A074789 -id:A074789 - cp's OEIS frontend

A061502 a(n) = Sum_{k<=n} tau(k)^2, where tau = number of divisors function A000005.

1, 5, 9, 18, 22, 38, 42, 58, 67, 83, 87, 123, 127, 143, 159, 184, 188, 224, 228, 264, 280, 296, 300, 364, 373, 389, 405, 441, 445, 509, 513, 549, 565, 581, 597, 678, 682, 698, 714, 778, 782, 846, 850, 886, 922, 938, 942, 1042, 1051, 1087

Offset: 1

N. J. A. Sloane, Jun 14 2001

nonn

R. Ayoub, An Introduction to the Analytic Theory of Numbers, Amer. Math. Soc., 1963; Chapter II, Problem 56.

N. J. A. Sloane, Table of n, a(n) for n = 1..1024
Adrian Dudek, On the Success of Mishandling Euclid's Lemma, arXiv:1602.03555 [math.HO], 2016. See B(n) p. 2.
Chaohua Jia and Ayyadurai Sankaranarayanan, The mean square of the divisor function, Acta Arithmetica 164 (2014), 181-208.
Michaela Cully-Hugill and Timothy Trudgian, Two explicit divisor sums, arXiv:1911.07369 [math.NT], Nov 19 2019
Vaclav Kotesovec, Graph - The asymptotic ratio (1000000 terms)
Florian Luca and László Tóth, The r-th moment of the divisor function: an elementary approach, Journal of Integer Sequences 20 (2017), Article 17.7.4, 8 pp.
Adolf Piltz, Über das Gesetz, nach welchem die mittlere Darstellbarkeit der natürlichen Zahlen als Produkte einer gegebenen Anzahl Faktoren mit der Grösse der Zahlen wächst, 1881.
Ramanujan's Papers, Some formulas in the analytic theory of numbers, Messenger of Mathematics, XLV, 1916, 81-84, Formula (3).
D. Suryanarayana and R. Rama Chandra Rao, On an Asymptotic Formula of Ramanujan, Mathematica Scandinavica, 32, 258-264, 1973.
B. M. Wilson, Proofs of some formulas enunciated by Ramanujan, Proc. London Math. Soc. (2) 21 (1922) 235-255.

Cf. A000005, A035116, A061503, A318755.
Cf. A092742 (A), A245074 (B), A319090 (C), A319091 (D).
Cf. A057434, A072379, A074789.

[&+[NumberOfDivisors(k^2)*Floor(n/k): k in [1..n]]: n in [1..60]]; // Vincenzo Librandi, Sep 10 2016

Table[Sum[DivisorSigma[0, k^2]*Floor[n/k], {k, 1, n}], {n, 1, 50}] (* Vaclav Kotesovec, Aug 30 2018 *)
Accumulate[Table[DivisorSigma[0, n]^2, {n, 1, 50}]] (* Vaclav Kotesovec, Sep 10 2018 *)

for (n=1, 1024, write("b061502.txt", n, " ", sum(k=1, n, numdiv(k)^2)) ) \\ Harry J. Smith, Jul 23 2009

vector(60, n, sum(k=1, n, numdiv(k)^2)) \\ Michel Marcus, Jul 23 2015

first(n)=my(v=vector(n),s); forfactored(k=1,n, v[k[1]] = s += numdiv(k)^2); v; \\ Charles R Greathouse IV, Nov 28 2018

a(n) = Sum_{k=1..n} tau(k^2)*floor(n/k).
Asymptotic to A*n*log(n)^3 + B*n*log(n)^2 + C*n*log(n) + D*n + O(n^(1/2+eps)) where A = 1/Pi^2 and B = (12*gamma-3)/Pi^2 - 36*zeta'(2)/Pi^4. [corrected by Vaclav Kotesovec, Aug 30 2018]
C = 36*gamma^2/Pi^2 - (288*z1/Pi^4 + 24/Pi^2)*gamma + (864*z1^2/Pi^6 + 72*z1/Pi^4 - 72/Pi^4*z2 + 6/Pi^2) - 24*g1/Pi^2 and D = 24*gamma^3/Pi^2 - (432*z1 /Pi^4+ 36/Pi^2)*gamma^2 + (3456*z1^2/Pi^6 + 288*(z1-z2)/Pi^4 + 24/Pi^2 - 72*g1/Pi^2)*gamma + g1*(288*z1/Pi^4 + 24/Pi^2)-10368*z1^3/Pi^8 - 864*z1^2/Pi^6 + 1728*z2*z1/Pi^6 + 72*(z2-z1)/Pi^4- 48*z3/Pi^4 + (12*g2-6)/Pi^2, where gamma is the Euler-Mascheroni constant A001620, z1 = Zeta'(2) = A073002, z2 = Zeta''(2) = A201994, z3 = Zeta'''(2) = A201995 and g1, g2 are the Stieltjes constants, see A082633 and A086279. - Vaclav Kotesovec, Sep 10 2018
See Cully-Hugill & Trudgian, Theorem 2, for an explicit version of the asymptotic given above. - Charles R Greathouse IV, Nov 19 2019

Definition corrected by N. J. A. Sloane, May 25 2008

A057434 a(n) = Sum_{k=1..n} phi(k)^2.

Original entry on oeis.org

1, 2, 6, 10, 26, 30, 66, 82, 118, 134, 234, 250, 394, 430, 494, 558, 814, 850, 1174, 1238, 1382, 1482, 1966, 2030, 2430, 2574, 2898, 3042, 3826, 3890, 4790, 5046, 5446, 5702, 6278, 6422, 7718, 8042, 8618, 8874, 10474, 10618, 12382, 12782

Offset: 1

N. J. A. Sloane, Sep 08 2000

nonn

Partial sums of A127473. - R. J. Mathar, Sep 29 2008

Ivan Neretin, Table of n, a(n) for n = 1..10000
U. Balakrishnan & Y.-F. S. Pétermann, The Dirichlet series of zeta(s)*zeta(s+1)^alpha*f(s+1): On an error term associated with its coefficients, Acta Arith. 75 (1996), 39--69.

Cf. A000010, A002088, A061502, A072379, A074789.

FoldList[Plus, 1, EulerPhi[Range[2, 50]]^2] (* Ivan Neretin, May 30 2015 *)

a(n) = sum(k=1, n, eulerphi(k)^2); \\ Michel Marcus, Dec 20 2015

We can derive an asymptotic formula from a general formula given in the reference, namely: a(n) = C*n^3 + O(log(x)^(4/3)log(log(x))^(8/3)) where C = (1/3)/zeta(2)^2*Product_{p prime}(1+1/(p-1)/(p+1)^2) = 0.142749835225698(...). - Benoit Cloitre, Dec 22 2015

a(n) ~ c * n^3 / 3, where c = A065464 = Product_{primes p} (1 - 2/p^2 + 1/p^3) = 0.4282495056770944402187657075818235461212985133559361440319... - Vaclav Kotesovec, Dec 18 2019

A072379 Sum_{k<=n} (sigma(k)^2), where sigma(k) denotes the sum of the divisors of k A000203.

Original entry on oeis.org

1, 10, 26, 75, 111, 255, 319, 544, 713, 1037, 1181, 1965, 2161, 2737, 3313, 4274, 4598, 6119, 6519, 8283, 9307, 10603, 11179, 14779, 15740, 17504, 19104, 22240, 23140, 28324, 29348, 33317, 35621, 38537, 40841, 49122, 50566, 54166, 57302

Offset: 1

Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Jul 20 2002

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000

Cf. A024916, A072861.

Cf. A057434, A061502, A074789.

A072379 := proc(n)
    add( numtheory[sigma](k)^2,k=0..n) ;
end proc:
seq(A072379(n),n=1..80) ; # R. J. Mathar, Jul 09 2024

Accumulate[Table[DivisorSigma[1, k]^2, {k, 1, 50}]] (* Vaclav Kotesovec, Sep 10 2018 *)

a(n) = sum(k=1, n, sigma(k)^2) \\ Michel Marcus, Jun 20 2013

Ramanujan's asymptotic formula: (5/6)*Zeta(3)*n^3+O(n^2*log(n)^2)

cp's OEIS Frontend

A061502 a(n) = Sum_{k<=n} tau(k)^2, where tau = number of divisors function A000005.

Views

Author

Keywords

References

Links

Crossrefs

Programs

Magma

Mathematica

PARI

PARI

PARI

Formula

Extensions

A057434 a(n) = Sum_{k=1..n} phi(k)^2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A072379 Sum_{k<=n} (sigma(k)^2), where sigma(k) denotes the sum of the divisors of k A000203.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula