A013928 -id:A013928 - cp's OEIS frontend

A173290 Partial sums of A001615.

1, 4, 8, 14, 20, 32, 40, 52, 64, 82, 94, 118, 132, 156, 180, 204, 222, 258, 278, 314, 346, 382, 406, 454, 484, 526, 562, 610, 640, 712, 744, 792, 840, 894, 942, 1014, 1052, 1112, 1168, 1240, 1282, 1378, 1422, 1494, 1566, 1638, 1686, 1782, 1838, 1928, 2000, 2084

Offset: 1

Jonathan Vos Post, Feb 15 2010

nonn

a(n) is even for n >= 2. - Jianing Song, Nov 24 2018

W. Hürlimann, Dedekind's arithmetic function and primitive four squares counting functions, Journal of Algebra, Number Theory: Advances and Applications, Volume 14, Number 2, 2015, Pages 73-88; http://scientificadvances.co.in; DOI: http://dx.doi.org/10.18642/jantaa_7100121599

Enrique Pérez Herrero, Table of n, a(n) for n = 1..5000
W. Hürlimann, Dedekind's arithmetic function and primitive four squares counting functions, Journal of Algebra, Number Theory: Advances and Applications, Volume 14, Number 2, 2015, Pages 73-88.

Cf. A001615, A063659.
Cf. A082020.
Cf. A175836 (partial products of the Dedekind psi function).

[(&+[MoebiusMu(k)^2*Floor(n/k)*Floor(1 + n/k): k in [1..n]])/2: n in [1..60]]; // G. C. Greubel, Nov 23 2018

with(numtheory): a:=n->(1/2)*add(mobius(k)^2*floor(n/k)*floor(1+n/k),k=1..n); seq(a(n),n=1..55); # Muniru A Asiru, Nov 24 2018

Table[Sum[DirichletConvolve[j, MoebiusMu[j]^2, j, k], {k,1,n}], {n,60}] (* G. C. Greubel, Nov 23 2018 *)
psi[n_] := If[n==1, 1, n*Times@@(1 + 1/FactorInteger[n][[;;,1]])]; Accumulate[Array[psi, 50]] (* Amiram Eldar, Nov 23 2018 *)

S(n) = sum(k=1, sqrtint(n), moebius(k)*(n\(k*k))); \\ see: A013928
a(n) = sum(k=1, sqrtint(n), k*(k+1) * (S(n\k) - S(n\(k+1))))/2 + sum(k=1, n\(1+sqrtint(n)), moebius(k)^2*(n\k)*(1+n\k))/2; \\ Daniel Suteu, Nov 23 2018

def A173290(n) :
    return add(k*mul(1+1/p for p in prime_divisors(k)) for k in (1..n))
[A173290(n) for n in (1..52)]  # Peter Luschny, Jun 10 2012

a(n) = Sum_{i=1..n} A001615(i) = Sum_{i=1..n} (n * Product_{p|n, p prime} (1 + 1/p)).
a(n) = 15*n^2/(2*Pi^2) + O(n*log(n)). - Enrique Pérez Herrero, Jan 14 2012
a(n) = Sum_{i=1..n} A063659(i) * floor(n/i). - Enrique Pérez Herrero, Feb 23 2013
a(n) = (1/2)*Sum_{k=1..n} mu(k)^2 * floor(n/k) * floor(1+n/k), where mu(k) is the Moebius function. - Daniel Suteu, Nov 19 2018
a(n) = (Sum_{k=1..floor(sqrt(n))} k*(k+1) * (A013928(1+floor(n/k)) - A013928(1+floor(n/(k+1)))) + Sum_{k=1..floor(n/(1+floor(sqrt(n))))} mu(k)^2 * floor(n/k) * floor(1+n/k))/2. - Daniel Suteu, Nov 23 2018

A179211 Number of squarefree numbers between n and 2*n (inclusive).

Original entry on oeis.org

2, 2, 3, 3, 4, 4, 5, 5, 6, 7, 8, 8, 9, 8, 9, 9, 11, 11, 13, 13, 15, 15, 15, 15, 15, 16, 16, 17, 19, 19, 20, 19, 21, 21, 22, 22, 24, 23, 24, 24, 25, 25, 26, 26, 27, 28, 29, 29, 30, 30, 32, 32, 34, 34, 36, 36, 38, 38, 38, 38, 39, 39, 38, 39, 41, 41, 42, 41, 43, 43, 44, 44, 46, 45, 45

Offset: 1

Reinhard Zumkeller, Jul 05 2010

nonn

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A005117, A008966, A013928, A035250, A061398, A179212, A179213, A179214.

a := n -> nops(select(issqrfree, [$n..(2*n)])):
seq(a(n), n=1..75); # Peter Luschny, Mar 02 2017

a[n_] := Select[Range[n, 2n], SquareFreeQ] // Length;
Array[a, 75] (* Jean-François Alcover, Jun 22 2018 *)

f(n)=my(s); forfactored(k=1,sqrtint(n), s += n\k[1]^2*moebius(k)); s
a(n)=f(2*n)-f(n-1) \\ Charles R Greathouse IV, Nov 05 2017

a(n) = Sum_{k=n..2*n} A008966(k).
a(n) > A035250(n) for n>2;
A179212(n) = a(n+1) - a(n);
a(n) = A013928(2*n+1) - A013928(n).
a(n) ~ (6/Pi^2) * n. - Amiram Eldar, Mar 03 2021

A243344 a(1) = 1, a(2n) = A013929(a(n)), a(2n+1) = A005117(1+a(n)).

Original entry on oeis.org

1, 4, 2, 12, 6, 8, 3, 32, 19, 18, 10, 24, 13, 9, 5, 84, 53, 50, 31, 49, 30, 27, 15, 63, 38, 36, 21, 25, 14, 16, 7, 220, 138, 136, 86, 128, 82, 81, 51, 126, 79, 80, 47, 72, 42, 44, 23, 162, 103, 99, 62, 96, 59, 54, 34, 64, 39, 40, 22, 45, 26, 20, 11, 564, 365

Offset: 1

Antti Karttunen, Jun 03 2014

nonn

This permutation entangles complementary pair odd/even numbers (A005408/A005843) with complementary pair A005117/A013929 (numbers which are squarefree/not squarefree).

Antti Karttunen, Table of n, a(n) for n = 1..4095
Index entries for sequences that are permutations of the natural numbers

Inverse: A243343.
Cf. A005843, A005408, A008966, A005117, A013929, A013928, A057627.
Similar permutations: A088610 (simple variant), A243345-A243346, A243347, A243287-A243288, A135141-A227413, A237126-A237427, A193231.

a(1) = 1, a(2n) = A013929(a(n)), a(2n+1) = A005117(1+a(n)).

For all n, A008966(a(n)) = A000035(n), or equally, mu(a(n)) = n modulo 2, where mu is Moebius mu (A008683). [The same property holds for A088610.]

A243345 a(1)=1; thereafter, if n is k-th squarefree number [i.e., n = A005117(k)], a(n) = 2a(k-1); otherwise, when n is k-th nonsquarefree number [i.e., n = A013929(k)], a(n) = 2a(k)+1.

Original entry on oeis.org

1, 2, 4, 3, 8, 6, 16, 5, 9, 12, 32, 7, 10, 18, 24, 17, 64, 13, 14, 33, 20, 36, 48, 11, 19, 34, 25, 65, 128, 26, 28, 15, 66, 40, 72, 21, 96, 22, 38, 37, 68, 50, 130, 49, 35, 256, 52, 129, 27, 29, 56, 67, 30, 41, 132, 73, 80, 144, 42, 97, 192, 44, 23, 39, 76, 74, 136, 69, 100

Offset: 1

Antti Karttunen, Jun 03 2014

Any other fixed points than 1, 2, 6, 9, 135, 147, 914, ... ?

Any other points than 4, 21, 39, 839, 4893, 12884, ... where a(n) = n-1 ?

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences that are permutations of the natural numbers

Inverse: A243346.
Cf. A005843, A005408, A008966, A005117, A013929, A013928, A057627.
Similar permutations: A243343-A243344, A243347, A243287-A243288, A135141-A227413, A237126-A237427, A193231.

a(1) = 1, and for n>1, if mu(n) = 0, a(n) = 1 + 2*a(A057627(n)), otherwise a(n) = 2*a(A013928(n)), where mu is Moebius mu function (A008683).

For all n > 1, A000035(a(n)+1) = A008966(n) = A008683(n)^2, or equally, a(n) = mu(n) + 1 modulo 2.

A378083 Nonsquarefree numbers appearing exactly twice in A377783 (least nonsquarefree number > prime(n)).

