A007053 -id:A007053 - cp's OEIS frontend

A126279 Triangle read by rows: T(k,n) is number of numbers <= 2^n that are products of k primes.

1, 2, 1, 4, 2, 1, 6, 6, 2, 1, 11, 10, 7, 2, 1, 18, 22, 13, 7, 2, 1, 31, 42, 30, 14, 7, 2, 1, 54, 82, 60, 34, 15, 7, 2, 1, 97, 157, 125, 71, 36, 15, 7, 2, 1, 172, 304, 256, 152, 77, 37, 15, 7, 2, 1, 309, 589, 513, 325, 168, 81, 37, 15, 7, 2, 1, 564, 1124, 1049, 669, 367, 177, 83, 37

Offset: 1

Jonathan Vos Post & Robert G. Wilson v, Dec 22 2006

			Triangle begins:
  1
  2 1
  4 2 1
  6 6 2 1
  11 10 7 2 1
  18 22 13 7 2 1
  31 42 30 14 7 2 1
  54 82 60 34 15 7 2 1
  97 157 125 71 36 15 7 2 1
  172 304 256 152 77 37 15 7 2 1

Adolf Hildebrand, On the number of prime factors of an integer. Ramanujan revisited (Urbana-Champaign, Ill., 1987), 167 - 185, Academic Press, Boston, MA, 1988.
Edmund Georg Hermann Landau, Handbuch der Lehre von der Verteilung der Primzahlen, Chelsea Publishing, NY 1953, pp. 205 - 211.

Jonathan Vos Post & Robert G. Wilson v, Table of n, a(n) for n = 1..1330
Jonathan Vos Post & Robert G. Wilson v, Regular Triangle of A126279 for the first 52 rows, with some holes.
Eric Weisstein's World of Mathematics, Semiprime.
Eric Weisstein's World of Mathematics, Almost Prime.

First column: A007053, second column: A125527, third column: A127396, 4th column: A334069. The last row reversed: A052130; the n-th row's sum: A000225 = 2^n -1.

Cf. A126280: same array but for powers of ten.

AlmostPrimePi[k_Integer, n_] := Module[{a, i}, a[0] = 1; If[k == 1, PrimePi[n], Sum[ PrimePi[ n/Times @@ Prime[ Array[a, k - 1]]] - a[k - 1] + 1, Evaluate[ Sequence @@ Table[{a[i], a[i - 1], PrimePi[(n/Times @@ Prime[ Array[a, i - 1]])^(1/(k - i + 1))]}, {i, k - 1}]] ]]]; (* Eric W. Weisstein, Feb 07 2006 *)
Table[ AlmostPrimePi[m, 2^n], {n, 16}, {m, n}] // Flatten

A182908 Rank of 2^n when all prime powers (A246655) p^n, for n>=1, are jointly ranked.

Original entry on oeis.org

1, 3, 6, 10, 18, 27, 44, 70, 117, 198, 340, 604, 1078, 1961, 3590, 6635, 12370, 23150, 43579, 82267, 155921, 296347, 564688, 1078555, 2064589, 3958999, 7605134, 14632960, 28195586, 54403835, 105102701, 203287169, 393625231, 762951922, 1480223716, 2874422303

Offset: 1

Clark Kimberling, Dec 13 2010

nonn

			a(3)=6 because 2^3 has rank 6 in the sequence (2,3,4,5,7,8,9,...).

Ray Chandler, Table of n, a(n) for n = 1..92 (using b-file file from A007053)

Cf. A000720, A024622, A025528, A246655.

Row 1 of A182869. Complement of A182909.

T[i_,j_]:=Sum[Floor[j*Log[Prime[i]]/Log[Prime[h]]],{h,1,PrimePi[Prime[i]^j]}]; Flatten[Table[T[i,j],{i,1,1},{j,1,22}]]
f[n_] := Sum[ PrimePi[ Floor[2^(n/k)]], {k, n + 1}]; Array[f, 34] (* Robert G. Wilson v, Jul 08 2011 *)

from sympy import primepi, integer_nthroot
def A182908(n):
    x = 1<Chai Wah Wu, Nov 05 2024

a(n) = A182908(n) = A024622(n) - 1 for n>=1.
a(n) = Sum_{i=1..n} pi(floor(2^(n/i))), where pi(n) = A000720(n). - Ridouane Oudra, Oct 26 2020
a(n) = A025528(2^n). - Pontus von Brömssen, Sep 27 2024

Minor edits by Ray Chandler, Aug 20 2021

A056606 Squarefree kernel of lcm(binomial(n,0), ..., binomial(n,n)).

Original entry on oeis.org

1, 1, 2, 3, 6, 10, 30, 105, 70, 42, 210, 2310, 2310, 4290, 6006, 15015, 30030, 170170, 510510, 1939938, 1385670, 881790, 9699690, 223092870, 44618574, 17160990, 74364290, 31870410, 223092870, 6469693230, 6469693230, 100280245065

Offset: 0

Labos Elemer, Aug 07 2000

nonn

Also squarefree kernel of A001142; row products in table A256113. - Reinhard Zumkeller, Mar 21 2015
a(2372) has 1001 decimal digits. - Michael De Vlieger, Jul 14 2017
Also the squarefree kernel of the cumulative product of n^n/n!. - Peter Luschny, Dec 21 2019
Conjecture: the few odd values belong to A070826. - Bill McEachen, Jun 24 2023
And their indices appear to be A007053. - Michel Marcus, Jul 01 2023

			a(7) = 105 because lcm(1, 7, 21, 35) = 105 is already squarefree.
a(0) = 1 because n^n/n! = 1 for the integer n = 0. - _Peter Luschny_, Dec 21 2019

Michael De Vlieger, Table of n, a(n) for n = 0..2371

Cf. A002944, A034386, A048633, A003418, A000142, A001405.
Cf. A007947, A001142, A256113, A002109, A000178, A329947.
Cf. A070826.

a056606 = a007947 . a001142  -- Reinhard Zumkeller, Mar 21 2015

h := n -> mul(k^k/factorial(k), k=0..n):
rad := n -> mul(k, k = numtheory[factorset](n)):
seq(rad(h(n)), n=0..31); # Peter Luschny, Dec 21 2019

Table[Apply[Times, FactorInteger[Product[k^(2 k - 1 - n), {k, n}]][[All, 1]]], {n, 0, 31}] (* or *)
Table[Apply[Times, FactorInteger[Apply[LCM, Range@ n]/n][[All, 1]]], {n, 1, 32}] (* Michael De Vlieger, Jul 14 2017 *)

rad(n) = factorback(factorint(n)[, 1]); \\ A007947
a(n) = rad(lcm(vector(n+1, k, binomial(n,k-1)))); \\ Michel Marcus, Jun 24 2023

a(n) = A007947(A002944(n+1)). - Michel Marcus, Dec 21 2019

a(n) = radical(hyperfactorial(n)/superfactorial(n)) = A007947(A002109(n)/ A000178(n)) for n >= 0. - Peter Luschny, Dec 21 2019

Extended with a(0) = 1 by Peter Luschny, Dec 21 2019

A024622 Position of 2^n among the powers of primes (A000961).

Original entry on oeis.org

1, 2, 4, 7, 11, 19, 28, 45, 71, 118, 199, 341, 605, 1079, 1962, 3591, 6636, 12371, 23151, 43580, 82268, 155922, 296348, 564689, 1078556, 2064590, 3959000, 7605135, 14632961, 28195587, 54403836, 105102702, 203287170, 393625232, 762951923, 1480223717, 2874422304

Offset: 0

Clark Kimberling

nonn

Number of prime powers <= 2^n. - Jon E. Schoenfield, Nov 06 2016

A000961(a(n)) = A000079(n); also position of record values in A192015: A001787(n) = A192015(a(n)). - Reinhard Zumkeller, Jun 26 2011

Ray Chandler, Table of n, a(n) for n = 0..92 (using b-file from A007053, first 61 terms from Hiroaki Yamanouchi)

Cf. A000079, A000720, A000961, A001787, A007053, A025528, A182908, A192015.

{1}~Join~Flatten[1 + Position[Select[Range[10^6], PrimePowerQ], k_ /; IntegerQ@ Log2@ k ]] (* Michael De Vlieger, Nov 14 2016 *)

lista(nn) = {v = vector(2^nn, i, i); vpp = select(x->ispp(x), v); print1(1, ", "); for (i=1, #vpp, if ((vpp[i] % 2) == 0, print1(i, ", ")););} \\ Michel Marcus, Nov 17 2014

a(n)=sum(k=1,n,primepi(sqrtnint(2^n,k)))+1 \\ Charles R Greathouse IV, Nov 21 2014

a(n)=my(s=0);for(i=1, 2^n, isprimepower(i) && s++);s+1 \\ Dana Jacobsen, Mar 23 2021

use ntheory ":all"; for my $n (0..20) { my $s=1; is_prime_power($) && $s++ for 1..2**$n; print "$n $s\n" } # _Dana Jacobsen, Mar 23 2021

use ntheory ":all"; for my $n (0..64) { my $s = ($n < 1) ? 1 : vecsum(map{prime_count(rootint(powint(2,$n)-1,$))}1..$n)+2; print "$n $s\n"; } # _Dana Jacobsen, Mar 23 2021

# with b-file for pi(2^n)
perl -Mntheory=:all -nE 'my($n,$pc)=split; say "$n ", addint($pc,vecsum( map{prime_count(rootint(powint(2,$n),$))} 2..$n )+1);'  b007053.txt  # _Dana Jacobsen, Mar 23 2021

from sympy import primepi, integer_nthroot
def A024622(n):
    x = 1<Chai Wah Wu, Nov 05 2024

def a(n): return sum(prime_pi(ZZ(2^n).nth_root(k+1,truncate_mode=1)[0]) for k in range(n))+1 # Dana Jacobsen, Mar 23 2021

From Ridouane Oudra, Oct 26 2020: (Start)
a(n) = 1 + Sum_{i=1..n} pi(floor(2^(n/i))), where pi(n) = A000720(n);
a(n) = 1 + A182908(n). (End)
a(n) = A025528(2^n)+1. - Pontus von Brömssen, Sep 28 2024

a(28)-a(36) from Hiroaki Yamanouchi, Nov 21 2014

a(46)-a(53) corrected by Hiroaki Yamanouchi, Nov 15 2016

A102210 Number of primes that are bitwise covered by n.

Original entry on oeis.org

0, 1, 2, 0, 1, 1, 4, 0, 0, 1, 3, 0, 2, 1, 6, 0, 1, 1, 4, 0, 2, 1, 7, 0, 1, 1, 5, 0, 4, 1, 11, 0, 0, 1, 2, 0, 2, 1, 5, 0, 1, 1, 5, 0, 4, 1, 10, 0, 1, 1, 4, 0, 4, 1, 9, 0, 2, 1, 8, 0, 8, 1, 18, 0, 0, 1, 3, 0, 1, 1, 6, 0, 1, 1, 5, 0, 3, 1, 10, 0, 1, 1, 6, 0, 2, 1, 10, 0, 3, 1, 9, 0, 6, 1, 17, 0, 1, 1, 4, 0, 4, 1

Offset: 1

Reinhard Zumkeller, Dec 30 2004

p is bitwise covered by n iff (p = (n AND p)) bitwise: A080099(n,p)=p.

			n=21->10101 -> a(21) = #{00101=5,10001=17} = 2.

Cf. A007053, A007088, A004676, A080099, A102211, A102212, A102213, A102550.

[#[p:p in PrimesUpTo(n)| p eq BitwiseAnd(n,p)] :n in [1..105] ]; // Marius A. Burtea, Jan 12 2020

a[n_] := Count[Range[n], ?(PrimeQ[#] && BitAnd[n, #] == # &)]; Array[a, 100] (* _Amiram Eldar, Jan 12 2020 *)

a(A102211(n)) = 0; a(A102212(n)) = 1; a(A102213(n)) > 1.

a(2^k-1) = A007053(k) for k > 1. - Amiram Eldar, Jan 12 2020

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

A126084 a(n) = XOR of first n primes.

