A059956 -id:A059956 - cp's OEIS frontend

A071172 Number of squarefree integers <= 10^n.

1, 7, 61, 608, 6083, 60794, 607926, 6079291, 60792694, 607927124, 6079270942, 60792710280, 607927102274, 6079271018294, 60792710185947, 607927101854103, 6079271018540405, 60792710185403794, 607927101854022750, 6079271018540280875, 60792710185402613302, 607927101854026645617

Offset: 0

Robert G. Wilson v, Jun 10 2002

nonn

The limit of a(n)/10^n is 6/Pi^2 (see A059956). - Gerard P. Michon, Apr 30 2009

J. Pawlewicz, Table of n, a(n) for n = 0..36
W. Hürlimann, A First Digit Theorem for Square-Free Integer Powers, Pure Mathematical Sciences, Vol. 3, 2014, no. 3, 129 - 139 HIKARI Ltd.
G. P. Michon, On the number of squarefree integers not exceeding N. - _Gerard P. Michon_, Apr 30 2009
J. Pawlewicz, Counting square-free numbers, arXiv preprint arXiv:1107.4890 [math.NT], 2011.
Eric Weisstein's World of Mathematics, Squarefree

Cf. A005117, A013928.
Apart from first two terms, same as A053462.
Binary counterpart is A143658. - Gerard P. Michon, Apr 30 2009

f[n_] := Sum[ MoebiusMu[i]Floor[n/i^2], {i, Sqrt@ n}]; Table[ f[10^n], {n, 0, 14}] (* Robert G. Wilson v, Aug 04 2012 *)

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

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

from math import isqrt
from sympy import mobius
def A071172(n): return sum(mobius(k)*(10**n//k**2) for k in range(1,isqrt(10**n)+1)) # Chai Wah Wu, May 10 2024

a(n) = Sum_{i=1..10^(n/2)} A008683(i)*floor(10^n/i^2). - Gerard P. Michon, Apr 30 2009

Extended by Eric W. Weisstein, Sep 14 2003
3 more terms from Jud McCranie, Sep 01 2005
4 more terms from Gerard P. Michon, Apr 30 2009

A280292 a(n) = sopfr(n) - sopf(n).

Original entry on oeis.org

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

Offset: 1

Michel Marcus, Dec 31 2016

nonn

Alladi and Erdős (1977) proved that for all numbers m>=0, m!=1, the sequence of numbers k such that a(k) = m has a positive asymptotic density which is equal to a rational multiple of 1/zeta(2) = 6/Pi^2 (A059956). For example, when m=0, the sequence is the squarefree numbers (A005117), whose density is 6/Pi^2, and when m=2 the sequence is A081770, whose density is 1/Pi^2. - Amiram Eldar, Nov 02 2020

Sum of prime factors minus sum of distinct prime factors. Counting partitions by this statistic (sum minus sum of distinct parts) gives A364916. - Gus Wiseman, Feb 21 2025

Jean-Marie De Koninck and Aleksandar Ivić, Topics in Arithmetical Functions: Asymptotic Formulae for Sums of Reciprocals of Arithmetical Functions and Related Fields, Amsterdam, Netherlands: North-Holland, 1980. See pp. 164-166.
Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 165.

Antti Karttunen, Table of n, a(n) for n = 1..20000
Krishnaswami Alladi and Paul Erdős, On an additive arithmetic function, Pacific Journal of Mathematics, Vol. 71, No. 2 (1977), pp. 275-294, alternative link.
Jean-Marie De Koninck, Paul Erdős and Aleksandar Ivić, Reciprocals of certain large additive functions, Canadian Mathematical Bulletin, Vol. 24, No. 2 (1981), pp. 225-231.
Aleksandar Ivić, On certain large additive functions, arXiv:math/0311505 [math.NT], 2003.

Cf. A001414 (sopfr), A008472 (sopf), A057521, A059956, A081770, A280163, A338559, A338560.
A multiplicative version is A003557, firsts A064549 (sorted A001694).
For length instead of sum we have A046660.
For product instead of sum we have A066503, firsts A381076.
Positions of first appearances are A280286 (sorted A381075).
For indices instead of factors we have A380955, firsts A380956 (sorted A380957).
For exponents instead of factors we have A380958, firsts A380989.
A000040 lists the primes, differences A001223.
A001222 counts prime factors (distinct A001221).
A003963 gives product of prime indices, distinct A156061, excess A380986.
A005117 lists squarefree numbers, complement A013929.
A007947 gives squarefree kernel.
A020639 gives least prime factor (index A055396), greatest A061395 (index A006530).
A027746 lists prime factors, distinct A027748.
A112798 lists prime indices (sum A056239), distinct A304038 (sum A066328).
Cf. A071625, A075255, A116861, A136565, A175508, A290106, A364916, A381077.

