A064610 -id:A064610 - cp's OEIS frontend

A064608 Partial sums of A034444: sum of number of unitary divisors from 1 to n.

1, 3, 5, 7, 9, 13, 15, 17, 19, 23, 25, 29, 31, 35, 39, 41, 43, 47, 49, 53, 57, 61, 63, 67, 69, 73, 75, 79, 81, 89, 91, 93, 97, 101, 105, 109, 111, 115, 119, 123, 125, 133, 135, 139, 143, 147, 149, 153, 155, 159, 163, 167, 169, 173, 177, 181, 185, 189, 191, 199, 201

Offset: 1

Labos Elemer, Sep 24 2001

a(n) = Sum_{k<=n} 2^omega(k) where omega(k) is the number of distinct primes in the factorization of k. - Benoit Cloitre, Apr 16 2002

a(n) is the number of (p, q) lattice points that are visible from (0, 0), where p and q satisfy: p >= 1, q >= 1, p * q <= n. - Luc Rousseau, Jul 09 2017

E. Landau, Handbuch der Lehre von der Verteilung der Primzahlen, Leipzig 1909 (Chelsea reprint 1953), p. 594.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harry J. Smith)
Masum Billal, Number of Ways To Write as Product of Co-prime Numbers, arXiv:1909.07823 [math.GM], 2019.
E. Cohen, The number of unitary divisors of an integer, Am. Math. Mon. 67, 879-880 (1960).
E. Landau, Handbuch der Lehre von der Verteilung der Primzahlen, vol. 2, Leipzig, Berlin, B. G. Teubner, 1909.
E. Landau, Handbuch der Lehre von der Verteilung der Primzahlen, vol. 1 and vol. 2, Leipzig, Berlin, B. G. Teubner, 1909.
F. Mertens, Uber einige asymptotische Gesetze der Zahlentheorie, J. Reine Angew. Math., 77 (1874), 289-338.
V. Sitaramaiah and M.V. Subbarao, Unitary divisor problem for arithmetic progressions, Annales Univ. Sci. Budapest., Sect. Comp. 32 (2010) 73-89.
D. Suryanarayana and V. Siva Rama Prasad, The number of k-free divisors of an integer, Acta Arithmetica XVII (1971), 345-354.
D. Zhang and W. Zhai, Mean Values of a Gcd-Sum Function Over Regular Integers Modulo n, J. Int. Seq. 13 (2010), 10.4.7. Eq (8) for asymptotics.

Cf. A001620, A001221, A034444, A064610, A069212.

with(numtheory): A064608:=n->add(mobius(k)^2*floor(n/k), k=1..n): seq(A064608(n), n=1..100); # Wesley Ivan Hurt, Dec 05 2015

a[n_] := Count[Divisors@ n, d_ /; GCD[d, n/d] == 1]; Accumulate@ Array[a, {61}] (* Michael De Vlieger, Oct 21 2015, after Jean-François Alcover at A034444 *)
Accumulate@ Array[2^PrimeNu[#] &, {61}] (* Amiram Eldar, Oct 21 2019 *)

{ for (n=1, 80, a=sum(k=1, n, moebius(k)^2*floor(n/k)); write("b064608.txt", n, " ", a) ) } \\ Harry J. Smith, Sep 20 2009

a(n)=sum(k=1,sqrtint(n),moebius(k)*(2*sum(l=1,sqrtint(n\(k*k)),n\(k*k*l))-sqrtint(n\(k*k))^2)); \\ More efficient formula for large n values (up to 10^14)
vector(80,i,a(i)) \\ Jerome Raulin, Nov 01 2015

from sympy.ntheory.factor_ import primenu
def A064608(n): return sum(1<Chai Wah Wu, Sep 07 2023

a(n) = a(n-1) + A034444(n) = a(n-1) + 2^A001221(n) Sum_{j=1..n} ud(j) where ud(j) = A034444(j) = 2^A001221(n).
a(n) = n*log(n)/zeta(2) + O(n) where zeta(2) = Pi^2/6. - Benoit Cloitre, Apr 16 2002
a(n) = Sum_{k=1..n} mu(k)^2*floor(n/k). - Benoit Cloitre, Apr 16 2002
Mertens's theorem (1874): a(n) = Sum_{k<=n} ud(k) = (n/Zeta(2))*(log(n) + 2*gamma - 1 - 2*Zeta'(2)/Zeta(2)) + O(sqrt(n)*log(n)), where gamma is the Euler-Mascheroni constant A001620. - Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Sep 07 2002
G.f.: (1/(1 - x))*Sum_{k>=1} mu(k)^2*x^k/(1 - x^k). - Ilya Gutkovskiy, Jan 03 2017

A064605 Numbers k such that A064602(k) is divisible by k.