Original entry on oeis.org

4, 8, 32, 44, 104, 140, 284, 464, 572, 620, 644, 824, 860, 1232, 1292, 1304, 1484, 1700, 1724, 1880, 2084, 2132, 2240, 2312, 2384, 2660, 2732, 2804, 3392, 3464, 3560, 3920, 3932, 4004, 4220, 4244, 4424, 4640, 4724, 5012, 5444, 5480, 5504, 5660, 6092, 6200

Offset: 1

Gus Wiseman, Nov 23 2024

nonn

Warning: do not confuse with A377783.

			The terms together with their prime indices begin:
     4: {1,1}
     8: {1,1,1}
    32: {1,1,1,1,1}
    44: {1,1,5}
   104: {1,1,1,6}
   140: {1,1,3,4}
   284: {1,1,20}
   464: {1,1,1,1,10}
   572: {1,1,5,6}
   620: {1,1,3,11}
   644: {1,1,4,9}
   824: {1,1,1,27}
   860: {1,1,3,14}
  1232: {1,1,1,1,4,5}

Subset of A377783 (union A378040, diffs A377784), restriction of A120327 (diffs A378039).
Terms appearing once are A378082.
Terms not appearing at all are A378084.
A000040 lists the primes, differences A001223, seconds A036263.
A005117 lists the squarefree numbers.
A013929 lists the nonsquarefree numbers, differences A078147.
A061398 counts squarefree numbers between primes, zeros A068360.
A061399 counts nonsquarefree numbers between primes, zeros A068361.
A071403(n) = A013928(prime(n)) counts squarefree numbers < prime(n).
A378086(n) = A057627(prime(n)) counts nonsquarefree numbers < prime(n).
Cf. A112926 (diffs A378037), opposite A112925 (diffs A378038).
Cf. A378032 (diffs A378034), restriction of A378033 (diffs A378036).
Cf. A053797, A053806, A070321, A072284, A112929, A120992, A224363, A337030, A377430, A377431, A377703.

y=Table[NestWhile[#+1&,Prime[n],SquareFreeQ[#]&],{n,1000}];
Select[Union[y],Count[y,#]==2&]

A379307 Positive integers whose prime indices include no squarefree numbers.

Original entry on oeis.org

1, 7, 19, 23, 37, 49, 53, 61, 71, 89, 97, 103, 107, 131, 133, 151, 161, 173, 193, 197, 223, 227, 229, 239, 251, 259, 263, 281, 307, 311, 337, 343, 359, 361, 371, 379, 383, 409, 419, 427, 433, 437, 457, 463, 479, 497, 503, 521, 523, 529, 541, 569, 593, 613, 623

Offset: 1

Gus Wiseman, Dec 27 2024

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The terms together with their prime indices begin:
    1: {}
    7: {4}
   19: {8}
   23: {9}
   37: {12}
   49: {4,4}
   53: {16}
   61: {18}
   71: {20}
   89: {24}
   97: {25}
  103: {27}
  107: {28}
  131: {32}
  133: {4,8}
  151: {36}
  161: {4,9}
  173: {40}

Partitions of this type are counted by A114374, strict A256012.
Positions of zero in A379306.
For a unique squarefree part we have A379316, counted by A379308 (strict A379309).
A000040 lists the primes, differences A001223.
A005117 lists the squarefree numbers, differences A076259.
A008966 is the characteristic function for the squarefree numbers.
A013929 lists the nonsquarefree numbers, differences A078147.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A061398 counts squarefree numbers between primes, zeros A068360.
A377038 gives k-th differences of squarefree numbers.
Other counts of prime indices:
- A257994 prime, see A002095, A096258, A320628, A331386, A331915, A379304, A379305.
- A257991 odd, see A000041, A000070, A066207, A349158.
- A257992 even, see A000009, A038348, A066208, A379317.
- A330944 nonprime, see A000586, A000607, A076610, A330945.
- A379300 composite, see A023895, A034891, A036497, A302540, A379301.
- A379310 nonsquarefree, see A302478.
- A379311 old prime, see A204389, A320629, A379312-A379315.
Cf. A000720, A013928, A038550, A057627, A068361, A070321, A071403, A072284, A087436, A112929, A377430, A378086.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1000],Length[Select[prix[#],SquareFreeQ]]==0&]

A379310 Number of nonsquarefree prime indices of n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Dec 27 2024

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The prime indices of 39 are {2,6}, so a(39) = 0.
The prime indices of 70 are {1,3,4}, so a(70) = 1.
The prime indices of 98 are {1,4,4}, so a(98) = 2.
The prime indices of 294 are {1,2,4,4}, a(294) = 2.
The prime indices of 1911 are {2,4,4,6}, so a(1911) = 2.
The prime indices of 2548 are {1,1,4,4,6}, so a(2548) = 2.

