A036386 -id:A036386 - cp's OEIS frontend

A060967 Number of squared primes <= 2^n.

0, 0, 1, 1, 2, 3, 4, 5, 6, 8, 11, 14, 18, 24, 31, 42, 54, 72, 97, 128, 172, 229, 309, 418, 564, 760, 1028, 1393, 1900, 2585, 3512, 4792, 6542, 8952, 12251, 16777, 23000, 31579, 43390, 59631, 82025, 112957, 155611, 214516, 295947, 408493, 564163, 779638

Offset: 0

Labos Elemer, May 09 2001

nonn

			For n = 8, the squared primes not exceeding 2^8 = 256 are 4, 9, 25, 49, 121, 169, so a(8) = 6.

Amiram Eldar, Table of n, a(n) for n = 0..150 (terms 0..63 from Harry J. Smith, terms 64..112 from Hiroaki Yamanouchi, terms 113..125 from Ray Chandler)
Index entries for sequences related to numbers of primes in various ranges.

Cf. A000720, A001248, A007053, A017910, A036386, A060969, A060970, A060971, A122121.

Table[ PrimePi[ Floor[ 2^(g/2)//N ] ], {g, 1, 75} ]

a(n) = { primepi(sqrtint(2^n)) } \\ Harry J. Smith, Jul 15 2009

a(2*n) = A007053(n). - Amiram Eldar, Jul 10 2024

a(n) = A000720(A017910(n)). - Amiram Eldar, Mar 22 2025

a(0) prepended by Harry J. Smith, Jul 15 2009

A060969 Number of cubes of primes <= 2^n.

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 6, 8, 9, 11, 12, 15, 18, 22, 26, 31, 37, 46, 54, 66, 79, 97, 117, 141, 172, 209, 257, 309, 376, 457, 564, 687, 842, 1028, 1266, 1549, 1900, 2327, 2861, 3512, 4323, 5320, 6542, 8072, 9936, 12251, 15104, 18640, 23000, 28428

Offset: 0

Labos Elemer, May 09 2001

nonn

			For n = 10, the cubes of primes not exceeding 2^10 = 1024 are 8, 27, 125, 343, so a(10) = 4.

Cf. A000720, A007053, A017979, A030078, A036386, A060967, A060970, A060971.

Table[ PrimePi[ Floor[ 2^(g/3)//N ] ], {g, 0, 90} ]

a(3*n) = A007053(n). - Chai Wah Wu, Jan 23 2025

a(n) = A000720(A017979(n)). - Amiram Eldar, Mar 22 2025

Missing a(0)=0 inserted by Sean A. Irvine, Jan 09 2023

A060970 Number of fourth powers of primes <= 2^n.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 1, 2, 2, 2, 3, 3, 4, 4, 5, 6, 6, 8, 8, 9, 11, 12, 14, 16, 18, 21, 24, 28, 31, 36, 42, 47, 54, 62, 72, 82, 97, 111, 128, 149, 172, 199, 229, 268, 309, 360, 418, 481, 564, 651, 760, 886, 1028, 1201, 1393, 1629, 1900, 2211, 2585, 3010, 3512, 4104, 4792

Offset: 0

Labos Elemer, May 09 2001

nonn

			For n = 12, the 4th powers of prime not exceeding 2^12 = 4096 are 16, 81, 625, 2401, so a(12) = 4.

Cf. A000720, A007053, A018048, A030514, A036386, A060967, A060969, A060971.

Table[ PrimePi[ Floor[ 2^(g/4)//N ] ], {g, 0, 100} ]

a(4*n) = A007053(n). - Chai Wah Wu, Jan 23 2025

a(n) = A000720(A018048(n)). - Amiram Eldar, Mar 22 2025

Missing a(0)=0 inserted by Sean A. Irvine, Jan 09 2023

A060971 Number of fifth powers of primes <= 2^n.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 4, 4, 4, 5, 6, 6, 7, 8, 9, 9, 11, 11, 13, 15, 16, 18, 21, 23, 25, 29, 31, 34, 39, 44, 47, 54, 62, 68, 76, 86, 97, 107, 122, 137, 154, 172, 193, 217, 244, 275, 309, 349, 393, 442, 499, 564, 635, 712, 807, 914, 1028, 1163, 1315, 1482

Offset: 0

Labos Elemer, May 09 2001

nonn

			For n = 10: the 5th powers of primes not exceeding 2^10 = 1024 are 32 and 243, so a(10) = 2.

