= 0, with equality if and only if n is cubefree (A004709).
a(n) <= A001222(n), with equality if and only if n is cubefull (A036966).
a(n) >= 3*A295659(n), with equality if and only if n is a biquadratefree number (A046100).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{p prime} (2/p^3 + 1/(p^2*(p-1))) = 2 * A085541 + A152441 = 0.67043452760761670220... .

A380688 Decimal expansion of Sum_{p prime} (p + 1)^3/((p - 1)^2*p^3).

Original entry on oeis.org

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

Offset: 1

Artur Jasinski, Mar 31 2025

			4.1586396688963117979214456647351...

Index to constants which are prime zeta sums {3,2,-3}.

Cf. A085541, A085548, A136141, A152441.

sumeulerrat((p + 1)^3/((p - 1)^2*p^3)) \\ Amiram Eldar, Apr 02 2025

Equals 5*A085548 + A085541 - 12*A136141 + 8*A152441.

Equals P(2) + 5*P(3) + Sum_{k>=4} (8*k - 20) * P(k), where P is the prime zeta function. - Amiram Eldar, Apr 02 2025

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A380840 Decimal expansion of Sum_{p prime} 1/(p-1)^3.

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A246549 Prime powers p^e where p is a prime and e >= 3 (prime powers without 1, the primes, or the squares of primes).

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A135177 a(n) = p^2*(p-1), where p = prime(n).

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A372695 Cubefull numbers that are not prime powers.

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A366076 The number of prime factors of the largest divisor of n that is a cubefull number (A036966), counted with multiplicity.

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A380688 Decimal expansion of Sum_{p prime} (p + 1)^3/((p - 1)^2*p^3).

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