Positions of first appearances are A000420.
Positions of zero are A302478, counted by A073576 (strict A087188).
No squarefree parts: A379307, counted by A114374 (strict A256012).
One squarefree part: A379316, counted by A379308 (strict A379309).
A000040 lists the primes, differences A001223.
A005117 lists the squarefree numbers, differences A076259.
A008966 is the characteristic function for the squarefree numbers.
A013929 lists the nonsquarefree numbers, differences A078147.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A061398 counts squarefree numbers between primes, zeros A068360.
A377038 gives k-th differences of squarefree numbers.
Other counts of prime indices:
- A257991 odd, see A000041, A000070, A066207, A349158.
- A257992 even, see A000009, A038348, A066208, A379317.
- A257994 prime, see A002095, A096258, A320628, A331386, A331915, A379304, A379305.
- A330944 nonprime, see A000586, A000607, A076610, A330945.
- A379300 composite, see A023895, A034891, A036497, A302540, A379301.
- A379311 old prime, see A204389, A320629, A379312-A379315.
Cf. A000720, A013928, A038550, A057627, A068361, A070321, A071403, A087436, A112929, A120327, A378086.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[Select[prix[n],Not@*SquareFreeQ]],{n,100}]

Totally additive with a(prime(k)) = A107078(k) = 1 - A008966(k).

A053462 Number of positive squarefree integers less than 10^n.

Original entry on oeis.org

0, 6, 61, 608, 6083, 60794, 607926, 6079291, 60792694, 607927124, 6079270942, 60792710280, 607927102274, 6079271018294, 60792710185947, 607927101854103, 6079271018540405, 60792710185403794, 607927101854022750, 6079271018540280875, 60792710185402613302

Offset: 0

Harvey P. Dale, Aug 01 2001

nonn

			There are 608 squarefree integers smaller than 1000.

Amiram Eldar, Table of n, a(n) for n = 0..36 (from the b-file at A071172; terms 0..20 from Charles R Greathouse IV)
Gérard P. Michon, On the number of squarefree integers not exceeding N.

Cf. A005117, A013928.
Cf. A059956, A063035.
Apart from first two terms, same as A071172.
Binary counterpart is A143658. - Gerard P. Michon, Apr 30 2009

a[n_] := Module[{t=10^n-1}, Sum[MoebiusMu[k]Floor[t/k^2], {k, 1, Sqrt[t]}]]

a(n)=sum(d=1,sqrtint(n=10^n-1), n\d^2*moebius(d)) \\ Charles R Greathouse IV, Nov 14 2012

a(n)=my(s); forsquarefree(d=1,sqrtint(n=10^n-1), s += n\d[1]^2 * moebius(d)); s \\ Charles R Greathouse IV, Jan 08 2018

from math import isqrt
from sympy import mobius
def A053462(n):
    m = 10**n-1
    return sum(mobius(k)*(m//k**2) for k in range(1, isqrt(m)+1)) # Chai Wah Wu, Jun 01 2024

a(n)/10^n = (6/Pi^2)*(1+o(1)), cf. A059956.

a(n) = A071172(n) - [n <= 1] where [] is the Iverson bracket. - Chai Wah Wu, Jun 01 2024

More terms from Dean Hickerson and Vladeta Jovovic, Aug 06 2001
One more term from Jud McCranie, Sep 01 2005
a(0)=0 and a(14)-a(17) from Gerard P. Michon, Apr 30 2009
a(18)-a(20) from Charles R Greathouse IV, Jan 08 2018

A066779 Sum of squarefree numbers <= n.

Original entry on oeis.org

1, 3, 6, 6, 11, 17, 24, 24, 24, 34, 45, 45, 58, 72, 87, 87, 104, 104, 123, 123, 144, 166, 189, 189, 189, 215, 215, 215, 244, 274, 305, 305, 338, 372, 407, 407, 444, 482, 521, 521, 562, 604, 647, 647, 647, 693, 740, 740, 740, 740, 791, 791, 844, 844, 899, 899

Offset: 1

Benoit Cloitre, Jan 18 2002

D. Suryanarayana, The number and sum of k-free integers <= x which are prime to n, Indian J. Math., Vol. 11 (1969), pp. 131-139.