Cf. A000720, A007053, A018117, A036386, A050997, A060967, A060969, A060970.

Table[ PrimePi[ Floor[ 2^(g/5)//N ] ], {g, 0, 150} ]

a(5*n) = A007053(n). - Chai Wah Wu, Jan 23 2025

a(n) = A000720(A018117(n)). - Amiram Eldar, Mar 22 2025

Missing a(0)=0 inserted by Sean A. Irvine, Jan 09 2023

A062762 Number of powerful numbers not exceeding 2^n.

Original entry on oeis.org

1, 1, 2, 3, 5, 8, 11, 18, 26, 38, 55, 80, 116, 166, 240, 345, 497, 710, 1016, 1453, 2073, 2955, 4211, 5992, 8523, 12111, 17202, 24423, 34648, 49152, 69694, 98795, 140009, 198378, 281016, 398002, 563612, 797999, 1129737, 1599166, 2263457, 3203381

Offset: 0

Labos Elemer, Jul 16 2001

nonn

Number of terms x from A001694 for which x <= 2^n.

			Below 128, the 18 powerful numbers {1,4,8,9,16,25,...,100,108,121,125,128} can be found, so a(7)=18.

Chai Wah Wu, Table of n, a(n) for n = 0..127 (terms 0..90 from Daniel Suteu)

Cf. A001694, A029837, A036380, A036386, A062761.

nn = 41; s = Union@ Flatten@ Table[a^2*b^3, {b, (2^nn)^(1/3)}, {a, Sqrt[(2^nn)/b^3]}]; Table[FirstPosition[s, 2^k][[1]], {k, 2, nn}] (* Michael De Vlieger, Oct 29 2023 *)

a(n) = my(s=0,N=2^n); forsquarefree(k=1, sqrtnint(N, 3), s += sqrtint(N\k[1]^3)); s; \\ Daniel Suteu, Feb 18 2020

from math import isqrt
from sympy import mobius, integer_nthroot
def A062762(n):
    def squarefreepi(n): return int(sum(mobius(k)*(n//k**2) for k in range(1, isqrt(n)+1)))
    m = 1<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, Sep 13 2024

a(n) = Sum_{k=0..n} A062761(k). - Daniel Suteu, Feb 18 2020

a(19)-a(41) from Donovan Johnson, Oct 01 2009

A380402 Number of proper prime powers (in A246547) that do not exceed primorial A002110(n).

Original entry on oeis.org

0, 0, 1, 6, 14, 34, 75, 187, 551, 1954, 8317, 38582, 200978, 1125541, 6562122, 40444003, 266832233, 1870169623, 13424553758, 101495825622, 793832121165, 6325729776075, 52616754936494, 450157758564742, 3999323787879764, 37180986240914714, 353667558431662474

Offset: 0

Michael De Vlieger, Jan 23 2025

			Let s = A246547.
a(0) = a(1) = 0 since P(0) = 1 and P(1) = 2, and the smallest number in s is 4.
a(2) = 1 since P(2) = 6, and s(1) = 4 is the only term in s <= 6.
a(3) = 6 since P(3) = 30, and the set s(1..6) = {4, 8, 9, 16, 25, 27} contains k <= 30.
a(4) = 14 since P(4) = 210, and the set s(1..14) = {4, 8, 9, 16, 25, 27, 32, 49, 64, 81, 121, 125, 128, 169} contains k <= 210, etc.

Cf. A036386, A246547.

Table[Sum[PrimePi@ Floor[#^(1/k)], {k, 2, Floor@ Log2[#]}] &[Product[Prime[i], {i, n}]], {n, 0, nn}]

from sympy import primorial, primepi, integer_nthroot
def A380402(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()))) # Chai Wah Wu, Jan 24 2025

a(n) = Sum_{k=2..floor(log_2(P(n)))} pi(floor(P(n)^(1/k))), where P(n) = A002110(n).

a(24) corrected by Chai Wah Wu, Jan 25 2025

a(26) from Jinyuan Wang, Feb 25 2025

A036380 Number of true prime powers whose binary order, ceiling(log_2(p^x)), is n.

Original entry on oeis.org

0, 1, 1, 2, 3, 2, 4, 3, 4, 6, 5, 9, 10, 11, 17, 15, 26, 31, 39, 53, 68, 90, 125, 159, 216, 290, 391, 536, 719, 971, 1329, 1812, 2477, 3386, 4626, 6351, 8729, 11995, 16459, 22669, 31259, 43049, 59388, 82024, 113275, 156558, 216560, 299566, 414821, 574654

Offset: 1

Labos Elemer

nonn

			There are 5 prime powers between 2^10 + 1 = 1025 and 2^11 = 2048 (inclusive): 1331 = 11^3, 1369 = 37^2, 1681 = 41^2, 1849 = 43^2, and 2048 = 2^11, so a(11) = 5.

Amiram Eldar, Table of n, a(n) for n = 1..150

Cf. A029837, A036378-A036390.

t=Table[Length[Union[Flatten[Table[Table[Prime[w]^s, {w, 1, PrimePi[2^(n/s)]}], {s, 2, g+1}]]] ], {n, 1, 42}]; Delete[t-RotateRight[t], 1]

a(n) = A036386(n) - A036386(n-1) for n >= 2. - Amiram Eldar, Mar 22 2025

More terms from Sean A. Irvine, Oct 29 2020

A372403 Number of k < 2^n that are neither squarefree nor prime powers.

Original entry on oeis.org

1, 5, 16, 37, 83, 178, 374, 772, 1565, 3160, 6361, 12770, 25599, 51265, 102634, 205374, 410873, 821924, 1644070, 3288433, 6577231, 13154868, 26310347, 52621521, 105244142, 210489792, 420981295, 841964929, 1683933254, 3367871086, 6735748322, 13471504796, 26943020642

Offset: 4

Michael De Vlieger, Jun 09 2024

Analogous to A143658 (number of squarefree k <= 2^n) and A182908 (position of 2^n among prime powers A246655).

			Let quality Q represent a number k that is neither squarefree nor prime power. For instance, Q(k) is true if and only if Omega(k) > omega(k) > 1, i.e., A001222(k) > A001221(k) > 1.
a(4) = 1 since there is one number k = 12 such that Q(k) is true; 12 < 2^4.
a(5) = 5 since there are 5 numbers k such that Q(k) is true; {12, 18, 20, 24, 28} are less than 2^5.
a(6) = 16 since A126706(16) < 2^6 < A126706(17), etc.

Chai Wah Wu, Table of n, a(n) for n = 4..70

Cf. A007053, A036386, A126706, A143658, A182908.

filter:= proc(n) local F;
  F:= ifactors(n)[2];
  nops(F) > 1 and max(F[..,2]) > 1
end proc:
R:= NULL: v:= 0:
for i from 4 to 20 do
  v:= v + nops(select(filter, [$2^(i-1)+1 .. 2^i-1]));
  R:= R,v;
od:
R; # Robert Israel, Jun 09 2024

Table[2^n - Sum[PrimePi@Floor[2^(n/k)], {k, 2, n}] - Sum[MoebiusMu[k]*Floor[#/(k^2)], {k, Floor[Sqrt[#]]}] &[2^n], {n, 4, 36} ] (* Michael De Vlieger, Jan 24 2025 *)

from math import isqrt
from sympy import mobius, nextprime, integer_log
def A372403(n):
    m, p = (1<Chai Wah Wu, Jun 10 2024

a(n) = 2^n - A036386(n) - A143658(n). - Michael De Vlieger, Jan 24 2025

a(30) onwards from Chai Wah Wu, Jun 10 2024

A062761 Number of powerful numbers between 2^(n-1)+1 and 2^n.

Original entry on oeis.org

1, 0, 1, 1, 2, 3, 3, 7, 8, 12, 17, 25, 36, 50, 74, 105, 152, 213, 306, 437, 620, 882, 1256, 1781, 2531, 3588, 5091, 7221, 10225, 14504, 20542, 29101, 41214, 58369, 82638, 116986, 165610, 234387, 331738, 469429, 664291, 939924, 1329876, 1881500, 2661826, 3765629

Offset: 0

Labos Elemer, Jul 16 2001

nonn

			64 < {72,81,100,108,121,125,128} <= 128, i.e., 7 powerful numbers are between 2^6 and 2^7, so a(7)=7.

Chai Wah Wu, Table of n, a(n) for n = 0..127 (terms 0..90 from Daniel Suteu)

Cf. A001694, A029837, A036380, A036386, A062762.

a(n) = my(ka = if (n==0, 1, 2^(n-1)+1)); #select(x->ispowerful(x), [ka..2^n]); \\ Michel Marcus, Aug 25 2019

Q(n) = my(s=0); forsquarefree(k=1, sqrtnint(n, 3), s += sqrtint(n\k[1]^3)); s;
a(n) = if(n==0, 1, Q(2^n) - Q(2^(n-1))); \\ Daniel Suteu, Feb 18 2020

# uses code from A062762
def A062761(n): return A062762(n)-A062762(n-1) if n else 1 # Chai Wah Wu, Sep 13 2024

Number of terms x from A001694 for which A029837(x)=n.

Sum_{k=0..n} a(k) = A062762(n). - Daniel Suteu, Feb 18 2020

a(19)-a(29) from Daniel Suteu, Aug 25 2019

a(30)-a(45) from Daniel Suteu, Feb 18 2020

A380431 Number of powerful numbers that are not powers of primes (i.e. are in A286708) that do not exceed 2^n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 4, 9, 17, 28, 48, 75, 115, 178, 266, 403, 590, 865, 1263, 1830, 2644, 3810, 5466, 7838, 11210, 16011, 22841, 32530, 46315, 65886, 93658, 133060, 188952, 268204, 380564, 539823, 765481, 1085224, 1538194, 2179816, 3088481, 4375308, 6197420, 8777222

Offset: 0

Michael De Vlieger, Jan 24 2025

			Let s = A286708 = A001694 \ A246547 \ {1}.
a(0..5) = 0 since the smallest number in s is 36.
a(6) = 1 since only s(1) = 36 is smaller than 2^6 = 64.
a(7) = 4 since s(1..4) = {36, 72, 100, 108} are smaller than 2^7 = 128.
a(8) = 9 since s(1..9) = {36, 72, 100, 108, 144, 196, 200, 216, 225} are smaller than 2^8 = 256, etc.

Amiram Eldar, Table of n, a(n) for n = 0..90

Cf. A001694, A036386, A062762, A246547, A286708, A372403.

Table[-1 + Sum[If[MoebiusMu[j] != 0, Floor[Sqrt[(2^n)/j^3]], 0], {j, 2^(n/3)}] - Sum[PrimePi@ Floor[2^(n/k)], {k, 2, n}], {n, 0, 45} ]

from math import isqrt
from sympy import mobius, integer_nthroot, primepi
def A380431(n):
    def squarefreepi(n): return int(sum(mobius(k)*(n//k**2) for k in range(1, isqrt(n)+1)))
    l, m = 0, 1<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, Jan 30 2025

a(n) = A062762(n) - A036386(n) - 1.

a(n) <= A372403(n), since A286708 is a proper subset of A126706.

cp's OEIS Frontend

A060967 Number of squared primes <= 2^n.

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A060969 Number of cubes of primes <= 2^n.

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A060970 Number of fourth powers of primes <= 2^n.

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A060971 Number of fifth powers of primes <= 2^n.

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A062762 Number of powerful numbers not exceeding 2^n.

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A380402 Number of proper prime powers (in A246547) that do not exceed primorial A002110(n).

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A036380 Number of true prime powers whose binary order, ceiling(log_2(p^x)), is n.

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