A064609 -id:A064609 - cp's OEIS frontend

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

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

A034448 usigma(n) = sum of unitary divisors of n (divisors d such that gcd(d, n/d)=1); also called UnitarySigma(n).

Original entry on oeis.org

1, 3, 4, 5, 6, 12, 8, 9, 10, 18, 12, 20, 14, 24, 24, 17, 18, 30, 20, 30, 32, 36, 24, 36, 26, 42, 28, 40, 30, 72, 32, 33, 48, 54, 48, 50, 38, 60, 56, 54, 42, 96, 44, 60, 60, 72, 48, 68, 50, 78, 72, 70, 54, 84, 72, 72, 80, 90, 60, 120, 62, 96, 80, 65, 84, 144, 68, 90, 96, 144

Offset: 1

N. J. A. Sloane, Dec 11 1999

Row sums of the triangle in A077610. - Reinhard Zumkeller, Feb 12 2002

Multiplicative with a(p^e) = p^e+1 for e>0. - Franklin T. Adams-Watters, Sep 11 2005

			Unitary divisors of 12 are 1, 3, 4, 12. Or, 12=3*2^2 hence usigma(12)=(3+1)*(2^2+1)=20.

James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 147.

T. D. Noe, Table of n, a(n) for n = 1..10000
Octavio A. Agustín-Aquino, Prime injections and quasipolarities, Matematiche 69 (2014) 159-168
Steven R. Finch, Unitarism and Infinitarism, February 25, 2004. [Cached copy, with permission of the author]
Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 50.
Carl Pomerance and Hee-Sung Yang, Variant of a theorem of Erdős on the sum-of-proper-divisors function, Math. Comp., to appear (2014).
Tim Trudgian, The sum of the unitary divisor function, Publications de l'Institut Mathématique 2015 Vol. 97, Issue 111, pp. 175-180.
Eric Weisstein's World of Mathematics, Unitary Divisor Function
Wikipedia, Unitary divisor

Cf. A000203, A034444, A034460, A047994, A048250, A064000, A064609, A191414.
Cf. A063937 (squares > 1).
Cf. A188999, A301981, A301982.

a034448 = sum . a077610_row  -- Reinhard Zumkeller, Feb 12 2012
(Python 3.8+)
from math import prod
from sympy import factorint
def A034448(n): return prod(p**e+1 for p, e in factorint(n).items()) # Chai Wah Wu, Jun 20 2021

A034448 := proc(n) local ans, i:ans := 1: for i from 1 to nops(ifactors(n)[ 2 ]) do ans := ans*(1+ifactors(n)[ 2 ][ i ][ 1 ]^ifactors(n)[ 2 ] [ i ] [ 2 ]): od: RETURN(ans) end:
a := proc(n) local i; numtheory[divisors](n); select(d -> igcd(d,n/d)=1, %); add(i,i=%) end; # Peter Luschny, May 03 2009

usigma[n_] := Block[{d = Divisors[n]}, Plus @@ Select[d, GCD[ #, n/# ] == 1 &]]; Table[ usigma[n], {n, 71}] (* Robert G. Wilson v, Aug 28 2004 *)
Table[DivisorSum[n, # &, CoprimeQ[#, n/#] &], {n, 70}] (* Michael De Vlieger, Mar 01 2017 *)
usigma[n_] := If[n == 1, 1, Times @@ (1 + Power @@@ FactorInteger[n])]; Array[usigma, 100] (* faster since avoids generating divisors, Giovanni Resta, Apr 23 2017 *)

A034448(n)=sumdiv(n,d,if(gcd(d,n/d)==1,d)) \\ Rick L. Shepherd

A034448(n) = {my(f=factorint(n)); prod(k=1, #f[,2], f[k,1]^f[k,2]+1)} \\ Andrew Lelechenko, Apr 22 2014

a(n)=sumdivmult(n,d,if(gcd(d,n/d)==1,d)) \\ Charles R Greathouse IV, Sep 09 2014

If n = Product p_i^e_i, usigma(n) = Product (p_i^e_i + 1). - Vladeta Jovovic, Apr 19 2001
Dirichlet generating function: zeta(s)*zeta(s-1)/zeta(2s-1). - Franklin T. Adams-Watters, Sep 11 2005
Conjecture: a(n) = sigma(n^2/rad(n))/sigma(n/rad(n)), where sigma = A000203 and rad = A007947. - Velin Yanev, Aug 20 2017
This conjecture is easily verified since all the functions involved are multiplicative and proving it for prime powers is straightforward. - Juan José Alba González, Mar 19 2021
From Amiram Eldar, May 29 2020: (Start)
Sum_{d|n, gcd(d, n/d) = 1} a(d) * (-1)^omega(n/d) = n.
a(n) <= sigma(n) = A000203(n), with equality if and only if n is squarefree (A005117). (End)
Sum_{k=1..n} a(k) ~ Pi^2 * n^2 / (12*zeta(3)). - Vaclav Kotesovec, May 20 2021
a(n) = uphi(n^2)/uphi(n) = A191414(n)/uphi(n), where uphi(n) = A047994(n). - Amiram Eldar, Sep 21 2024

More terms from Erich Friedman

A307159 Partial sums of the bi-unitary divisors sum function: Sum_{k=1..n} bsigma(k), where bsigma is A188999.