Array[Total@ # - Total@ Union@ # &@ Flatten[ConstantArray[#1, #2] & @@@ FactorInteger@ #] &, 105] (* Michael De Vlieger, Feb 25 2019 *)

sopfr(n) = my(f=factor(n)); sum(j=1, #f~, f[j, 1]*f[j, 2]);
sopf(n) = my(f=factor(n)); sum(j=1, #f~, f[j, 1]);
a(n) = sopfr(n) - sopf(n);

a(n) = A001414(n) - A008472(n).
a(A005117(n)) = 0.
a(n) = A001414(A003557(n)). - Antti Karttunen, Oct 07 2017
Additive with a(1) = 0 and a(p^e) = p*(e-1) for prime p and e > 0. - Werner Schulte, Feb 24 2019
From Amiram Eldar, Nov 02 2020: (Start)
a(n) = a(A057521(n)).
Sum_{n<=x} a(n) ~ x*log(log(x)) + O(x) (Alladi and Erdős, 1977).
Sum_{n<=x, n nonsquarefree} 1/a(n) ~ c*x + O(sqrt(x)*log(x)), where c = Integral_{t=0..1} (F(t)-6/Pi^2)/t dt, and F(t) = Product_{p prime} (1-1/p)*(1-1/(t^p - p)) (De Koninck et al., 1981; Finch, 2018), or, equivalently c = Sum_{k>=2} d(k)/k = 0.1039..., where d(k) = (6/Pi^2)*A338559(k)/A338560(k) is the asymptotic density of the numbers m with a(m) = k (Alladi and Erdős, 1977; Ivić, 2003). (End)

More terms from Antti Karttunen, Oct 07 2017

A132696 Decimal expansion of 6/Pi.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Aug 26 2007, Nov 02 2007

6/Pi = Volume of the cuboid (If L1>L2>L3) / Volume of the inscribed ellipsoid.
6/Pi = Volume of the cuboid (If L1>(L2=L3)) / Volume of the inscribed spheroid.
6/Pi = Volume of the regular hexahedron (or cube) / Volume of the inscribed Sphere.
6/Pi = 1 / Arc of 30 degrees.
6/Pi = Volume of the cuboid (If L1<(L2=L3)) / Volume of the inscribed spheroid.
6/Pi = Surface area of the regular hexahedron (or cube) / surface area of the inscribed sphere.

			1.90985931710274402922660516047... .

Index entries for transcendental numbers

Cf. A049541, A060294, A089491, A088538, A086201, A059956.

RealDigits[N[6/Pi, 200]] (* Erich Friedman, Mar 22 2008 *)

6/Pi \\ Charles R Greathouse IV, Dec 31 2011

Equals Product_{k>=1} (2k+1)^3 / ( (2k)^2*(2k+3) ). - Federico Provvedi, Nov 09 2024

More terms from Erich Friedman, Mar 22 2008

A070258 Smallest of 3 consecutive numbers each divisible by a square.

Original entry on oeis.org

48, 98, 124, 242, 243, 342, 350, 423, 475, 548, 603, 724, 774, 844, 845, 846, 1024, 1250, 1274, 1323, 1375, 1420, 1448, 1519, 1664, 1674, 1680, 1681, 1682, 1848, 1862, 1924, 2007, 2023, 2056, 2106, 2150, 2223, 2275, 2348, 2366, 2523, 2527, 2574, 2644

Offset: 1

Sharon Sela (sharonsela(AT)hotmail.com), May 09 2002

nonn

The sequence includes an infinite family of arithmetic progressions. Such AP's can be constructed to each term, with large differences [like e.g. square of primorials, A061742]. It is necessary to solve suitable systems of linear Diophantine equations. E.g.: subsequences of triples of terms = {900a+548, 900a+549, 900a+550} = {4(225f+137), 9(100f+61), 25(36f+22)}; starting terms in this sequence = {548, 1448, 2348, ...}; difference = A002110(3)^2. - Labos Elemer, Nov 25 2002
The numbers of terms not exceeding 10^k, for k = 1, 2, ..., are 0, 2, 16, 180, 1868, 18649, 186335, 1863390, 18634236, 186340191, ... . Apparently, the asymptotic density of this sequence exists and equals 0.01863... . - Amiram Eldar, Jan 18 2023
The asymptotic density of this sequence is 1 - 3/zeta(2) + 3 * Product_{p prime} (1 - 2/p^2) - Product_{p prime} (1 - 3/p^2) = 1 - 3 * A059956 + 3 * A065474 - A206256 = 0.018634010349844827414... . - Amiram Eldar, Sep 12 2024

Jean-Marie De Koninck, Ces nombres qui nous fascinent, Entry 48, p. 18, Ellipses, Paris, 2008.

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

Subsequence of A013929 and A068781.
Cf. A002110, A061742, A235578.
Cf. A059956, A065474, A206256.

f[n_] := Union[ Transpose[ FactorInteger[n]] [[2]]] [[ -1]]; a = 0; b = 1; Do[c = f[n]; If[a> 1 && b > 1 && c > 1, Print[n - 2]]; a = b; b = c, {n, 3, 10^6}]
Flatten[Position[Partition[SquareFreeQ/@Range[3000],3,1],?(Union[#] == {False}&),{1},Heads->False]] (* _Harvey P. Dale, May 24 2014 *)
f@n_ := Flatten@  Position[Partition[SquareFreeQ /@ Range@2000, n, 1], Table[False, {n}]]; f@3 (* Hans Rudolf Widmer, Aug 30 2022 *)

a(n) = A235578(n) - 1. - Amiram Eldar, Feb 09 2021

More terms from Jason Earls and Robert G. Wilson v, May 10 2002

Offset corrected by Amiram Eldar, Feb 09 2021

A206256 Decimal expansion of Product_{p prime} (1 - 3/p^2).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Feb 05 2012, based on a posting by Warren Smith to the Math Fun Mailing List, Feb 04 2012

For a randomly selected number k, this is the probability that k, k+1, k+2 all are squarefree.

			0.1254869809058...

Leon Mirsky, Note on an asymptotic formula connected with r-free integers, The Quarterly Journal of Mathematics, Vol. os-18, No. 1 (1947), pp. 178-182.
Leon Mirsky, Arithmetical pattern problems relating to divisibility by rth powers, Proceedings of the London Mathematical Society, Vol. s2-50, No. 1 (1949), pp. 497-508.

Cf. A059956, A065474, A007675, A335131, A370600.

# See A175640 using efact := 1-3/p^2. - R. J. Mathar, Mar 22 2012

$MaxExtraPrecision = 500; m = 500; c = LinearRecurrence[{0, 3}, {0, -6}, m]; RealDigits[(1/4) * Exp[NSum[Indexed[c, n]*(PrimeZetaP[n] - 1/2^n)/n, {n, 2, m}, NSumTerms -> m, WorkingPrecision -> m]], 10, 100][[1]] (* Amiram Eldar, Oct 01 2019 *)

prodeulerrat(1 - 3/p^2) \\ Amiram Eldar, Mar 16 2021

More terms from Amiram Eldar, Oct 01 2019

More terms from Vaclav Kotesovec, Dec 17 2019

A327837 Decimal expansion of the asymptotic mean of the number of exponential divisors function (A049419).

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Sep 27 2019

			1.602317102305418052349626315621161003776939495785572...

Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 52 (constant Z3).
V. Sita Ramaiah and D. Suryanarayana, Sums of reciprocals of some multiplicative functions - II, Indian J. Pure Appl. Math., Vol. 11 (1980), pp. 1334-1355 (eq. 2.37 and 3.18, pp. 1346 and 1354).
Abdelhakim Smati and Jie Wu, On the exponential divisor function, Publications de l'Institut Mathématique, Vol. 61 (1997), pp. 21-32.
László Tóth, An order result for the exponential divisor function, Publ. Math. Debrecen, Vol. 71, No. 1-2 (2007), pp. 165-171, arXiv preprint,, arXiv:0708.3552 [math.NT], 2007.
László Tóth, Alternating sums concerning multiplicative arithmetic functions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1 (section 4.10, p. 30).
Jie Wu, Problème de diviseurs exponentiels et entiers exponentiellement sans facteur carré, Journal de théorie des nombres de Bordeaux, Vol. 7, No. 1, (1995), pp. 133-141.

Cf. A000005, A047994, A049419, A145353, A361013, A361967, A379517, A379518.

Cf. A059956 (constant for unitary divisors), A306071 (bi-unitary), A327576 (infinitary).

$MaxExtraPrecision = 1500; m = 1500; em = 500; f[x_] := 1 + Log[1 + Sum[x^e * (DivisorSigma[0, e] - DivisorSigma[0, e - 1]), {e, 2, em}]]; c = Rest[ CoefficientList[Series[f[x], {x, 0, m}], x] * Range[0, m] ]; RealDigits[ Exp[NSum[Indexed[c, k] * PrimeZetaP[k]/k, {k, 2, m}, NSumTerms -> m, WorkingPrecision -> m]], 10, 100][[1]]

Equals lim_{k->oo} A145353(k)/k.
Equals Product_{p prime} (1 + Sum_{e >= 2} p^(-e) * (d(e) - d(e-1))), where d(e) is the number of divisors of e (A000005).
Equals Product_{p prime} (1 - 1/p) * (2 - (log(p-1) + QPolyGamma(0, 1, 1/p)) / log(p)). - Vaclav Kotesovec, Feb 27 2023
From Amiram Eldar, Dec 24 2024: (Start)
Equals lim_{m->oo} (1/m) * Sum_{k=1..m} k/uphi(k) = lim_{m->oo} (1/m) * Sum_{k=1..m} A319677(k)/A319676(k), where uphi(k) is the unitary totient function (A047994).
Equals lim_{m->oo} (1/log(m)) * Sum_{k=1..m} 1/uphi(k) = lim_{m->oo} (1/log(m)) * A379517(m)/A379518(m).
Equals lim_{m->oo} (1/m) * Sum_{k=1..m} A361967(k).
Equals Product_{p prime} ((1-1/p) * (1 + Sum_{k>=1} 1/(p^k-1))).
Equals Product_{p prime} (1 + (1-1/p) * Sum_{k>=1} 1/(p^k*(p^k-1))). (End)

More digits from Vaclav Kotesovec, Jun 13 2021

A094387 Numbers k such that gcd(k, A000120(k)) = 1.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 14, 15, 16, 17, 19, 22, 23, 25, 26, 27, 28, 29, 31, 32, 33, 35, 37, 38, 39, 41, 43, 44, 45, 47, 49, 50, 51, 52, 53, 56, 57, 59, 61, 62, 64, 65, 67, 70, 71, 73, 74, 75, 76, 77, 79, 82, 83, 85, 87, 88, 89, 91, 93, 94, 95, 97, 98, 99, 100, 101, 103

Offset: 1

Benoit Cloitre, Jun 08 2004

This sequence has density 6/Pi^2 (Olivier, 1975).

All primes, powers of 2, and powers of two plus one are terms of this sequence. - William Boyles, Jan 27 2022

Jean-Paul Allouche and Jeffrey Shallit, Automatic Sequences, Cambridge University Press, 2003, p. 117.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Christian Mauduit, Carl Pomerance and András Sárközy, On the distribution in residue classes of integers with a fixed sum of digits, The Ramanujan Journal, Vol. 9, No. 1-2 (2005), pp. 45-62; alternative link.
Michel Olivier, Sur la probabilité que n soit premier à la somme de ses chiffres, C. R. Math. Acad. Sci. Paris, Série A, Vol. 280 (1975), pp. 543-545.
Michel Olivier, Fonctions g-additives et formule asymptotique pour la propriété (n, f(n)) = q, Acta Arithmetica, Vol. 31, No. 4 (1976), pp. 361-384; alternative link.

Cf. A000120, A059956.

Select[Range[100], CoprimeQ[#, DigitCount[#, 2, 1]] &] (* Amiram Eldar, Nov 22 2020 *)

lista(nn) = {for (n=1, nn, if (gcd(n, norml2(binary(n))) == 1, print1(n, ", ")););} \\ Michel Marcus, May 25 2013

from math import gcd
def ok(n): return gcd(n, bin(n).count('1')) == 1
print([k for k in range(104) if ok(k)]) # Michael S. Branicky, Jan 25 2022

A271971 Decimal expansion of (6/Pi^2) Sum_{p prime} 1/(p(p+1)), a Meissel-Mertens constant related to the asymptotic density of certain sequences of integers.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Apr 17 2016

This is the density of A060687, the numbers with one 2 and the rest 1s in the exponents of its prime factorization. - Charles R Greathouse IV, Aug 03 2016

			0.200755722019265986996250723114404765853535555352561916...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.2 Meissel-Mertens Constants, p. 95.

