A182908 -id:A182908 - cp's OEIS frontend

A080101 Number of prime powers in all composite numbers between n-th prime and next prime.

0, 1, 0, 2, 0, 1, 0, 0, 2, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0

Offset: 1

Reinhard Zumkeller, Jan 28 2003

nonn

The maximum value of terms in the sequence, through the (10^5)th term, is 2. - Harvey P. Dale, Aug 24 2014

This is conjectured to be the maximum, see also A366833. - Gus Wiseman, Nov 06 2024

			There are two prime powers between 2179 = A000040(327) and 2203 = A000040(328): 2187 = 3^7 and 2197 = 13^3, therefore a(327) = 2, A080102(327) = 2187 and A080103(327) = 2197.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harvey P. Dale)

Cf. A080102, A080103, A025475, A000961.
For powers of 2 instead of primes we have A244508, see also A013597, A014210, A014234, A304521.
Adding one gives A366833.
For non-prime-powers instead of prime-powers we have A368748.
Positions of positive terms are A377057, primes A053607.
Positions of 0 are A377286.
Positions of 1 are A377287.
Positions of 2 are A377288, primes A053706.
For perfect-powers (instead of prime-powers) we have A377432.
A000015 gives the least prime-power >= n, difference A377282.
A000040 lists the primes, differences A001223.
A000961 lists the powers of primes, differences A057820, seconds A376596.
A031218 gives the greatest prime-power <= n, difference A276781.
A046933(n) counts the interval from A008864(n) to A006093(n+1).
A065514 gives the greatest prime-power < prime(n), difference A377289.
A246655 lists the prime-powers not including 1, complement A361102.
A345531 gives the least prime-power > prime(n), difference A377281.
Cf. A001597, A002808, A024619, A065890, A182908, A224363, A376597, A377051, A377054, A377436.

a := proc(n) local c, k, p: c, p := 0, ithprime(n): for k from p+1 to nextprime(p)-1 do if nops(numtheory:-factorset(k)) = 1 then c := c+1: fi: od: c: end:
seq(a(n), n = 1 .. 105); # Lorenzo Sauras Altuzarra, Jul 08 2022

prpwQ[n_]:=Module[{fi=FactorInteger[n]},Length[fi]==1&&fi[[1,2]]>1]; nn=600;With[{pwrs=Table[If[prpwQ[n],1,0],{n,nn}]},Table[Total[ Take[ pwrs,{Prime[n],Prime[n+1]}]],{n,PrimePi[nn]-1}]] (* Harvey P. Dale, Aug 24 2014 *)
Table[Length[Select[Range[Prime[n]+1,Prime[n+1]-1],PrimePowerQ]],{n,30}] (* Gus Wiseman, Nov 06 2024 *)

a(n) = A366833(n) - 1. - Gus Wiseman, Nov 06 2024

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

A182869 Joint-rank array of prime powers: p(i)^j, i>=1, j>=1, read by antidiagonals.

Original entry on oeis.org

1, 3, 2, 6, 7, 4, 10, 15, 14, 5, 18, 32, 42, 23, 8, 27, 68, 136, 86, 41, 9, 44, 152, 482, 392, 244, 53, 11, 70, 359, 1880, 2001, 1773, 360, 91, 12, 117, 893, 7771, 11211

Offset: 1

Clark Kimberling, Dec 09 2010

Joint-rank arrays are defined in the first comment at A182801. A182869 is a permutation of the positive integers.

			First, arrange the prime powers in rows:
2....4....8....16....32...
3....9...27....81...243...
5...25..125...625..3125...
Then replace each prime power by its rank when they are all jointly ranked:
1....3....6....10.....18...
2....7...15....32.....68...
4...14...42...136....482...
5...23...86...392...2001...
8...41..244..1773..14901...

Cf. A000961, A027883, A182801, A182870, A182908.

T[i_,j_]:=Sum[Floor[j*Log[Prime[i]]/Log[Prime[h]]],{h,1,PrimePi[Prime[i]^j]}];
TableForm[Table[T[i,j],{i,1,6},{j,1,6}]]

T(i,j) = Sum_{h>=1} floor(j*log(p(i))/log(p(h))), where p(i) denotes the i-th prime.

Corrected and extended by Clark Kimberling, Dec 13 2010

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

A333235 a(n) is the product of indices of unitary prime power divisors of n.

Original entry on oeis.org

1, 1, 2, 3, 4, 2, 5, 6, 7, 4, 8, 6, 9, 5, 8, 10, 11, 7, 12, 12, 10, 8, 13, 12, 14, 9, 15, 15, 16, 8, 17, 18, 16, 11, 20, 21, 19, 12, 18, 24, 20, 10, 21, 24, 28, 13, 22, 20, 23, 14, 22, 27, 24, 15, 32, 30, 24, 16, 25, 24, 26, 17, 35, 27, 36, 16, 28, 33, 26, 20

Offset: 1

Ilya Gutkovskiy, Mar 12 2020

Equivalently: replace each prime power p^e in the prime factorization of n by its index in A246655. - M. F. Hasler, Jun 16 2021

			a(600) = a(2^3 * 3 * 5^2) = a(A246655(6) * A246655(2) * A246655(14)) = 6 * 2 * 14 = 168.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A003963, A025528, A027883, A141128, A141809, A156061, A182908, A246655.

Cf. A322981 (the index of n = p^e in A246655).

N:= 1000: # for a(1)..a(N)
R:= NULL: p:= 2:
while p < N do
  R:= R,  seq(p^k,k=1..ilog[p](N));
  p:= nextprime(p);
od:
L:= sort([R]):
f:= proc(n) local F, t;
  F:= ifactors(n)[2];
  mul(ListTools:-BinarySearch(L,t[1]^t[2]),t=F)
end proc:
map(f, [$1..N]); # Robert Israel, Feb 11 2021

PrimePowerPi[n_] := Sum[Boole[PrimePowerQ[k]], {k, 1, n}]; a[1] = 1; a[n_] := Times @@ (PrimePowerPi[#[[1]]^#[[2]]] & /@ FactorInteger[n]); Table[a[n], {n, 1, 70}]

apply( {A333235(n)=vecprod([A322981(f[1]^f[2])|f<-factor(n)~])}, [1..99]) \\ M. F. Hasler, Jun 16 2021

If n = Product (p_j^k_j) then a(n) = Product (A025528(p_j^k_j)).
a(prime(n)) = A027883(n).
a(2^n) = A182908(n).
a(A246655(n)) = n.

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

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

Michael De Vlieger, Jan 24 2025

			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.

Cf. A000849, A002110, A182908, A246655, A380402.

Table[Sum[PrimePi[Floor[#^(1/k)]], {k, Floor@ Log2[#]}] &[Product[Prime[i], {i, n}]], {n, 0, 14}]

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

a(n) = A000849(n) + A380402(n).

cp's OEIS Frontend

A080101 Number of prime powers in all composite numbers between n-th prime and next prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula

Extensions

A182869 Joint-rank array of prime powers: p(i)^j, i>=1, j>=1, read by antidiagonals.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

Extensions

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

PARI

PARI

Perl

Perl

Perl

Python

SageMath

Formula

Extensions

A333235 a(n) is the product of indices of unitary prime power divisors of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica