A065515 -id:A065515 - cp's OEIS frontend

A379771 Number of k <= n that are neither squarefree nor prime powers.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 7, 7, 7, 7, 8, 9, 9, 9, 10, 10, 11, 11, 12, 12, 13, 13, 14, 14, 14, 14, 15, 15, 15, 16, 16, 16, 16, 16, 17, 17, 17, 17, 18, 18, 18, 19, 20, 20

Offset: 1

Michael De Vlieger, Jan 10 2025

Partial sums of A355447.

			a(n) = 0 for n = 1..11, since 12 is the smallest number that is neither squarefree nor a prime power.
a(n) = 1 for n = 12..17, since the only k <= n that is neither squarefree nor a prime power is 12.
a(n) = 2 for n = 18..19, since 12 and 18 are the only numbers in A126706 that do not exceed n.
a(n) = 3 for n = 20..23, since 12, 18, and 20 are the only numbers in A126706 that do not exceed n, etc.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^20.

Cf. A013928, A065515, A126706, A355447.

Array[Count[Range[#], _?(Nor[SquareFreeQ[#], PrimePowerQ[#]] &)] &, 120]

a(n) = sum(k=1, n, !issquarefree(k) && !isprimepower(k)); \\ Michel Marcus, Jan 11 2025

from math import isqrt
from sympy import mobius, integer_nthroot, primepi
def A379771(n): return -sum(mobius(k)*(n//k**2) for k in range(2,isqrt(n)+1))-sum(primepi(integer_nthroot(n,k)[0]) for k in range(2,n.bit_length())) # Chai Wah Wu, Jan 22 2025

A126706(n) = m = positions of 1's in A355447; a(m) - a(m-1) = 1.

A380757 Powers of primes that have a primitive root.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 9, 11, 13, 17, 19, 23, 25, 27, 29, 31, 37, 41, 43, 47, 49, 53, 59, 61, 67, 71, 73, 79, 81, 83, 89, 97, 101, 103, 107, 109, 113, 121, 125, 127, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229

Offset: 1

Michael De Vlieger, Feb 01 2025

Proper subset of A033948.

A046022 is a proper subset of this sequence.

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

Cf. A000079, A000961, A033948, A046022, A061345, A278568.

With[{nn = 2^8},
  Complement[#, Array[2^# &, Floor@ Log2[#[[-1]]] + 2, 3]] &@
  Union[{1}, Prime@ Range@ PrimePi[#[[-1]] ], #] &@
  Select[Union@ Flatten@
    Table[a^2*b^3, {b, Surd[#, 3]}, {a, Sqrt[nn/b^3]}],
    PrimePowerQ] ]

from sympy import primepi, integer_nthroot
def A380757(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 n if x<6 else int(n+x-3-sum(primepi(integer_nthroot(x,k)[0])-1 for k in range(1,x.bit_length())))
    return bisection(f,n,n) # Chai Wah Wu, Feb 03 2025

Union of {1, 2, 4} and A061345.
This sequence is A000961 without A000079(k) for k > 2.
A033948 = union of {a(n)} and {2*a(n)} without 8 = union of {a(n)} and A278568, where {a(n)} represents this sequence.
Intersection of A000961 and A033948.
Define c(m) to be the number of terms that do not exceed m. Then for m >= 4, c(m) = 3 + (Sum_{k = 1..floor(log_2(m))} pi(floor(m^(1/k)))) - floor(log_2(m)) = 3 + A065515(m) - A113473(m).

A025471 Position of 5^n among the powers of primes (A000961).

Original entry on oeis.org

1, 5, 15, 43, 137, 483, 1881, 7772, 33292, 146014, 650688, 2935909, 13377899, 61448617, 284148766, 1321470026, 6175996065, 28988110806, 136575209239, 645621236243, 3061130340023, 14552991533455, 69354804657255, 331251331834172, 1585299655420744, 7600886382041555, 36504944105963117

Offset: 0

Clark Kimberling

nonn

Cf. A000351 (powers of 5), A000961, A065515.

a(n) = n=5^n; 1+sum(k=1, logint(n,2), primepi(sqrtnint(n,k))); \\ Daniel Suteu, Dec 15 2019

a(n) = A065515(5^n). - Michel Marcus, Mar 24 2013

a(13)-a(20) from Robert G. Wilson v, Dec 13 2019

a(21)-a(26) from Daniel Suteu, Dec 15 2019

A025472 a(n) is the position of 7^n among the powers of primes (A000961).

Original entry on oeis.org

1, 6, 24, 87, 393, 2002, 11212, 65907, 398221, 2454882, 15355483, 97168563, 620657849, 3995177593, 25885249943, 168651040452, 1104127513216, 7259026757936, 47901525960522, 317140345586299, 2105877518905780, 14020561598521781, 93570332415595735, 625829903112713938

Offset: 0

Clark Kimberling

nonn

The number of prime powers <= 7^n. - Robert G. Wilson v, Dec 16 2019

Cf. A000420 (7^n), A000961, A065515.

a(n)= n=7^n+.5; 1+sum(k=1, log(n)\log(2), primepi(n^(1/k))) \\ Michel Marcus, Mar 25 2013

a(n) = A065515(7^n). - Michel Marcus, Mar 24 2013

a(11)-a(17) from Robert G. Wilson v, Dec 13 2019
a(18)-a(22) from Daniel Suteu, Dec 15 2019
a(23) from Robert G. Wilson v, Dec 15 2019

A327247 Number of odd prime powers <= n (with exponents > 0).

Original entry on oeis.org

0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 6, 6, 7, 7, 8, 8, 8, 8, 9, 9, 10, 10, 11, 11, 12, 12, 13, 13, 13, 13, 13, 13, 14, 14, 14, 14, 15, 15, 16, 16, 16, 16, 17, 17, 18, 18, 18, 18, 19, 19, 19, 19, 19, 19, 20, 20, 21, 21, 21, 21, 21, 21, 22, 22, 22, 22, 23, 23, 24, 24, 24

Offset: 1

Ilya Gutkovskiy, Sep 14 2019

nonn

Cf. A000523, A025528, A033270, A061345, A065515, A069637, A085501, A174275 (first differences), A246655, A285879.

Table[Sum[Boole[OddQ[k] && PrimePowerQ[k]], {k, 1, n}], {n, 1, 75}]

a(n) = {sum(k=2, primepi(n), logint(n, prime(k)))} \\ Andrew Howroyd, Sep 14 2019

a(n) = A025528(n) - A000523(n).

cp's OEIS Frontend

A379771 Number of k <= n that are neither squarefree nor prime powers.

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A380757 Powers of primes that have a primitive root.

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A025471 Position of 5^n among the powers of primes (A000961).

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A025472 a(n) is the position of 7^n among the powers of primes (A000961).

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A327247 Number of odd prime powers <= n (with exponents > 0).

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