Cf. A059956, A060687, A085548, A136141, A179119, A222056.

digits = 100; S = (6/Pi^2)*NSum[(-1)^n PrimeZetaP[n], {n, 2, Infinity}, Method -> "AlternatingSigns", WorkingPrecision -> digits+5]; RealDigits[ S, 10, digits] // First

eps()=2.>>bitprecision(1.)
primezeta(s)=my(t=s*log(2)); sum(k=1, lambertw(t/eps())\t, moebius(k)/k*log(abs(zeta(k*s))))
sumalt(k=2, (-1)^k*primezeta(k))*6/Pi^2 \\ Charles R Greathouse IV, Aug 03 2016

sumeulerrat(1/(p*(p+1)))/zeta(2) \\ Amiram Eldar, Mar 18 2021

Equals (6/Pi^2)*A179119.

A195089 Numbers k such that (number of prime factors of k counted with multiplicity) less (number of distinct prime factors of k) = 5.

Original entry on oeis.org

64, 192, 288, 320, 432, 448, 648, 704, 729, 800, 832, 960, 972, 1088, 1216, 1344, 1440, 1458, 1472, 1568, 1856, 1984, 2000, 2016, 2112, 2160, 2240, 2368, 2400, 2496, 2624, 2752, 3008, 3024, 3168, 3240, 3264, 3392, 3520, 3600, 3645, 3648, 3744, 3776, 3872, 3904

Offset: 1

Harvey P. Dale, Sep 08 2011

The asymptotic density of this sequence is (6/Pi^2) * Sum_{k>=1} f(a(k)) = 0.0118439..., where f(k) = A112526(k) * Product_{p|k} p/(p+1). - Amiram Eldar, Sep 24 2024

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

Cf. A060687, A195069, A195086, A195087, A195088, A195090, A195091, A195092, A195093.

Cf. A025487, A046660, A059956, A112526, A257851, A261256, A264959.

a195089 n = a195089_list !! (n-1)
a195089_list = filter ((== 5) . a046660) [1..]
-- Reinhard Zumkeller, Nov 29 2015

Select[Range[4000],PrimeOmega[#]-PrimeNu[#]==5&]

is(n)=bigomega(n)-omega(n)==5 \\ Charles R Greathouse IV, Sep 14 2015

A046660(a(n)) = 5. - Reinhard Zumkeller, Nov 29 2015

A195091 Numbers k such that (number of prime factors of k counted with multiplicity) less (number of distinct prime factors of k) = 7.

Original entry on oeis.org

256, 768, 1152, 1280, 1728, 1792, 2592, 2816, 3200, 3328, 3840, 3888, 4352, 4864, 5376, 5760, 5832, 5888, 6272, 6561, 7424, 7936, 8000, 8064, 8448, 8640, 8748, 8960, 9472, 9600, 9984, 10496, 11008, 12032, 12096, 12672, 12960, 13056, 13122, 13568

Offset: 1

Harvey P. Dale, Sep 08 2011

The asymptotic density of this sequence is (6/Pi^2) * Sum_{k>=1} f(a(k)) = 0.0029589..., where f(k) = A112526(k) * Product_{p|k} p/(p+1). - Amiram Eldar, Sep 24 2024

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

Cf. A060687, A195069, A195086, A195087, A195088, A195089, A195090, A195092, A195093.

Cf. A025487, A046660, A059956, A112526, A257851, A261256, A264959.

a195091 n = a195091_list !! (n-1)
a195091_list = filter ((== 7) . a046660) [1..]
-- Reinhard Zumkeller, Nov 29 2015

Select[Range[14000],PrimeOmega[#]-PrimeNu[#]==7&]

is(n)=bigomega(n)-omega(n)==7 \\ Charles R Greathouse IV, Sep 14 2015

A046660(a(n)) = 7. - Reinhard Zumkeller, Nov 29 2015

cp's OEIS Frontend

A071172 Number of squarefree integers <= 10^n.

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A280292 a(n) = sopfr(n) - sopf(n).

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A132696 Decimal expansion of 6/Pi.

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A070258 Smallest of 3 consecutive numbers each divisible by a square.

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A206256 Decimal expansion of Product_{p prime} (1 - 3/p^2).

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A327837 Decimal expansion of the asymptotic mean of the number of exponential divisors function (A049419).

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Extensions

A094387 Numbers k such that gcd(k, A000120(k)) = 1.

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