Original entry on oeis.org

0, 2, 1, 4, 3, 8, 5, 20, 7, 16, 13, 18, 55, 30, 53, 26, 47, 20, 41, 106, 45, 100, 43, 120, 33, 64, 37, 66, 41, 68, 53, 74, 201, 64, 203, 94, 201, 84, 247, 80, 253, 78, 251, 68, 133, 64, 135, 84, 139, 104, 141, 100, 139, 122, 129, 384, 135, 394, 133, 400, 137, 402, 183, 388, 179

Offset: 0

Esko Ranta, Mar 02 2007

The values at odd positive indices are even and the values at even positive indices are odd.
Does this sequence contain any zeros for n > 0? Probabilistically, one would expect so; but none in first 10000 terms. - Franklin T. Adams-Watters, Jul 17 2011
None below 1.5 * 10^11: any prime p such that a(pi(p)) = 0 is 43 bits or longer. Heuristic chances that a prime below 2^100 yields 0 are about 45%. Note that an n-bit prime can yield 0 only if a(pi(p)) is odd, where p is the smallest n-bit prime. That is, for n > 1, there are no zeros from pi(2^n) to pi(2^(n+1)) if A007053(n) is even. - Charles R Greathouse IV, Jul 17 2011

			a(4) = 3 because ((2 XOR 3) XOR 5) XOR 7 = (1 XOR 5) XOR 7 = 4 XOR 7 = 3
[Or, in base 2]
((10 XOR 11) XOR 101) XOR 111 = (1 XOR 101) XOR 111 = 100 XOR 111 = 11

Franklin T. Adams-Watters, Table of n, a(n) for n = 0..10000

Cf. A003815, A004767, A112591.

Module[{nn=70,prs},prs=Prime[Range[nn]];Table[BitXor@@Take[prs,n],{n,0,nn}]] (* Harvey P. Dale, Jun 23 2016 *)

al(n)=local(m);vector(n,k,m=bitxor(m,prime(k))) /* Produces a vector without a(0) = 0; Franklin T. Adams-Watters, Jul 17 2011 */

v=primes(300); for(i=2,#v,v[i]=bitxor(v[i],v[i-1])); concat(0, v) \\ Charles R Greathouse IV, Aug 26 2014

q=0; forprime(p=2, 313, print1(q, ","); q=bitxor(q, p)) /* Klaus Brockhaus, Mar 06 2007; adapted by Rémy Sigrist, Oct 23 2017 */

from operator import xor
from functools import reduce
from sympy import primerange, prime
def A126084(n): return reduce(xor,primerange(2,prime(n)+1)) if n else 0 # Chai Wah Wu, Jul 09 2022

a(0) = 0; a(n) = a(n-1) XOR prime(n).

More terms from Klaus Brockhaus, Mar 06 2007
Edited by N. J. A. Sloane, Oct 22 2017 (merging old entry A193174 with this)
Edited by Rémy Sigrist, Oct 23 2017

A308710 Primitive practical numbers of the form 2^i * prime(k).

Original entry on oeis.org

6, 20, 28, 88, 104, 272, 304, 368, 464, 496, 1184, 1312, 1376, 1504, 1696, 1888, 1952, 4288, 4544, 4672, 5056, 5312, 5696, 6208, 6464, 6592, 6848, 6976, 7232, 8128, 16768, 17536, 17792, 19072, 19328, 20096, 20864, 21376, 22144, 22912, 23168, 24448, 24704, 25216

Offset: 1

Miko Labalan, Jun 19 2019

nonn

Intersection of A267124 and A100368.
a(n) is a number of the form 2^i * prime(k) for i > 0 and A007053(i) < k <= A007053(i+1).
Terms are pseudoperfect numbers, A005835 and are also primitive pseudoperfect numbers, A006036.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000040, A000079, A005153, A005835, A006036, A007053, A100368, A267124.

a[n_] := (p = Prime[n+1]) * 2^Floor[Log2[p]]; Array[a, 50] (* Amiram Eldar, Sep 22 2019 *)

isok(n) = is_A100368(n) && is_A267124(n); \\ Michel Marcus, Jun 19 2019

a(n) = 2^floor(log_2(prime(n+1))) * prime(n+1).

cp's OEIS Frontend

A126279 Triangle read by rows: T(k,n) is number of numbers <= 2^n that are products of k primes.

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A182908 Rank of 2^n when all prime powers (A246655) p^n, for n>=1, are jointly ranked.

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A056606 Squarefree kernel of lcm(binomial(n,0), ..., binomial(n,n)).

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A024622 Position of 2^n among the powers of primes (A000961).

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A102210 Number of primes that are bitwise covered by n.

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Magma

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

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