A065515 -id:A065515 - cp's OEIS frontend

A305976 Filter sequence for a(prime^k) = constant sequences.

1, 2, 2, 2, 2, 3, 2, 2, 2, 4, 2, 5, 2, 6, 7, 2, 2, 8, 2, 9, 10, 11, 2, 12, 2, 13, 2, 14, 2, 15, 2, 2, 16, 17, 18, 19, 2, 20, 21, 22, 2, 23, 2, 24, 25, 26, 2, 27, 2, 28, 29, 30, 2, 31, 32, 33, 34, 35, 2, 36, 2, 37, 38, 2, 39, 40, 2, 41, 42, 43, 2, 44, 2, 45, 46, 47, 48, 49, 2, 50, 2, 51, 2, 52, 53, 54, 55, 56, 2, 57, 58, 59, 60, 61, 62, 63, 2, 64, 65, 66, 2

Offset: 1

Antti Karttunen, Jul 02 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..100000

Cf. A000961 (A246655), A085970, A305800, A305975, A305979.

up_to = 100000;
partialsums(f,up_to) = { my(v = vector(up_to), s=0); for(i=1,up_to,s += f(i); v[i] = s); (v); }
v065515 = partialsums(n -> (omega(n)<=1), up_to);
A065515(n) = v065515[n];
A085970(n) = (n - A065515(n));
A305976(n) = if(1==n,n,if(isprimepower(n),2,2+A085970(n)));

a(1) = 1, for n > 1, if A010055(n) = 1 [when n is in A246655], a(n) = 2, otherwise a(n) = 2+A085970(n) = running count from 3 onward.

A024620 Positions of primes among the powers of primes (A000961).

Original entry on oeis.org

2, 3, 5, 6, 9, 10, 12, 13, 14, 17, 18, 20, 21, 22, 23, 25, 26, 27, 29, 30, 31, 32, 34, 35, 36, 37, 38, 39, 40, 41, 44, 46, 47, 48, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 70, 72, 73, 74, 75, 76, 77, 78, 80, 81, 82, 83, 84, 85, 86, 88, 89, 90, 91, 93, 94

Offset: 1

Clark Kimberling

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Complement of A024621.
Cf. A001222 (bigomega), A025474, A056604, A027883.
Cf. A025528, A065515, A000040.

a024620 n = a024620_list !! (n-1)
a024620_list = filter ((== 1) . a025474) [1..]
-- Reinhard Zumkeller, May 01 2015

a[n_] := PrimeOmega[LCM @@ Range@Prime@n] + 1; Array[a, 100] (* Amiram Eldar, Dec 02 2018 *)

lista(nn) = my(powpr = select((i->((omega(i)==1) || (i==1))), [1..nn])); for (i = 1, #powpr, if (isprime(powpr[i]), print1(i, ", ")); ); \\ Michel Marcus, Jun 03 2021

from sympy import prime, primepi, integer_nthroot
def A024620(n):
    x = prime(n)
    return n+1+sum(primepi(integer_nthroot(x,k)[0]) for k in range(2,x.bit_length())) # Chai Wah Wu, Nov 05 2024

A025474(a(n)) = 1. - Reinhard Zumkeller, May 01 2015
a(n) = A001222(A056604(n)) + 1. - Eric Desbiaux, Dec 02 2018
From Ridouane Oudra, Oct 18 2020: (Start)
a(n) = A027883(n) + 1;
a(n) = A025528(A000040(n)) + 1;
a(n) = A065515(A000040(n)). (End)

A085501 Number of prime powers p^k <= n that are not prime (k = 0 or k > 1).

Original entry on oeis.org

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

Offset: 1

Reinhard Zumkeller, Jul 03 2003

a(n) = Max{k: A025475(k)<=n};
a(n)=A065515(n)-A000720(n)=A069637(n)+1;
for n<36=(2*3)^2: a(n) = A069623(n).

