A007053 -id:A007053 - 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

A244508 Number of odd prime powers (A246655) between 2^n and 2^(n+1).

Original entry on oeis.org

0, 1, 2, 3, 7, 8, 16, 25, 46, 80, 141, 263, 473, 882, 1628, 3044, 5734, 10779, 20428, 38687, 73653, 140425, 268340, 513866, 986033, 1894409, 3646134, 7027825, 13562625, 26208248, 50698865, 98184467, 190338061, 369326690, 717271793, 1394198586, 2712112561

Offset: 0

Michel Marcus, Nov 17 2014

nonn

			Between 2 and 4, there is just 1 prime power: 3, so a(1) = 1.
Between 4 and 8, there are 2 prime powers: 5 and 7, so a(2) = 2.

Ray Chandler, Table of n, a(n) for n = 0..91 (using b-file from A007053, corrected n = 45..52, n = 0..52 from Hiroaki Yamanouchi)

Cf. A246655 (prime powers), A182908 (positions of 2^n among prime powers).

Table[Count[Range[2^n + 1, 2^(n + 1) - 1], ?PrimePowerQ], {n, 0, 27}] (* _Ivan N. Ianakiev, Nov 18 2014 *)

a(n) = sum(i=2^n+1, 2^(n+1)-1, isprimepower(i)>0);

from sympy import primepi, integer_nthroot
def A244508(n):
    def f(x): return int(1+sum(primepi(integer_nthroot(x, k)[0]) for k in range(1, x.bit_length())))
    return f((1<Chai Wah Wu, Nov 05 2024

a(n) = A182908(n+1) - A182908(n). - Ray Chandler, Aug 20 2021

a(28)-a(36) from Hiroaki Yamanouchi, Nov 20 2014

Minor edits by Ray Chandler, Aug 20 2021

A373406 Sum of the n-th maximal run of odd primes differing by two.

Original entry on oeis.org

15, 24, 36, 23, 60, 37, 84, 47, 53, 120, 67, 144, 79, 83, 89, 97, 204, 216, 113, 127, 131, 276, 300, 157, 163, 167, 173, 360, 384, 396, 211, 223, 456, 233, 480, 251, 257, 263, 540, 277, 564, 293, 307, 624, 317, 331, 337, 696, 353, 359, 367, 373, 379, 383

Offset: 1

Gus Wiseman, Jun 05 2024

nonn

The length of this run is given by A251092.
For this sequence we define a run to be an interval of positions at which consecutive terms differ by two. Normally, a run has consecutive terms differing by one, but odd prime numbers already differ by at least two.
Contains A054735 (sums of twin prime pairs) without its first two terms and A007510 (non-twin primes) as subsequences. - R. J. Mathar, Jun 07 2024

			Row-sums of:
   3   5   7
  11  13
  17  19
  23
  29  31
  37
  41  43
  47
  53
  59  61
  67
  71  73
  79
  83
  89
  97

Gus Wiseman, Four statistics for runs and antiruns of prime, nonprime, squarefree, and nonsquarefree numbers

The partial sums are a subset of A071148 (partial sums of odd primes).
Functional neighbors: A025584, A054265, A067774, A251092 (or A175632), A373405, A373413, A373414.
A000040 lists the primes, differences A001223.
A046933 counts composite numbers between primes.
A065855 counts composite numbers up to n.
Cf. A005117, A007053, A029707, A038664, A293697, A350842, A371201.

Total/@Split[Select[Range[3,100],PrimeQ],#1+2==#2&]//Most

A373669 Least k such that the k-th maximal run of non-prime-powers has length n. Position of first appearance of n in A110969, and the sequence ends if there is none.

Original entry on oeis.org

1, 5, 7, 12, 18, 190, 28, 109, 40, 28195574, 53

Offset: 1

Gus Wiseman, Jun 14 2024

A run of a sequence (in this case A361102) is an interval of positions at which consecutive terms differ by one.
Are there only 9 terms?
From David A. Corneth, Jun 14 2024: (Start)
No. a(10) exists.
Between the prime 144115188075855859 and 144115188075855872 = 2^57 there are 12 non-prime-powers so a(12) exists. (End)

			The maximal runs of non-prime-powers begin:
   1
   6
  10
  12
  14  15
  18
  20  21  22
  24
  26
  28
  30
  33  34  35  36
  38  39  40
  42
  44  45  46
  48
  50  51  52
  54  55  56  57  58
  60

Gus Wiseman, Four statistics for runs and antiruns of prime, nonprime, squarefree, and nonsquarefree numbers.