Original entry on oeis.org

1, 2, 8, 74, 146, 150, 158, 307, 526, 541, 16157, 20289, 271343, 953614, 1002122, 2233204, 3015123, 15988923, 48033767, 85110518238

Offset: 1

Labos Elemer, Sep 24 2001

Analogous sequences for various arithmetical functions are A050226, A056650, A064605, A064606, A064607, A064610, A064611, A048290, A062982, A045345.
a(20) > 3*10^10. - Donovan Johnson, Aug 31 2012
a(21) > 10^11, if it exists. - Amiram Eldar, Jan 18 2024

			Summing divisor-square sums for j = 1..8 gives 1+5+10+21+26+50+50+85 = 248, which is divisible by 8, so 8 is a term and the integer quotient is 31.

Cf. A001157, A064602, A050226, A056650, A064606, A064607, A064610, A064611, A064612, A048290, A062982, A045345.

k = 1; lst = {}; s = 0; While[k < 1000000001, s = s + DivisorSigma[2, k]; If[ Mod[s, k] == 0, AppendTo[lst, k]; Print@ k]; k++]; lst (* Robert G. Wilson v, Apr 25 2011 *)

(Sum_{j=1..k} sigma_2(j)) mod k = A064602(k) mod k = 0.

a(15)-a(19) from Donovan Johnson, Jun 21 2010

a(20) from Amiram Eldar, Jan 18 2024

A064611 Partial sum of usigma is divisible by n, where usigma(n) = A034448(n) and summatory-usigma(n) = A064609(n).

Original entry on oeis.org

1, 2, 8, 11, 12, 174, 212, 524, 1567, 14096, 19795, 38466, 42114, 55575, 338809, 498001, 1175281, 2424880, 3994532, 7908519, 48453784, 696840720, 5497869355, 7479239685

Offset: 1

Labos Elemer, Sep 24 2001

Analogous sequences for various arithmetical functions are A050226, A056550, A064605-A064607, A064610, A064612, A048290, A062982, A045345.

			udivisor sums[=usigma(j) values] from 1 to 8 are added: 1+3+4+5+6+12+8+9=48; it is divisible by 8, thus 8 is here.

Amiram Eldar, Table of n, a(n), A064609(a(n))/a(n) for n=1..24.

Cf. A034448, A064609, A050226, A056550, A064605, A064606, A064607, A064610, A064612, A048290, A062982, A045345.

s = Table[DivisorSum[n, # &, CoprimeQ[#, n/#] &], {n, 10^6}]; Module[{a = First@ s, b = {First@ s}}, Do[a += s[[i]]; If[Divisible[a, i], AppendTo[b, i]], {i, 2, Length@ s}]; b] (* Michael De Vlieger, Mar 18 2017 *)

A064609(n) mod n = 0.

a(17)-a(22) from Donovan Johnson, Jul 20 2012

a(23)-a(24) from Amiram Eldar, Mar 17 2019

A064607 Numbers k such that A064604(k) is divisible by k.

Original entry on oeis.org

1, 2, 7, 151, 257, 1823, 3048, 5588, 6875, 7201, 8973, 24099, 5249801, 9177919, 18926164, 70079434, 78647747, 705686794, 2530414370, 3557744074, 25364328389, 32487653727, 66843959963

Offset: 1

Labos Elemer, Sep 24 2001

Analogous sequences for various arithmetical functions are A050226, A056650, A064605-A064607, A064610, A064611, A048290, A062982, A045345.
a(19) > 2*10^9. - Donovan Johnson, Jun 21 2010
a(24) > 10^11, if it exists. - Amiram Eldar, Jan 18 2024

			Adding 4th-power divisor-sums for j = 1..7 gives 1+17+82+273+626+1394+2402 = 4795 which is divisible by 7, so 7 is a term and the integer quotient is 655.

