A319592 -id:A319592 - cp's OEIS frontend

A074787 Sum of squares of the number of unitary divisors of k from 1 to n.

1, 5, 9, 13, 17, 33, 37, 41, 45, 61, 65, 81, 85, 101, 117, 121, 125, 141, 145, 161, 177, 193, 197, 213, 217, 233, 237, 253, 257, 321, 325, 329, 345, 361, 377, 393, 397, 413, 429, 445, 449, 513, 517, 533, 549, 565, 569, 585, 589, 605, 621, 637, 641, 657, 673

Offset: 1

Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Sep 07 2002

nonn

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A034444, A064608, A069212, A285052, A319592.

Equals 4*A069811(n) + 1, for n <= 29.

with(numtheory): seq(add(2^(2*nops(ifactors(k)[2])),k=1..n),n=1..100);

Accumulate[Table[Count[Divisors[n],?(GCD[#,n/#]==1&)],{n,60}]^2] (* _Harvey P. Dale, Dec 06 2012 *)
Accumulate[Table[4^PrimeNu[n], {n, 1, 50}]] (* Amiram Eldar, Jul 02 2022 *)

a(n) = Sum_{k=1..n} ud(k)^2 = Sum_{k=1..n} A034444(k)^2 . a(n) = Sum_{k=1..n} 2^(2*omega(k)) = Sum_{k=1..n} 2^(2*A001221(k)).

a(n) ~ c * n * log(n)^3, where c = (1/6) * Product_{p prime} ((1-1/p)^3*(1+3/p)) = A319592 / 6. - Amiram Eldar, Jul 02 2022

A320027 Decimal expansion of the probability that an integer 4-tuple is pairwise unitary coprime.

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, Aug 27 2019

Two numbers are unitary coprime if their largest common unitary divisor is 1.

			0.137310651809073591871587470612435012319854472214516...

Steven R. Finch, Mathematical Constants II, Cambridge University Press, 2018, p. 54.

László Tóth, Multiplicative arithmetic functions of several variables: a survey, in Themistocles M. Rassias and Panos M. Pardalos (eds.), Mathematics Without Boundaries, Springer, New York, NY, 2014, pp. 483-514 (see p. 509), preprint, arXiv:1310.7053 [math.NT], 2013-2014 (see p. 22).

Cf. A077610, A306071, A319592.

$MaxExtraPrecision = 1000; nm = 1000; f[x_] := 1 - 8*x^2 + 3*x^3 + 27*x^4 - 24*x^5 - 14*x^6 - 3*x^7 + 37*x^8 - 30*x^9 + 42*x^10 - 33*x^11 - 41*x^12 + 78*x^13 - 44*x^14 + 9*x^15; c = LinearRecurrence[{-3, 2, 11, -3, -16, -14, 6, 7, 19, 0, -17, 9}, {0, -16, 9, -20, 0, 161, -588, 2116, -5859, 15104, -34716, 70609}, nm]; RealDigits[Zeta[2]^2*Zeta[3]*Zeta[4]*f[1/2]*f[1/3]*Exp[NSum[Indexed[c, k]*(PrimeZetaP[k] - 1/2^k - 1/3^k)/k, {k, 2, nm}, NSumTerms -> nm, WorkingPrecision -> nm]], 10, 100][[1]]

zeta(2)^2 * zeta(3) * zeta(4) * prodeulerrat(1-8/p^2+3/p^3+27/p^4-24/p^5-14/p^6-3/p^7+37/p^8-30/p^9+42/p^10-33/p^11-41/p^12+78/p^13-44/p^14+9/p^15) \\ Amiram Eldar, Jun 29 2023

Equals zeta(2)^2 * zeta(3) * zeta(4) * Product_{p prime} (1 - 8/p^2 + 3/p^3 + 27/p^4 - 24/p^5 - 14/p^6 - 3/p^7 + 37/p^8 - 30/p^9 + 42/p^10 - 33/p^11 - 41/p^12 + 78/p^13 - 44/p^14 + 9/p^15).

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A074787 Sum of squares of the number of unitary divisors of k from 1 to n.

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A320027 Decimal expansion of the probability that an integer 4-tuple is pairwise unitary coprime.

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