For composite runs we have A073051, sorted A373400, firsts of A176246.
For squarefree runs we have firsts of A120992.
For prime-powers runs we have firsts of A174965.
For prime runs we have firsts of A251092 or A175632.
For squarefree antiruns we have A373128, firsts of A373127.
For nonsquarefree runs we have A373199, firsts of A053797.
The sorted version is A373670.
For antiruns we have firsts of A373672.
For runs of non-prime-powers:
- length A110969
- min A373676
- max A373677
- sum A373678
A000961 lists the powers of primes (including 1).
A005117 lists the squarefree numbers, first differences A076259.
A013929 lists the nonsquarefree numbers, first differences A078147.
A057820 gives first differences of consecutive prime-powers, gaps A093555.
A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
A361102 lists the non-prime-powers, without 1 A024619.
Cf. A007053, A008864, A014963, A027833, A038664, A054265, A067774, A356068, A373401, A373403, A373671.

q=Length/@Split[Select[Range[10000],!PrimePowerQ[#]&],#1+1==#2&]//Most;
spna[y_]:=Max@@Select[Range[Length[y]],SubsetQ[y,Range[#1]]&];
Table[Position[q,k][[1,1]],{k,spna[q]}]

A055729 Number of primes <= 3^n.

Original entry on oeis.org

0, 2, 4, 9, 22, 53, 129, 327, 847, 2227, 5968, 16097, 43934, 120739, 334349, 931260, 2607165, 7332159, 20700806, 58648288, 166677978, 475023803, 1357200840, 3886548158, 11152818693, 32064929886, 92349038518, 266398236486, 769616513836, 2226457080707, 6449247674296, 18703411700669, 54301968156067, 157820174545456

Offset: 0

Robert G. Wilson v, Jun 09 2000

nonn

Robert Price, Table of n, a(n) for n = 0..51 (terms 0..49 from Henri Lifchitz)
Index entries for sequences related to numbers of primes in various ranges

Cf. A006880 and A007053.

Mathematica
```
Table[PrimePi[3^n], {n, 0, 29}]
```

a(n) = primepi(3^n); \\ Michel Marcus, Aug 25 2014

a(n) = A000720(A000244(n)). - Michel Marcus, Aug 25 2014

a(28)-a(40) from Henri Lifchitz, Nov 11 2012

A036386 Number of prime powers (p^2, p^3, ...) <= 2^n.

Original entry on oeis.org

0, 1, 2, 4, 7, 9, 13, 16, 20, 26, 31, 40, 50, 61, 78, 93, 119, 150, 189, 242, 310, 400, 525, 684, 900, 1190, 1581, 2117, 2836, 3807, 5136, 6948, 9425, 12811, 17437, 23788, 32517, 44512, 60971, 83640, 114899, 157948, 217336, 299360, 412635, 569193, 785753, 1085319, 1500140, 2074794, 2870849, 3974425, 5504966

Offset: 1

Labos Elemer

nonn

			For n = 6, there are 9 prime powers not exceeding 2^6 = 64: 4, 8, 9, 16, 25, 27, 32, 49, 64, so a(6) = 9.
For n = 25, a(25) = 900, pi(5792) + pi(322) + pi(76) + pi(32) + pi(17) + pi(11) + pi(8) + pi(6) + pi(5) + pi(4) + pi(4) + pi(3) + pi(3) + pi(3) + pi(2) + pi(2) + pi(2) + pi(2) + pi(2) + pi(2) + pi(2) + pi(2) + pi(2) + pi(2) + pi(1) = 760 + 66 + 21 + 11 + 7 + 5 + 4 + 3 + 3 + 2 + 2 + 2 + 2 + 2 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 0 = 900.

Amiram Eldar, Table of n, a(n) for n = 1..150 (terms 1..112 from Hiroaki Yamanouchi, terms 113..125 from Ray Chandler)
Index entries for sequences related to numbers of primes in various ranges.

