A380403
Number of integers k that are neither squarefree nor prime powers (in A126706) and that do not exceed primorial A002110(n).
Original entry on oeis.org
0, 0, 0, 5, 67, 871, 11693, 199976, 3802411, 87466676, 2536583089, 78634293907, 2909470106300, 119288281458176, 5129396144497507, 241081619059363357, 12777325812023481231, 753862222923258499554
Offset: 0
Let s = A126706 and let P(n) = A002110(n).
a(0..2) = 0 since P(0..2) = {1, 2, 6}, and the smallest number in s is 12.
a(3) = 5 since P(3) = 30, and the set s(1..6) = {12, 18, 20, 24, 28} contains k <= 30.
a(4) = 67 since P(4) = 210, and the set s(1..67) = {12, 18, 20, ..., 207, 208} contains k <= 210, etc.
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Table[# - Sum[PrimePi@ Floor[#^(1/k)], {k, 2, Floor[Log2[#]]}] - Sum[MoebiusMu[k]*Floor[#/(k^2)], {k, Floor[Sqrt[#]]}] &[Product[Prime[i], {i, n}]], {n, 0, 12}]
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a(n) = my(q=vecprod(primes(n))); q - sum(k=2, logint(q, 2), primepi(sqrtnint(q, k))) - sum(k=1, sqrtint(q), q\k^2*moebius(k)); \\ Jinyuan Wang, Feb 25 2025
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from math import isqrt
from sympy import primorial, primepi, integer_nthroot, mobius
def A380403(n):
if n == 0: return 0
m = primorial(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, Jan 24 2025
A380430
Number of powerful numbers k that are not powers of primes (i.e., k is in A286708) that do not exceed the primorial number A002110(n).
Original entry on oeis.org
0, 0, 0, 0, 7, 50, 254, 1245, 5898, 29600, 163705, 925977, 5690175, 36681963, 241663896, 1662446097, 12134853382, 93406989325, 730785520398, 5990426525483, 50538885715526, 432266550168097, 3845700235189327, 35065304557027821, 334652745159828239, 3262707438761612651
Offset: 0
Let P = A002110 and let s = A286708 = A001694 \ A246547 \ {1}.
a(0..3) = 0 since the smallest number in s is 36.
a(4) = 7 since P(4) = 210 and numbers in s that are less than 210 include {36, 72, 100, 108, 144, 196, 200}, etc.
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Table[-1 + Sum[If[MoebiusMu[j] != 0, Floor[Sqrt[#/j^3]], 0], {j, #^(1/3)}] - Sum[PrimePi@ Floor[#^(1/k)], {k, 2, Floor[Log2[#]]}] &[Product[Prime[i], {i, n}]], {n, 0, 12} ]
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from math import isqrt
from sympy import primorial, primepi, integer_nthroot, mobius
def A380430(n):
def squarefreepi(n): return int(sum(mobius(k)*(n//k**2) for k in range(1, isqrt(n)+1)))
if n == 0: return 0
m = primorial(n)
c, l, j = int(squarefreepi(integer_nthroot(m, 3)[0])-sum(primepi(integer_nthroot(m,k)[0]) for k in range(2,m.bit_length()))-1), 0, isqrt(m)
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-l # Chai Wah Wu, Feb 25 2025
A380404
Number of prime powers that do not exceed the primorial number A002110(n).
Original entry on oeis.org
0, 1, 4, 16, 60, 377, 3323, 42518, 646580, 12285485, 300378113, 8028681592, 259488951722, 9414917934636, 362597756958862, 15397728568256861, 742238179325555125, 40068968503380861518, 2251262473065725514585, 139566579946046888545036
Offset: 0
Let P = A002110 and let s = A246655.
a(0) = 0 since P(0) = 1, and the smallest term in s is 2.
a(1) = 1 since P(1) = 2.
a(2) = 4 since P(2) = 6 and the terms in s that do not exceed 6 are {2, 3, 4, 5}.
a(3) = 16 since P(3) = 30; the numbers 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, and 29 are less than 30, etc.
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Table[Sum[PrimePi[Floor[#^(1/k)]], {k, Floor@ Log2[#]}] &[Product[Prime[i], {i, n}]], {n, 0, 14}]
Showing 1-3 of 3 results.