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Prime Power.
Eric Weisstein's World of Mathematics, Perfect Powers.

a(n)=sum(k=2,logint(n,2), primepi(sqrtnint(n,k)))+1 \\ Charles R Greathouse IV, Jul 21 2017

first(n)=my(v=vector(n),s=1); for(e=2,logint(n,2), forprime(p=2,sqrtnint(n,e), v[p^e]=1)); for(i=1,n, s+=v[i]; v[i]=s); v \\ Charles R Greathouse IV, Jul 21 2017

from sympy import primepi, integer_nthroot
def A085501(n): return 1+sum(primepi(integer_nthroot(n,k)[0]) for k in range(2,n.bit_length())) # Chai Wah Wu, Aug 15 2024

A305975 Filter sequence: All prime powers p^k, k >= 1, are allotted to distinct equivalence classes according to their exponent k, while all other numbers occur in singular equivalence classes of their own.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 5, 3, 6, 2, 7, 2, 8, 9, 10, 2, 11, 2, 12, 13, 14, 2, 15, 3, 16, 5, 17, 2, 18, 2, 19, 20, 21, 22, 23, 2, 24, 25, 26, 2, 27, 2, 28, 29, 30, 2, 31, 3, 32, 33, 34, 2, 35, 36, 37, 38, 39, 2, 40, 2, 41, 42, 43, 44, 45, 2, 46, 47, 48, 2, 49, 2, 50, 51, 52, 53, 54, 2, 55, 10, 56, 2, 57, 58, 59, 60, 61, 2, 62, 63, 64, 65, 66, 67, 68, 2, 69, 70, 71

Offset: 1

Antti Karttunen, Jul 02 2018

nonn

Restricted growth sequence transform of A305974.

For all i, j: A305800(i) = A305800(j) => a(i) = a(j) => A305976(i) = A305976(j).

Antti Karttunen, Table of n, a(n) for n = 1..100000

Cf. A000961 (A246655), A085970, A305800, A305974, A305976, A305979.

up_to = 100000;
partialsums(f,up_to) = { my(v = vector(up_to), s=0); for(i=1,up_to,s += f(i); v[i] = s); (v); }
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
v065515 = partialsums(n -> (omega(n)<=1), up_to);
A065515(n) = v065515[n];
A085970(n) = (n - A065515(n));
A305974(n) = if(1==n,n,my(e = isprimepower(n)); if(e,-e,1+A085970(n)));
v305975 = rgs_transform(vector(up_to,n,A305974(n)));
A305975(n) = v305975[n];

a(prime) = 2, a(prime^2) = 3, a(prime^3) = 5, a(prime^4) = 10, a(prime^5) = 19.

A085972 Number of numbers <= n that are primes or not prime powers.

Original entry on oeis.org

0, 1, 2, 2, 3, 4, 5, 5, 5, 6, 7, 8, 9, 10, 11, 11, 12, 13, 14, 15, 16, 17, 18, 19, 19, 20, 20, 21, 22, 23, 24, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 54, 55, 56, 57, 58, 59, 60, 61

Offset: 1

Reinhard Zumkeller, Jul 06 2003

nonn

a(n) = Max{k: A085971(k)<=n};

a(n) = n-A085501(n) = A000720(n)+n-A065515(n) = A085970(n)+A000720(n).

Accumulate[Table[If[PrimeQ[n]||(!PrimePowerQ[n]),1,0],{n,80}]]-1 (* Harvey P. Dale, Oct 13 2022 *)

from sympy import primepi, integer_nthroot
def A085972(n): return n-1-sum(primepi(integer_nthroot(n,k)[0]) for k in range(2,n.bit_length())) # Chai Wah Wu, Aug 20 2024

A024623 a(n) is the number of prime powers <= 3^n.

Original entry on oeis.org

1, 3, 8, 16, 33, 69, 153, 360, 894, 2294, 6060, 16229, 44126, 121021, 334771, 931905, 2608160, 7333717, 20703268, 58652234, 166684333, 475034096, 1357217584, 3886575564, 11152863757, 32065004164, 92349161271, 266398439961, 769616851854, 2226457643329, 6449248612381, 18703413267318

Offset: 0

Clark Kimberling

nonn

a(n) is the position of 3^n among the powers of primes (A000961). - Robert G. Wilson v, Dec 16 2019

David A. Corneth, Table of n, a(n) for n = 0..47 (calculated using the bfile in A055729)

Cf. A000961 (powers of primes), A000244 (powers of 3), A055729, A065515.

a(n) = A065515(3^n). - Michel Marcus, Dec 15 2019

a(18)-a(31) from Robert G. Wilson v, Dec 15 2019

A081063 Number of numbers <= n that are 3-smooth or prime powers.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 9, 10, 11, 12, 12, 12, 13, 14, 15, 16, 16, 16, 16, 17, 18, 19, 19, 20, 20, 21, 21, 22, 23, 23, 23, 23, 24, 25, 25, 25, 25, 26, 26, 27, 27, 27, 27, 28, 29, 30, 30, 30, 30, 31, 32, 32, 32, 32, 32, 33, 33, 34, 34, 34, 35, 35, 35, 36, 36, 36, 36, 37, 38, 39

Offset: 1

Reinhard Zumkeller, Mar 04 2003

nonn

a(n) = #{A081061(k): 1<=k<=n}.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A003586, A000961.

Accumulate[Table[If[PrimePowerQ[n]||Max[FactorInteger[n][[All,1]]]<5,1,0],{n,80}]] (* Harvey P. Dale, Nov 29 2020 *)

from sympy import integer_log, primepi, integer_nthroot
def A081063(n): return int(1-(a:=n.bit_length())-(b:=integer_log(n,3)[0])+sum((n//3**i).bit_length() for i in range(b+1))+sum(primepi(integer_nthroot(n, k)[0]) for k in range(1, a))) # Chai Wah Wu, Sep 16 2024

a(n)=A071521(n)+A065515(n)-A000523(n)-A062153(n)+1.

A305974 a(1) = 1; for n > 1, if n = p^k for some prime p and exponent k >= 1, then a(n) = -k, otherwise a(n) = 1+A085970(n).

Original entry on oeis.org

1, -1, -1, -2, -1, 2, -1, -3, -2, 3, -1, 4, -1, 5, 6, -4, -1, 7, -1, 8, 9, 10, -1, 11, -2, 12, -3, 13, -1, 14, -1, -5, 15, 16, 17, 18, -1, 19, 20, 21, -1, 22, -1, 23, 24, 25, -1, 26, -2, 27, 28, 29, -1, 30, 31, 32, 33, 34, -1, 35, -1, 36, 37, -6, 38, 39, -1, 40, 41, 42, -1, 43, -1, 44, 45, 46, 47, 48, -1, 49, -4, 50, -1, 51, 52

Offset: 1

Antti Karttunen, Jul 02 2018

sign

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A000961, A065515, A085970, A095874, A305975 (rgs-transform).

up_to = 65537;
partialsums(f,up_to) = { my(v = vector(up_to), s=0); for(i=1,up_to,s += f(i); v[i] = s); (v); }
v065515 = partialsums(n -> (omega(n)<=1), up_to);
A065515(n) = v065515[n];
A085970(n) = (n - A065515(n));
A305974(n) = if(1==n,n,my(e = isprimepower(n)); if(e,-e,1+A085970(n)));

a(1) = 1; for n > 1, if n = p^k for some prime p and exponent k >= 1, then a(n) = -k, otherwise [when n is not a prime power], a(n) = 1+A085970(n) = running count from 2 onward.

A308819 Product of prime powers <= n.

Original entry on oeis.org

1, 1, 2, 6, 24, 120, 120, 840, 6720, 60480, 60480, 665280, 665280, 8648640, 8648640, 8648640, 138378240, 2352430080, 2352430080, 44696171520, 44696171520, 44696171520, 44696171520, 1028011944960, 1028011944960, 25700298624000, 25700298624000, 693908062848000

Offset: 0

Ilya Gutkovskiy, Jun 26 2019

nonn

a(n) is the smallest number that's divisible by all numbers less than or equal to n. - Keith F. Lynch, Apr 24 2025

			a(9) = 60480 because 2, 3, 4, 5, 7, 8, 9 are the prime powers less than or equal to 9 and 2 * 3 * 4 * 5 * 7 * 8 * 9 = 60480.

Cf. A000961, A024923, A034386, A065515, A074793, A100994, A179215, A246655.

a[n_] := Product[If[PrimePowerQ[k], k, 1], {k, 1, n}]; Table[a[n], {n, 0, 27}]
Module[{nn=50,ppwr},ppwr=Select[Range[nn],PrimePowerQ[#]&];Table[Times@@ Select[ ppwr,#<= n&],{n,0,nn}]] (* Harvey P. Dale, Apr 03 2024 *)

a(n) = prod(k=1, n, if (isprime(k) || isprimepower(k), k, 1)); \\ Michel Marcus, Jun 27 2019

A330669 The prime indices of the prime powers (A000961).

Original entry on oeis.org

0, 1, 2, 1, 3, 4, 1, 2, 5, 6, 1, 7, 8, 9, 3, 2, 10, 11, 1, 12, 13, 14, 15, 4, 16, 17, 18, 1, 19, 20, 21, 22, 2, 23, 24, 25, 26, 27, 28, 29, 30, 5, 3, 31, 1, 32, 33, 34, 35, 36, 37, 38, 39, 6, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49

Offset: 1

Grant E. Martin and Robert G. Wilson v, Dec 23 2019

			a(16) is 2 since A000961(16) is 27 which is 3^3 = (p_2)^3, i.e., the prime index of 3 is 2.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A000040, A000961, A000720, A025473, A025474, A065515.

b:= proc(n) option remember; local k; for k from
      1+b(n-1) while nops(ifactors(k)[2])>1 do od; k
    end: b(1):=1:
a:= n-> `if`(n=1, 0, numtheory[pi](ifactors(b(n))[2, 1$2])):
seq(a(n), n=1..100);  # Alois P. Heinz, Feb 20 2020

mxn = 500; Join[{0}, Transpose[ Sort@ Flatten[ Table[ {Prime@n^ex, n}, {n, PrimePi@ mxn}, {ex, Log[Prime@n, mxn]}], 1]][[2]]]

lista(nn) = {print1(0); for(n=2, nn, if(isprimepower(n, &p), print1(", ", primepi(p)))); } \\ Jinyuan Wang, Feb 19 2020

from sympy import primepi, integer_nthroot, primefactors
def A330669(n):
    if n == 1: return 0
    def f(x): return int(n-2+x-sum(primepi(integer_nthroot(x,k)[0]) for k in range(1,x.bit_length())))
    kmin, kmax = 1,2
    while f(kmax) >= kmax:
        kmax <<= 1
    while True:
        kmid = kmax+kmin>>1
        if f(kmid) < kmid:
            kmax = kmid
        else:
            kmin = kmid
        if kmax-kmin <= 1:
            break
    return int(primepi(primefactors(kmax)[0])) # Chai Wah Wu, Aug 20 2024

a(n) = A000720(A025473(n)). - Michel Marcus, Dec 24 2019

A000040(a(n))^A025474(n) = A000961(n) for n > 0. - Alois P. Heinz, Feb 20 2020

cp's OEIS Frontend

A305976 Filter sequence for a(prime^k) = constant sequences.

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A024620 Positions of primes among the powers of primes (A000961).

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A085501 Number of prime powers p^k <= n that are not prime (k = 0 or k > 1).

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A305975 Filter sequence: All prime powers p^k, k >= 1, are allotted to distinct equivalence classes according to their exponent k, while all other numbers occur in singular equivalence classes of their own.

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A085972 Number of numbers <= n that are primes or not prime powers.

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A024623 a(n) is the number of prime powers <= 3^n.

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A081063 Number of numbers <= n that are 3-smooth or prime powers.

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A305974 a(1) = 1; for n > 1, if n = p^k for some prime p and exponent k >= 1, then a(n) = -k, otherwise a(n) = 1+A085970(n).

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A308819 Product of prime powers <= n.