A013940 -id:A013940 - cp's OEIS frontend

A056170 Number of non-unitary prime divisors of n.

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

Offset: 1

Labos Elemer, Jul 27 2000

A prime factor of n is unitary iff its exponent is 1 in the prime factorization of n. (Of course for any prime p, GCD(p, n/p) is either 1 or p. For a unitary prime factor it must be 1.)
Number of squared primes dividing n. - Reinhard Zumkeller, May 18 2002
a(A005117(n)) = 0; a(A013929(n)) > 0; a(A190641(n)) = 1. - Reinhard Zumkeller, Dec 29 2012
First differences of A013940. - Jason Kimberley, Feb 01 2017
Number of exponents larger than 1 in the prime factorization of n. - Antti Karttunen, Nov 28 2017

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for sequences computed from exponents in factorization of n.

Cf. A000188, A001221, A003557, A013940, A034444, A046660, A048105, A056169, A085548, A124010, A162641, A212177, A275812, A295659, A295666.

Cf. A057427(a(n)) = 1 - A008966(n).

a056170 = length . filter (> 1) . a124010_row
-- Reinhard Zumkeller, Dec 29 2012

A056170:=func;
[A056170(n):n in[1..105]];
// Jason Kimberley, Jan 22 2017

A056170 := n -> nops(select(t -> (t[2]>1), ifactors(n)[2]));
seq(A056170(n),n=1..100); # Robert Israel, Jun 03 2014

a[n_] := Count[FactorInteger[n], {, k /; k > 1}]; Table[a[n], {n, 105}]  (* Jean-François Alcover, Mar 23 2011 *)
Table[Count[FactorInteger[n][[All,2]],?(#>1&)],{n,110}] (* _Harvey P. Dale, Jul 08 2019 *)

a(n)=my(f=factor(n)[,2]); sum(i=1,#f,f[i]>1) \\ Charles R Greathouse IV, May 18 2015

from sympy import factorint
def a(n):
    f = factorint(n)
    return sum([1 for i in f if f[i]!=1]) # Indranil Ghosh, Apr 24 2017

Additive with a(p^e) = 0 if e = 1, 1 otherwise.
G.f.: Sum_{k>=1} x^(prime(k)^2)/(1 - x^(prime(k)^2)). - Ilya Gutkovskiy, Jan 01 2017
a(n) = log_2(A000005(A071773(n))). - observed by Velin Yanev, Aug 20 2017, confirmed by Antti Karttunen, Nov 28 2017
From Antti Karttunen, Nov 28 2017: (Start)
a(n) = A001221(n) - A056169(n).
a(n) = omega(A000188(n)) = omega(A003557(n)) = omega(A057521(n)) = omega(A295666(n)), where omega = A001221.
For all n >= 1 it holds that:
a(A003557(n)) = A295659(n).
a(n) >= A162641(n).
(End)
Dirichlet g.f.: primezeta(2s)*zeta(s). - Benedict W. J. Irwin, Jul 11 2018
Asymptotic mean: lim_{n->oo} (1/n) * Sum_{k=1..n} a(k) = Sum_{p prime} 1/p^2 = 0.452247... (A085548). - Amiram Eldar, Nov 01 2020
a(n) = A275812(n) - A046660(n). - Amiram Eldar, Jan 09 2024

Minor edits by Franklin T. Adams-Watters, Mar 23 2011

A057627 Number of nonsquarefree numbers not exceeding n.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 1, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 6, 6, 7, 7, 7, 7, 8, 9, 9, 10, 11, 11, 11, 11, 12, 12, 12, 12, 13, 13, 13, 13, 14, 14, 14, 14, 15, 16, 16, 16, 17, 18, 19, 19, 20, 20, 21, 21, 22, 22, 22, 22, 23, 23, 23, 24, 25, 25, 25, 25, 26, 26, 26, 26, 27, 27, 27, 28, 29, 29, 29

Offset: 1

Labos Elemer, Oct 10 2000

Number of integers k in A013929 in the range 1 <= k <= n.
This sequence is different from A013940, albeit the first 35 terms are identical.
Asymptotic to k*n where k = 1 - 1/zeta(2) = 1 - 6/Pi^2 = A229099. - Daniel Forgues, Jan 28 2011
This sequence is the sequence of partial sums of A107078 (not of A056170). - Jason Kimberley, Feb 01 2017
Number of partitions of 2n into two parts with the smallest part nonsquarefree. - Wesley Ivan Hurt, Oct 25 2017

			a(36)=13 because 13 nonsquarefree numbers exist which do not exceed 36:{4,8,9,12,16,18,20,24,25,27,28,32,36}.

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

Cf. A005117, A008683, A013929, A013940, A013928, A002321, A028442, A107078, A229099, A294242.

N:= 1000: # to get terms up to a(N)
B:= Array(1..N, numtheory:-issqrfree):
C:= map(`if`,B,0,1):
A:= map(round,Statistics:-CumulativeSum(C)):
seq(A[n],n=1..N); # Robert Israel, Jun 03 2014

Accumulate[Table[If[SquareFreeQ[n],0,1],{n,80}]] (* Harvey P. Dale, Jun 04 2014 *)

a(n) = my(s=0); forsquarefree(k=1, sqrtint(n), s += (-1)^(#k[2]~) * (n\k[1]^2)); n - s; \\ Charles R Greathouse IV, May 18 2015; corrected by Daniel Suteu, May 11 2023

from math import isqrt
from sympy import mobius
def A057627(n): return n-sum(mobius(k)*(n//k**2) for k in range(1,isqrt(n)+1)) # Chai Wah Wu, May 10 2024

(define (A057627 n) (- n (A013928 (+ n 1))))

a(n) = n - A013928(n+1) = n - Sum_{k=1..n} mu(k)^2.

G.f.: Sum_{k>=1} (1 - mu(k)^2)*x^k/(1 - x). - Ilya Gutkovskiy, Apr 17 2017

Offset and formula corrected by Antti Karttunen, Jun 03 2014

A322068 a(n) = (1/2)Sum_{p prime <= n} floor(n/p) floor(1 + n/p).

Original entry on oeis.org

0, 0, 1, 2, 4, 5, 10, 11, 15, 18, 25, 26, 36, 37, 46, 54, 62, 63, 78, 79, 93, 103, 116, 117, 137, 142, 157, 166, 184, 185, 216, 217, 233, 247, 266, 278, 308, 309, 330, 346, 374, 375, 416, 417, 443, 467, 492, 493, 533, 540, 575, 595, 625, 626, 671, 687, 723, 745

Offset: 0

Daniel Suteu, Nov 25 2018

Partial sums of A069359.

Wikipedia, Prime Zeta Function

Cf. A000217, A000720, A013939, A013940, A069359.

with(numtheory): seq(add(i*pi(floor(n/i)), i=1..n), n=0..60); # Ridouane Oudra, Oct 16 2019

a[n_] := Module[{s=0, p=2}, While[p<=n, s += (Floor[n/p] * Floor[1 + n/p]); p=NextPrime[p]]; s]/2; Array[a, 100, 0] (* Amiram Eldar, Nov 25 2018 *)

a(n) = my(s=0); forprime(p=2, n, s+=(n\p)*(1+n\p)); s/2;

a(n) = sum(k=1, sqrtint(n), k*(k+1) * (primepi(n\k) - primepi(n\(k+1))))/2 + sum(k=1, n\(sqrtint(n)+1), if(isprime(k), (n\k)*(1+n\k), 0))/2;

a(n) ~ A085548 * n*(n+1)/2.
a(n) = Sum_{p prime <= n} A000217(floor(n/p)).
a(n) = (Sum_{k=1..floor(sqrt(n))} k*(k+1) * (pi(floor(n/k)) - pi(floor(n/(k+1)))) + Sum_{p prime <= floor(n/(1+floor(sqrt(n))))} floor(n/p)*floor(1+n/p))/2, where pi(x) is the prime-counting function (A000720).
a(n) = Sum_{i=1..n} i*pi(floor(n/i)), where pi(n) = A000720(n). - Ridouane Oudra, Oct 16 2019

A013941 a(n) = Sum_{k = 1..n} floor(n/prime(k)^3).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Henri Lifchitz

Cf. A013940.

Partial sums of A295659.

A013941:=n->add(floor(n/ithprime(k)^3), k=1..n): seq(A013941(n), n=1..150); # Wesley Ivan Hurt, Jan 29 2017

Table[Floor[Sum[n/Prime[k]^3, {k, n}]], {n, 75}] (* Alonso del Arte, Jan 29 2017 *)

a(n) = sum(k = 1, n, n\prime(k)^3); \\ Michel Marcus, Aug 24 2013

G.f.: (1/(1 - x))*Sum_{k>=1} x^(prime(k)^3)/(1 - x^(prime(k)^3)). - Ilya Gutkovskiy, Feb 11 2017

A057639 First differences of zero-sites (A028442) of Mertens's function A002321.

Original entry on oeis.org

37, 1, 18, 7, 28, 8, 44, 4, 1, 9, 1, 3, 1, 2, 48, 17, 1, 3, 1, 2, 16, 75, 2, 1, 1, 20, 2, 1, 2, 4, 1, 1, 2, 27, 8, 2, 1, 1, 2, 1, 5, 1, 5, 1, 2, 1, 1, 1, 2, 1, 109, 4, 66, 1, 27, 1, 1, 144, 4, 8, 2, 1, 2, 13, 1, 2, 9, 1, 1, 24, 1, 3, 16, 8, 6, 1, 2, 3, 4, 2, 1, 2, 5, 1, 2, 4, 3, 2, 1, 3, 1, 82, 3, 5

Offset: 1

Labos Elemer, Oct 11 2000

nonn

Mertens's function (A002321) is oscillating. The width of its waves is given here.

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

Cf. A002321, A013928, A028442, A013940, A051400, A051401, A051402.

Differences[Position[Accumulate[Array[MoebiusMu,1500]],0]//Flatten] (* Harvey P. Dale, Nov 10 2016 *)

lista(kmax) = {my(s = 0, k1 = 2); for(k2 = 3, kmax, s += moebius(k2); if(s == 0, print1(k2 - k1, ", "); k1 = k2));} \\ Amiram Eldar, Jun 09 2024

a(n) = A028442(n+1) - A028442(n).

Offset corrected by Amiram Eldar, Jun 09 2024

cp's OEIS Frontend

A056170 Number of non-unitary prime divisors of n.

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A057627 Number of nonsquarefree numbers not exceeding n.

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A322068 a(n) = (1/2)*Sum_{p prime <= n} floor(n/p) * floor(1 + n/p).

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A013941 a(n) = Sum_{k = 1..n} floor(n/prime(k)^3).

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A057639 First differences of zero-sites (A028442) of Mertens's function A002321.

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A322068 a(n) = (1/2)Sum_{p prime <= n} floor(n/p) floor(1 + n/p).