A267712
Number of nontrivial prime powers p^k (k > 0) less than 10^n.
Original entry on oeis.org
7, 35, 193, 1280, 9700, 78734, 665134, 5762859, 50851223, 455062595, 4118082969, 37607992088, 346065767406, 3204942420923, 29844572385358, 279238346816392, 2623557174778438, 24739954338671299, 234057667428388198, 2220819603016308079, 21127269487386615271, 201467286693435354626, 1925320391619238700024, 18435599767386814628355, 176846309399257764978954, 1699246750872783231673649
Offset: 1
For n=1, there are 4 primes plus 3 prime powers less than 10^1: 2, 3, 4, 5, 7, 8, 9; 7 in total.
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Table[Count[Range[10^n], k_ /; PrimePowerQ@ k], {n, 6}] (* Michael De Vlieger, Jan 20 2016 *)
f[n_] := Sum[PrimePi[10^(n/k)], {k, n*Log2[10]}]; Array[f, 14] (* Robert G. Wilson v, Aug 17 2017, after Giovanni Resta in A267574 *)
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def A267712(n):
gen = (p for p in srange(2, 10^n) if p.is_prime_power())
return sum(1 for _ in gen)
print([A267712(n) for n in range(1, 7)]) # Peter Luschny, Sep 16 2023
A381391
Number of k <= 10^n that are neither squarefree nor prime powers (i.e., k is in A126706).
Original entry on oeis.org
0, 29, 367, 3866, 39098, 391838, 3920154, 39205902, 392069187, 3920718974, 39207261564, 392072817656, 3920728751139, 39207289143932, 392072896183208, 3920728975677128, 39207289797472001, 392072898095046811, 3920728981307675534, 39207289814141997459, 392072898144605471040
Offset: 1
Let S = A126706.
a(1) = 0 since the smallest term in S is 12.
a(2) = 29 since S(1..29) = {12, 18, 20, 24, ..., 99, 100}, etc.
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Table[10^n - Sum[PrimePi@ Floor[10^(n/k)], {k, 2, Floor[Log2[10^n]]}] - Sum[MoebiusMu[k]*Floor[10^n/(k^2)], {k, Floor[Sqrt[10^n]]}], {n, 10}]
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from math import isqrt
from sympy import primepi, integer_nthroot, mobius
def A381391(n):
m = 10**n
return int(-sum(primepi(integer_nthroot(m,k)[0]) for k in range(2,m.bit_length()))-sum(mobius(k)*(m//k**2) for k in range(2, isqrt(m)+1))) # Chai Wah Wu, Feb 23 2025
A303220
a(n) is the number of n-digit proper prime powers.
Original entry on oeis.org
3, 7, 15, 26, 57, 128, 319, 849, 2285, 6395, 18072, 51914, 150497, 439554, 1292568, 3819778, 11341738, 33806234, 101113152, 303345648, 912494104, 2751564993, 8315282765, 25179029388, 76381806785, 232094778772, 706331084162, 2152626447195, 6569037508556
Offset: 1
a(1) = 3 because there are 3 1-digit proper prime powers: 2^2 = 4, 2^3 = 8, and 3^2 = 9.
a(2) = 7 because there are 7 2-digit proper prime powers: 2^4 = 16, 2^5 = 32, 2^6 = 64; 3^3 = 27, 3^4 = 81; 5^2 = 25; and 7^2 = 49.
A365670
Number of perfect powers k which are not prime powers, and 1 < k < 10^n.
Original entry on oeis.org
0, 1, 14, 72, 257, 873, 2838, 9085, 28979, 92145, 292832, 930124, 2953569, 9376798, 29760901, 94434276, 299569798, 950072891, 3012393832, 9549260877, 30264906899, 95902117819, 303839485659, 962486295193, 3048497625289, 9654373954803, 30571355398031, 96797106918709
Offset: 1
There are 14 perfect powers less than 1000 which are not prime powers:
6^2, 10^2, 12^2, 14^2, 6^3, 15^2, 18^2, 20^2, 21^2, 22^2, 24^2, 26^2, 28^2, 30^2.
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from sympy import mobius, integer_nthroot, primepi
def A365670(n): return int(sum(mobius(x)*(1-(a:=integer_nthroot(10**n,x)[0]))-primepi(a) for x in range(2,(10**n).bit_length())))-1 if n>1 else n-1 # Chai Wah Wu, Aug 14 2024
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def A365670(n):
gen = (p for p in srange(2, 10^n)
if p.is_perfect_power() and not p.is_prime_power())
return sum(1 for _ in gen)
print([A365670(n) for n in range(1, 7)])
A381496
Number of powerful numbers that are not prime powers that do not exceed 10^n.
Original entry on oeis.org
0, 0, 3, 28, 133, 510, 1790, 5997, 19639, 63541, 204037, 652173, 2078320, 6609816, 20993381, 66612867, 211222374, 669428537, 2120835892, 6717184256, 21270247404, 67341572823, 213173925948, 674739560651, 2135491756895, 6758117426102, 21385762133815, 67670426242420
Offset: 0
Let S = A286708 = A001694 \ A246547 = A126706 \ A001694.
a(0) = a(1) = 0 since 36 is the smallest term in S.
a(2) = 3 since S(1..3) = {36, 72, 100}.
a(3) = 28 since S(4..28) = {108, 144, ..., 972, 1000}.
a(4) = 133 since S(29..133) = {1089, 1125, ..., 9801, 10000}, etc.
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Table[Sum[Boole[SquareFreeQ[k]]*Floor[Sqrt[10^n/k^3]], {k, 10^(n/3)}] - Sum[PrimePi[10^(n/k)], {k, 2, n*Log2[10]}] - 1, {n, 0, 12}]
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from math import isqrt
from sympy import primepi, integer_nthroot, mobius
def A381496(n):
def squarefreepi(n): return int(sum(mobius(k)*(n//k**2) for k in range(1, isqrt(n)+1)))
m, l = 10**n, 0
j, c = isqrt(m), -1-sum(primepi(integer_nthroot(m,k)[0]) for k in range(2,m.bit_length())),
while j>1:
k2 = integer_nthroot(m//j**2,3)[0]+1
w = squarefreepi(k2-1)
c += j*(w-l)
l, j = w, isqrt(m//k2**3)
return c+squarefreepi(integer_nthroot(m,3)[0])-l # Chai Wah Wu, Feb 25 2025
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