A255165 -id:A255165 - cp's OEIS frontend

A089723 a(1)=1; for n>1, a(n) gives number of ways to write n as n = x^y, 2 <= x, 1 <= y.

1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1

Offset: 1

Naohiro Nomoto, Jan 07 2004

This function depends only on the prime signature of n. - Franklin T. Adams-Watters, Mar 10 2006
a(n) is the number of perfect divisors of n. Perfect divisor of n is divisor d such that d^k = n for some k >= 1. a(n) > 1 for perfect powers n = A001597(m) for m > 2. - Jaroslav Krizek, Jan 23 2010
Also the number of uniform perfect integer partitions of n - 1. An integer partition of n is uniform if all parts appear with the same multiplicity, and perfect if every nonnegative integer up to n is the sum of a unique submultiset. The Heinz numbers of these partitions are given by A326037. The a(16) = 3 partitions are: (8,4,2,1), (4,4,4,1,1,1), (1,1,1,1,1,1,1,1,1,1,1,1,1,1,1). - Gus Wiseman, Jun 07 2019
The record values occur at 1 and at 2^A002182(n) for n > 1. - Amiram Eldar, Nov 06 2020

			144 = 2^4 * 3^2, gcd(4,2) = 2, d(2) = 2, so a(144) = 2. The representations are 144^1 and 12^2.
From _Friedjof Tellkamp_, Jun 14 2025: (Start)
n:          1, 2, 3, 4, 5, 6, 7, 8, 9, ...
----------------------------------------------------
1st powers: 1, 1, 1, 1, 1, 1, 1, 1, 1, ... (A000012)
Squares:    1, 0, 0, 1, 0, 0, 0, 0, 1, ... (A010052)
Cubes:      1, 0, 0, 0, 0, 0, 0, 1, 0, ... (A010057)
Quartics:   1, 0, 0, 0, 0, 0, 0, 0, 0, ... (A374016)
...
Sum:       oo, 1, 1, 2, 1, 1, 1, 2, 2, ...
a(1)=1:     1, 1, 1, 2, 1, 1, 1, 2, 2, ... (= this sequence). (End)

N. J. A. Sloane, Table of n, a(n) for n = 1..20000
Solomon W. Golomb, A new arithmetic function of combinatorial significance, J. Number Theory, Vol. 5, No. 3 (1973), pp. 218-223. 1973JNT.....5..218G
Jan Mycielski, Sur les représentations des nombres naturels par des puissances à base et exposant naturels, Colloquium Mathematicum, Vol. 2 (1951), pp. 254-260. See gamma(n).
N. J. A. Sloane, Maple programs for A007916, A278028, A278029, A052409, A089723, A277564

Cf. A000005, A277564, A278028.
Cf. A000961, A002033, A002182, A007916, A047966, A070941, A108917, A126796, A276024, A326035, A326036, A326037.
Cf. A010052, A010057, A374016, A259362.

with(numtheory):
A089723 := proc(n) local t1,t2,g,j;
if n=1 then 1 else
t1:=ifactors(n)[2]; t2:=nops(t1); g := t1[1][2];
for j from 2 to t2 do g:=gcd(g,t1[j][2]); od:
tau(g); fi; end;
[seq(A089723(n),n=1..100)]; # N. J. A. Sloane, Nov 10 2016

Table[DivisorSigma[0, GCD @@ FactorInteger[n][[All, 2]]], {n, 100}] (* Gus Wiseman, Jun 12 2017 *)

a(n) = if (n==1, 1, numdiv(gcd(factor(n)[,2]))); \\ Michel Marcus, Jun 13 2017

from math import gcd
from sympy import factorint, divisor_sigma
def a(n):
    if n == 1: return 1
    e = list(factorint(n).values())
    g = e[0]
    for ei in e[1:]: g = gcd(g, ei)
    return divisor_sigma(g, 0)
print([a(n) for n in range(1, 105)]) # Michael S. Branicky, Jul 15 2021

If n = Product p_i^e_i, a(n) = d(gcd()). - Franklin T. Adams-Watters, Mar 10 2006
Sum_{n=1..m} a(n) = A255165(m) + 1. - Richard R. Forberg, Feb 16 2015
Sum_{n>=2} a(n)/n^s = Sum_{n>=2} 1/(n^s-1) = Sum_{k>=1} (zeta(s*k)-1) for all real s with Re(s) > 1 (Golomb, 1973). - Amiram Eldar, Nov 06 2020
For n > 1, a(n) = Sum_{i=1..floor(n/2)} floor(n^(1/i))-floor((n-1)^(1/i)). - Wesley Ivan Hurt, Dec 08 2020
Sum_{n>=1} (a(n)-1)/n = 1 (Mycielski, 1951). - Amiram Eldar, Jul 15 2021
From Friedjof Tellkamp, Jun 14 2025: (Start)
a(n) = 1 + A259362(n) = 1 + A010052(n) + A010057(n) + A374016(n) + (...), for n > 1.
G.f.: x + Sum_{j>=2, k>=1} x^(j^k). (End)

A043000 Number of digits in all base-b representations of n, for 2 <= b <= n.

Original entry on oeis.org

2, 4, 7, 9, 11, 13, 16, 19, 21, 23, 25, 27, 29, 31, 35, 37, 39, 41, 43, 45, 47, 49, 51, 54, 56, 59, 61, 63, 65, 67, 70, 72, 74, 76, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99, 101, 103, 106, 108, 110, 112, 114, 116, 118, 120, 122, 124, 126

Offset: 2

Clark Kimberling

From A.H.M. Smeets, Dec 14 2019: (Start)
a(n)-a(n-1) >= 2 due to the fact that n = 10_n, so there is an increment of at least 2. If n can be written as a perfect power m^s, an additional +1 comes to it for the representation of n in each base m.
For instance, for n = 729 we have 729 = 3^6 = 9^3 = 27^2, so there is an additional increment of 3. For n = 1296 we have 1296 = 6^4 = 36^2, so there is an additional increment of 2. For n = 4096 we have 4096 = 2^12 = 4^6 = 8^4 = 16^3= 64^2, so there is an additional increment of 5. (End)

			5 = 101_2 = 12_3 = 11_4 = 10_5. Thus a(5) = 3+2+2+2 = 9.

A.H.M. Smeets, Table of n, a(n) for n = 2..20000
Vaclav Kotesovec, Plot of a(n)/(2*n) for n = 2..1000000

Cf. A001597, A043306, A068953, A191322, A253642, A255165.

[&+[Floor(Log(i,i*n)):k in [2..n]]:n in [1..70]]; // Marius A. Burtea, Nov 13 2019

A043000 := proc(n) add( nops(convert(n,base,b)),b=2..n) ; end proc: # R. J. Mathar, Jun 04 2011

Table[Total[IntegerLength[n,Range[2,n]]],{n,2,60}] (* Harvey P. Dale, Apr 23 2019 *)

a(n)=sum(b=2,n,#digits(n,b)) \\ Jeppe Stig Nielsen, Dec 14 2019

a(n)= n-1 +sum(b=2,n,logint(n,b)) \\ Jeppe Stig Nielsen, Dec 14 2019

a(n) = {2*n-2+sum(i=2, logint(n, 2), sqrtnint(n, i)-1)} \\ David A. Corneth, Dec 31 2019

first(n) = my(res = vector(n)); res[1] = 2; for(i = 2, n, inc = numdiv(gcd(factor(i+1)[,2]))+1; res[i] = res[i-1]+inc); res \\ David A. Corneth, Dec 31 2019

def count(n,b):
    c = 0
    while n > 0:
        n, c = n//b, c+1
    return c
n = 0
while n < 50:
    n = n+1
    a, b = 0, 1
    while b < n:
        b = b+1
        a = a + count(n,b)
    print(n,a) # A.H.M. Smeets, Dec 14 2019

a(n) = Sum_{i=2..n} floor(log_i(i*n)); a(n) ~ 2*n. - Vladimir Shevelev, Jun 03 2011 [corrected by Vaclav Kotesovec, Apr 05 2021]
a(n) = A070939(n) + A081604(n) + A110591(n) + ... + 1. - R. J. Mathar, Jun 04 2011
From Ridouane Oudra, Nov 13 2019: (Start)
a(n) = Sum_{i=1..n-1} floor(n^(1/i));
a(n) = n - 1 + Sum_{i=1..floor(log_2(n))} floor(n^(1/i) - 1);
a(n) = n - 1 + A255165(n). (End)
If n is in A001597 then a(A001597(m)) - a(A001597(m)-1) = 2 + A253642(m), otherwise a(n) - a(n-1) = 2. - A.H.M. Smeets, Dec 14 2019

A342871 a(n) = Sum_{k=1..n} floor(n^(1/k)), n >= 1.

Original entry on oeis.org

1, 3, 5, 8, 10, 12, 14, 17, 20, 22, 24, 26, 28, 30, 32, 36, 38, 40, 42, 44, 46, 48, 50, 52, 55, 57, 60, 62, 64, 66, 68, 71, 73, 75, 77, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, 100, 102, 104, 107, 109, 111, 113, 115, 117, 119, 121, 123, 125, 127, 129, 131, 133

Offset: 1

Avid Rajai, Mar 28 2021

nonn

Avid Rajai and David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A043000, A255165, A089361, A001597, A253642.

Table[Sum[Floor[n^(1/k)],{k,n}],{n,100}] (* Giorgos Kalogeropoulos, Mar 31 2021 *)

a(n)=sum(k=1, n, sqrtnint(n,k)) \\ Andrew Howroyd, Mar 28 2021

a(n) = if(n < 2, return(n)); my(c = logint(n, 2)); 2*n + sum(i = 2, c, sqrtnint(n, i)) - c \\ David A. Corneth, Mar 28 2021

from sympy import integer_nthroot
def A342871(n):
    c = 0
    for k in range(1,n+1):
        m = integer_nthroot(n,k)[0]
        if m == 1:
            return c+n-k+1
        else:
            c += m
    return c # Chai Wah Wu, Apr 06 2021

Lim_{n->infinity} a(n)/n = 2.
a(n) = 2*n + sqrt(n) + O(n^(1/3)).
Lim_{n->infinity} (a(n)/n - 2)*sqrt(n) = 1.
a(n) = A043000(n) + 1 for n >= 2.
a(n) = A255165(n) + n for n >= 2.
a(n) = A089361(n) + 2*n - 1 for n >= 2.
a(n) = n + Sum_{i=1..floor(log_2(n))} floor(n^(1/i) - 1).
If n is in A001597 then a(A001597(m)) - a(A001597(m)-1) = 2 + A253642(m), otherwise a(n) - a(n-1) = 2.
2 <= a(n)/n <= 9/4 iff n >= 4.
1 <= (a(n)/n - 2)*sqrt(n) <= 27/16 iff n >= 27.
2*n + sqrt(n) < a(n) <= 2*n + (27/16)*sqrt(n) iff n >= 27.

cp's OEIS Frontend

A089723 a(1)=1; for n>1, a(n) gives number of ways to write n as n = x^y, 2 <= x, 1 <= y.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Python

Formula

A043000 Number of digits in all base-b representations of n, for 2 <= b <= n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

PARI

PARI

PARI

Python

Formula

A342871 a(n) = Sum_{k=1..n} floor(n^(1/k)), n >= 1.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Python

Formula