Cf. A001159, A064604, A050226, A056650, A064605, A064606, A064610-A064612, A048290, A062982, A045345.

k = 1; lst = {}; s = 0; While[k < 1000000001, s = s + DivisorSigma[4, k]; If[ Mod[s, k] == 0, AppendTo[lst, k]; Print@ k]; k++]; lst (* Robert G.Wilson v, Aug 25 2011 *)

(Sum_{j=1..k} sigma_4(j)) mod k = A064604(k) mod k = 0.

a(13)-a(18) from Donovan Johnson, Jun 21 2010

a(19)-a(23) from Amiram Eldar, Jan 18 2024

A064612 Partial sum of bigomega is divisible by n, where bigomega(n)=A001222(n) and summatory-bigomega(n)=A022559(n).

Original entry on oeis.org

1, 4, 5, 2178, 416417176, 416417184, 416417185, 416417186, 416417194, 416417204, 416417206, 416417208, 416417213, 416417214, 416417231, 416417271, 416417318, 416417319, 416417326, 416417335, 416417336, 416417338, 416417339, 416417374

Offset: 1

Labos Elemer, Sep 24 2001

nonn

Analogous sequences for various arithmetical functions are A050226, A056650, A064605-A064607, A064610, A064611, A048290, A062982, A045345.
Partial sums of A001222, similarly to summatory A001221 increases like loglog(n), explaining small quotients.
a(25) > 10^13. - Giovanni Resta, Apr 25 2017

			Sum of bigomega values from 1 to 5 is: 0+0+1+1+2+1=5, which is divisible by n=5, so 5 is here, with quotient=1. For the last value,2178,below 1000000 the quotient is only 3.

A001222, A022559, A050226, A056650, A064602-A064611, A048290, A062982, A045345.

Mod[A022559(n), n]=0

a(5)-a(24) from Donovan Johnson, Nov 15 2009

A064606 Numbers k such that A064603(k) is divisible by k.

Original entry on oeis.org

1, 2, 7, 45, 184, 210, 267, 732, 1282, 3487, 98374, 137620, 159597, 645174, 3949726, 7867343, 13215333, 14153570, 14262845, 317186286, 337222295, 2788845412, 10937683400, 72836157215, 95250594634

Offset: 1

Labos Elemer, Sep 24 2001

Analogous sequences for various arithmetical functions are A050226, A056650, A064605-A064607, A064610, A064611, A048290, A062982, A045345.
a(22) > 2*10^9. - Donovan Johnson, Jun 21 2010
a(26) > 10^11, if it exists. - Amiram Eldar, Jan 18 2024

			Adding divisor-cube sums for j = 1..7 gives 1+9+28+73+126+252+344 = 833 = 7*119, which is divisible by 7, so 7 is a term and the integer quotient is 119.

Cf. A001158, A064603, A050226, A056650, A064605, A064607, A064610-A064612, A048290, A062982, A045345.

A064603 = Accumulate[Table[DivisorSigma[3, k], {k, 1, 1000000}]]; Select[Range[Length[A064603]], Divisible[A064603[[#]], #] &] (* Vaclav Kotesovec, Jul 11 2021 *)

(Sum_{j=1..k} sigma_3(j)) mod k = A064603(k) mod k = 0.

a(15)-a(21) from Donovan Johnson, Jun 21 2010

a(22)-a(25) from Amiram Eldar, Jan 18 2024

A344731 Numbers k such that k divides A306069(k).

Original entry on oeis.org

1, 275, 277, 3337, 3353, 3359, 39675, 39689, 472467, 797806459, 9501109507

Offset: 1

Amiram Eldar, May 27 2021

The corresponding quotients A306069(k)/k are 1, 5, 5, 7, 7, 7, 9, 9, 11, 17, 19, ...

a(12) > 7.5*10^10, if it exists.

			a(1) = 1 since A306069(1) = 1 is divisible by 1.
a(2) = 275 since A306069(275) = 1375 = 5 * 275 is divisible by 275.

Cf. A286324, A306069.
The bi-unitary version of A050226.
Similar sequences: A064610, A344732, A344733.

f[p_, e_] := If[Mod[e, 2] == 1, (e + 1), e]; s[1] = 1; s[n_] := s[n] = s[n - 1] + Times @@ f @@@ FactorInteger[n]; Select[Range[40000], Divisible[s[#], #] &]

A344732 Numbers k such that k divides Sum_{j=1..k} A048105(j).

Original entry on oeis.org

1, 2, 3, 54, 58, 62, 71, 10535, 10541, 10579, 135242, 135243, 1733777, 1733781, 1733895, 1733905, 1733999, 22216757, 22216765, 22216790, 22216808, 22216814, 46745561148, 46745561156

Offset: 1

Amiram Eldar, May 27 2021

The corresponding quotients Sum_{j=1..k} A048105(j)/k are 0, 0, 0, 1, 1, 1, 1, 3, 3, 3, 4, 4, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 9, 9, ...

a(25) > 7.5*10^10, if it exists.

			a(1) = 1 since A048105(1) = 0 is divisible by 1.
a(4) = 54 since Sum_{j=1..54} A048105(j) = 54 is divisible by 54.

Cf. A048105.
The non-unitary version of A050226.
Similar sequences: A064610, A344731, A344733.

s[1] = 0; s[n_] := s[n] = s[n - 1] + DivisorSigma[0, n] - 2^PrimeNu[n]; Select[Range[140000], Divisible[s[#], #] &]

A344733 Numbers k such that k divides A327573(k).

Original entry on oeis.org

1, 25, 387, 6063, 1416379, 1416403, 1416411, 331362359, 5068450527

Offset: 1

Amiram Eldar, May 27 2021

The corresponding quotients A327573(k)/k are 1, 3, 5, 7, 11, 11, 11, 15, 17, ...

a(10) > 7.5*10^10, if it exists.

			a(1) = 1 since A327573(1) = 1 is divisible by 1.
a(2) = 25 since A327573(25) = 75 = 3 * 25 is divisible by 25.

Cf. A037445, A327573.
The infinitary version of A050226.
Similar sequences: A064610, A344731, A344732.

f[p_, e_] := 2^DigitCount[e, 2, 1]; s[1] = 1; s[n_] := s[n] = s[n - 1] + Times @@ f @@@ FactorInteger[n]; Select[Range[1.5*10^6], Divisible[s[#], #] &]

A355544 Numbers k such that the arithmetic mean of the first k squarefree numbers is an integer.

Original entry on oeis.org

1, 3, 6, 37, 75, 668, 1075, 37732, 742767, 1811865, 3140083, 8937770, 108268896, 282951249, 633932500, 1275584757, 60455590365

Offset: 1

Amiram Eldar, Jul 06 2022

Numbers k such that A173143(k) is divisible by k.

The corresponding quotients A173143(k)/k are 1, 2, 4, 29, ..., and the corresponding values of A005117(k) are 1, 3, 7, 59, ... (see the link for more values).

			3 is a term since the arithmetic mean of the first 3 squarefree numbers, (1+2+3)/3 = 2, is an integer.

Amiram Eldar, Table of n, a(n), A173143(a(n))/a(n), A005117(a(n)) for n = 1..16.

Cf. A005117, A173143.

Similar sequences: A045345, A050226, A056550, A064605, A064606, A064607, A064610, A064611, A064612, A048290.

s = Select[Range[10^6], SquareFreeQ]; r = Accumulate[s]/Range[Length[s]]; ind = Position[r, _?IntegerQ] // Flatten

upto(n) = my(s=0,k=0); forsquarefree(m=1, n, s+=m[1]; k+=1; if(s%k == 0, print1(k, ", "))); \\ Daniel Suteu, Jul 06 2022

a(17) from Daniel Suteu, Jul 06 2022

cp's OEIS Frontend

A064608 Partial sums of A034444: sum of number of unitary divisors from 1 to n.

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A064605 Numbers k such that A064602(k) is divisible by k.

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A064611 Partial sum of usigma is divisible by n, where usigma(n) = A034448(n) and summatory-usigma(n) = A064609(n).

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A064607 Numbers k such that A064604(k) is divisible by k.

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A064612 Partial sum of bigomega is divisible by n, where bigomega(n)=A001222(n) and summatory-bigomega(n)=A022559(n).

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A064606 Numbers k such that A064603(k) is divisible by k.

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A344731 Numbers k such that k divides A306069(k).

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A344732 Numbers k such that k divides Sum_{j=1..k} A048105(j).