Cf. A000720, A007053, A029837, A036378-A036390, A060967, A060969, A060970, A060971, A122121, A246547.

f[n_] := Length@ Union@ Flatten@ Table[ Prime[j]^k, {k, 2, n + 1}, {j, PrimePi[2^(n/k)]}]; Array[f, 46] (* Robert G. Wilson v, Jul 08 2011 *)

from sympy import primepi, integer_nthroot
def A036386(n):
    m = 1<Chai Wah Wu, Jan 23 2025

a(n) = Sum_{j=2..n+1} pi(floor(2^(n/j))). The summation starts with squares (j=2); for arbitrary range (=y), the y^(1/j) argument has to be used.

More terms from Labos Elemer, May 07 2001

Terms a(47) and beyond from Hiroaki Yamanouchi, Nov 15 2016

A065843 Let u be any string of n digits from {0,1}; let f(u) = number of distinct primes, not beginning with 0, formed by permuting the digits of u to a base-2 number; then a(n) = max_u f(u).

Original entry on oeis.org

0, 1, 1, 2, 2, 3, 5, 12, 11, 24, 34, 79, 105, 194, 362, 734, 1143, 2045, 3872, 7758, 13001, 23902, 45539, 90436, 159510, 296210, 563833, 1110387, 2030754, 3876871, 7333827, 14353074, 26730538, 51246344, 97529176, 190928828, 358117285, 694240090, 1324674524, 2587693929, 4903604087, 9547001123

Offset: 1

Sascha Kurz, Nov 24 2001

			a(4)=2 because 1011 and 1101 in base-2 notation are primes (11 and 13) and there is no set of three or more 4-digit primes with a common number of ones.

Cf. A007053, A065844, A065845, A065846, A065847 A065848, A065849, A065850, A065851, A065852, A065853.

A065843 := proc(n)
    local b,u,udgs,uperm,a;
    b :=2 ;
    a := 0 ;
    for u from b^(n-1) to b^n-1 do
        udgs := convert(u,base,b) ;
        prs := {} ;
        for uperm in combinat[permute](udgs) do
            if op(-1,uperm) <> 0 then
                p := add( op(i,uperm)*b^(i-1),i=1..nops(uperm)) ;
                if isprime(p) then
                    prs := prs union {p} ;
                end if;
            end if;
        end do:
        a := max(a,nops(prs)) ;
    end do:
    a ;
end proc:
for n from 1 do
    print(n,A065843(n)) ;
end do: # R. J. Mathar, Apr 23 2016

c[x_] := Module[{},
   Length[Select[Permutations[x],
     First[#] != 0 && PrimeQ[FromDigits[#, 2]] &]]];
A065843[n_] := Module[{i},
   Return[Max[Map[c, DeleteDuplicatesBy[Tuples[Range[0, 1], n],
       Table[Count[#, i], {i, 0, 1}] &]]]]];
Table[A065843[n], {n, 1, 19}] (* Robert Price, Mar 30 2019 *)

lista(n) = {my(m = matrix(n,n),c); forprime(i=2,2^n, b = binary(i); m[#b,hammingweight(b)]++);vector(n,i,vecmax(m[i,]))} \\ David A. Corneth, Apr 23 2016

from sympy import isprime
from itertools import combinations_with_replacement as mc
from sympy.utilities.iterables import multiset_permutations as mp
def a(n): return n-1 if n < 3 else max(sum(1 for p in mp(c) if isprime(int("1"+"".join(p)+"1", 2))) for c in mc("01", n-2))
print([a(n) for n in range(1, 21)]) # Michael S. Branicky, Oct 09 2022

6 more terms from Sean A. Irvine, Sep 06 2009
a(37)-a(39) from Michael S. Branicky, May 30 2024
a(40)-a(42) from Michael S. Branicky, Jun 14 2024

A145592 a(n)=number of numbers removed in each step of Eratosthenes's sieve for 2^14.

Original entry on oeis.org

8191, 2730, 1091, 623, 340, 260, 182, 154, 121, 94, 89, 74, 66, 62, 55, 48, 43, 39, 35, 31, 28, 25, 23, 19, 15, 12, 11, 9, 7, 5, 1

Offset: 1

Artur Jasinski with assistance from Bob Hanlon (hanlonr(AT)cox.net), Oct 14 2008

Number of steps in Eratosthenes's sieve for 2^n is A060967(n).