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A008966, A013928, A034387, A179213, A179215.

Table[ n*Boole[ SquareFreeQ[n] ], {n, 1, 56}] // Accumulate (* Jean-François Alcover, Jun 18 2013 *)

s=0; for (n=1, 1000, write("b066779.txt", n, " ", s+=moebius(n)^2*n) ) \\ Harry J. Smith, Mar 24 2010

a(n)=sum(d=1,sqrtint(n),moebius(d)*d^2*binomial(n\d^2+1,2)) \\ Charles R Greathouse IV, Apr 26 2012

a(n)=my(s,k2); forsquarefree(k=1,sqrtint(n), k2=k[1]^2; s+= k2*binomial(n\k2+1,2)*moebius(k)); s \\ Charles R Greathouse IV, Jan 08 2018

from sympy.ntheory.factor_  import core
def a(n): return sum ([i for i in range(1, n + 1) if core(i) == i]) # Indranil Ghosh, Apr 16 2017

a(n) = Sum_{i=1..n} mu(i)^2*i.
a(n) = Sum_{k=1..n} k*A008966(k). - Reinhard Zumkeller, Jul 05 2010
a(n) = Sum_{d=1..sqrt(n)} mu(d)*d^2*floor(n/d^2)*floor(n/d^2+1)/2. - Charles R Greathouse IV, Apr 26 2012
G.f.: Sum_{k>=1} mu(k)^2*k*x^k/(1 - x). - Ilya Gutkovskiy, Apr 16 2017
a(n) ~ (3/Pi^2) * n^2 + O(n^(3/2)) (Suryanarayana, 1969). - Amiram Eldar, Mar 07 2021

A179215 Product of squarefree numbers less than n+1.

Original entry on oeis.org

1, 1, 2, 6, 6, 30, 180, 1260, 1260, 1260, 12600, 138600, 138600, 1801800, 25225200, 378378000, 378378000, 6432426000, 6432426000, 122216094000, 122216094000, 2566537974000, 56463835428000, 1298668214844000, 1298668214844000, 1298668214844000, 33765373585944000

Offset: 0

Reinhard Zumkeller, Jul 05 2010

nonn

G. C. Greubel, Table of n, a(n) for n = 0..680

Cf. A013928, A066779, A005117, A008966, A179214.

Cf. A000142, A000720, A001221, A025487, A034386.

a:= proc(n) option remember; `if`(n=0, 1,
      a(n-1)*`if`(issqrfree(n), n, 1))
    end:
seq(a(n), n=0..27);  # Alois P. Heinz, Sep 20 2021

With[{sfnos=Select[Range[50],SquareFreeQ]},Table[Times@@Select[sfnos, #Harvey P. Dale, Jun 13 2011 *)

a(n) = prod(k=1, n, if (issquarefree(k), k, 1)); \\ Michel Marcus, Sep 20 2021

a(n) = my(p=1); forsquarefree(x=1, n, p*=x[1]); p; \\ Michel Marcus, Sep 20 2021

a(n) = Product_{k=1..n} k^A008966(k).
A001221(a(n)) = A000720(n).
Subsequence of A025487.
A034386(n) <= a(n) <= A000142(n).
A179214(n) = a(2*n)/a(n-1) for n>0.

Definition corrected by Harvey P. Dale, Jun 13 2011

cp's OEIS Frontend

A173290 Partial sums of A001615.

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A179211 Number of squarefree numbers between n and 2*n (inclusive).

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A243344 a(1) = 1, a(2n) = A013929(a(n)), a(2n+1) = A005117(1+a(n)).

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A243345 a(1)=1; thereafter, if n is k-th squarefree number [i.e., n = A005117(k)], a(n) = 2*a(k-1); otherwise, when n is k-th nonsquarefree number [i.e., n = A013929(k)], a(n) = 2*a(k)+1.

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A378083 Nonsquarefree numbers appearing exactly twice in A377783 (least nonsquarefree number > prime(n)).

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A379307 Positive integers whose prime indices include no squarefree numbers.

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A379310 Number of nonsquarefree prime indices of n.

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A053462 Number of positive squarefree integers less than 10^n.

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A243345 a(1)=1; thereafter, if n is k-th squarefree number [i.e., n = A005117(k)], a(n) = 2a(k-1); otherwise, when n is k-th nonsquarefree number [i.e., n = A013929(k)], a(n) = 2a(k)+1.