Number of primes less than 2^14 is equal to 2^14 - (sum all of numbers in this sequence) - 1 = A007053(14).

Cf. A006880, A122121, A145532-A145540, A145583-A145592.

f3[k_Integer?Positive, i_Integer?Positive] := Module[{f, m, r, p}, p = Transpose[{r = Range[2, i], Prime[r]}];f[x_] := Catch[Fold[If[Mod[x, #2[[2]]] == 0, Throw[m[ #2[[1]]] = m[ #2[[1]]] + 1], #1] &, If[Mod[x, 2] == 0, Throw[m[1] = m[1] + 1]], p]]; Table[m[n] = -1, {n, i}]; f /@ Range[k]; Table[m[n], {n, i}]];nn = 14; kk = PrimePi[Sqrt[2^nn]]; t3 = f3[2^nn, kk] (* Bob Hanlon (hanlonr(AT)cox.net) *)

A125527 Number of semiprimes <= 2^n.

Original entry on oeis.org

0, 1, 2, 6, 10, 22, 42, 82, 157, 304, 589, 1124, 2186, 4192, 8110, 15658, 30253, 58546, 113307, 219759, 426180, 827702, 1608668, 3129211, 6091437, 11868599, 23140878, 45150717, 88157689, 172235073, 336717854, 658662065, 1289149627, 2524532330

Offset: 1

Robert G. Wilson v, Dec 29 2006

nonn

Robert G. Wilson v, Table of n, a(n) for n = 1..63 (using data from A120033, terms n=48, 50..57 from Dana Jacobsen)
Eric Weisstein's World of Mathematics, Semiprime.

Cf. A126279, A066265, A007053, A120033, A127396.

SemiPrimePi[n_] := Sum[ PrimePi[ n/Prime@i] - i + 1, {i, PrimePi@ Sqrt@n}]; Table[ SemiPrimePi[2^n], {n, 47}]

a(n)=my(s,i,N=2^n); forprime(p=2, sqrtint(N), s+=primepi(N\p); i++); s - i * (i-1)/2 \\ Charles R Greathouse IV, May 12 2013

use ntheory ":all"; print "$ ",semiprime_count(1 << $),"\n" for 1..48; # Dana Jacobsen, Sep 10 2018

a(n) = A072000(2^n). - R. J. Mathar, Aug 26 2011

A040014 Number of primes < e^n.

Original entry on oeis.org

0, 1, 4, 8, 16, 34, 79, 183, 429, 1019, 2466, 6048, 14912, 37128, 93117, 234855, 595341, 1516233, 3877186, 9950346, 25614562, 66124777, 171141897, 443963543, 1154106844, 3005936865, 7842921261, 20496470801, 53645077679, 140599114669, 368973074565, 969455391690, 2550043255883

Offset: 0

Jud McCranie

nonn

a(n) = A000720(A000149(n)). - Reinhard Zumkeller, Mar 17 2015

David Baugh, Table of n, a(n) for n = 0..59 (terms n = 0..39 from Giovanni Resta, terms n = 40..59 found using Kim Walisch's primecount program).
Index entries for sequences related to numbers of primes in various ranges

Cf. A006880 and A007053.

Cf. A000720, A000149.

a040014 = a000720 . a000149  -- Reinhard Zumkeller, Mar 17 2015

Mathematica
```
Table[PrimePi[Exp[n]], {n, 0, 33}]
```

a(27)-a(29) from Robert G. Wilson v, Jun 09 2000

a(30)-a(32) from Seiichi Manyama, May 04 2016

cp's OEIS Frontend

A060967 Number of squared primes <= 2^n.

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A244508 Number of odd prime powers (A246655) between 2^n and 2^(n+1).

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A373406 Sum of the n-th maximal run of odd primes differing by two.

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A373669 Least k such that the k-th maximal run of non-prime-powers has length n. Position of first appearance of n in A110969, and the sequence ends if there is none.

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Mathematica

A055729 Number of primes <= 3^n.

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A036386 Number of prime powers (p^2, p^3, ...) <= 2^n.

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Mathematica

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Extensions

A065843 Let u be any string of n digits from {0,1}; let f(u) = number of distinct primes, not beginning with 0, formed by permuting the digits of u to a base-2 number; then a(n) = max_u f(u).

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