Original entry on oeis.org

1, 4, 8, 13, 19, 31, 39, 54, 64, 82, 94, 114, 128, 152, 176, 203, 221, 251, 271, 301, 333, 369, 393, 453, 479, 521, 561, 601, 631, 703, 735, 798, 846, 900, 948, 998, 1036, 1096, 1152, 1242, 1284, 1380, 1424, 1484, 1544, 1616, 1664, 1772, 1822, 1900, 1972, 2042

Offset: 1

Amiram Eldar, Mar 27 2019

nonn

D. Suryanarayana and M. V. Subbarao, Arithmetical functions associated with the biunitary k-ary divisors of an integer, Indian J. Math., Vol. 22 (1980), pp. 281-298.

Amiram Eldar, Table of n, a(n) for n = 1..10000
László Tóth, Alternating sums concerning multiplicative arithmetic functions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1, section 4.13.

Cf. A024916, A064609, A188999, A306069, A307042, A307160.

fun[p_,e_] := If[OddQ[e],(p^(e+1)-1)/(p-1),(p^(e+1)-1)/(p-1)-p^(e/2)]; bsigma[1] = 1; bsigma[n_] := Times @@ (fun @@@ FactorInteger[n]); Accumulate[Array[bsigma, 60]]

a(n) ~ c * n^2, where c = (zeta(2)*zeta(3)/2) * Product_{p}(1 - 2/p^3 + 1/p^4 + 1/p^5 - 1/p^6) (A307160).

A370898 Partial alternating sums of the sum of unitary divisors function (A034448).

Original entry on oeis.org

1, -2, 2, -3, 3, -9, -1, -10, 0, -18, -6, -26, -12, -36, -12, -29, -11, -41, -21, -51, -19, -55, -31, -67, -41, -83, -55, -95, -65, -137, -105, -138, -90, -144, -96, -146, -108, -168, -112, -166, -124, -220, -176, -236, -176, -248, -200, -268, -218, -296, -224, -294, -240, -324, -252, -324, -244, -334, -274, -394

Offset: 1

Amiram Eldar, Mar 05 2024

Amiram Eldar, Table of n, a(n) for n = 1..10000
László Tóth, Alternating Sums Concerning Multiplicative Arithmetic Functions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1.

Cf. A002117, A034448, A064609.

Similar sequences: A068762, A068773, A307704, A357817, A362028.

usigma[n_] := Times @@ (1 + Power @@@ FactorInteger[n]); usigma[1] = 1; Accumulate[Array[(-1)^(# + 1) * usigma[#] &, 100]]

usigma(n) = {my(f = factor(n)); prod(i = 1, #f~, 1 + f[i, 1]^f[i, 2]);}
lista(kmax) = {my(s = 0); for(k = 1, kmax, s += (-1)^(k+1) * usigma(k); print1(s, ", "))};

a(n) = Sum_{k=1..n} (-1)^(k+1) * A034448(k).

a(n) = -c * n^2 + O(n * log(n)^(5/3)), where c = Pi^2/(84*zeta(3)) = 0.0977451984014... (Tóth, 2017).

A307042 Partial sums of the exponential divisors sum function: Sum_{k=1..n} esigma(k), where esigma is A051377.

Original entry on oeis.org

1, 3, 6, 12, 17, 23, 30, 40, 52, 62, 73, 91, 104, 118, 133, 155, 172, 196, 215, 245, 266, 288, 311, 341, 371, 397, 427, 469, 498, 528, 559, 593, 626, 660, 695, 767, 804, 842, 881, 931, 972, 1014, 1057, 1123, 1183, 1229, 1276, 1342, 1398, 1458, 1509, 1587, 1640

Offset: 1

Amiram Eldar, Mar 21 2019

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000
J. Fabrykowski and M. V. Subbarao, The maximal order and the average order of multiplicative function sigma^(e)(n), in Théorie des nombres/Number theory (Quebec, PQ, 1987), 201-206, de Gruyter, Berlin, 1989.

Cf. A024916, A051377, A064609, A275480.

esigma[n_] := Times @@ (Sum[ First[#]^d, {d, Divisors[Last[#]]}] & ) /@ FactorInteger[n]; Accumulate[Array[esigma, 60]] (* after Jean-François Alcover at A051377 *)

a(n) ~ B * n^2, where B = 0.5682854937... (A275480).

A074789 Partial sums of usigma(n)^2: square of the sum of unitary divisors of n.

Original entry on oeis.org

1, 10, 26, 51, 87, 231, 295, 376, 476, 800, 944, 1344, 1540, 2116, 2692, 2981, 3305, 4205, 4605, 5505, 6529, 7825, 8401, 9697, 10373, 12137, 12921, 14521, 15421, 20605, 21629, 22718, 25022, 27938, 30242, 32742, 34186, 37786, 40922, 43838

Offset: 1

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

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000
Vaclav Kotesovec, Graph - the asymptotic ratio.
László Tóth, An asymptotic formula concerning the unitary divisor sum function, Stud. Univ. Babes-Bolyai, Math., Vol. 34, No. 2 (1989), pp. 3-10; entire volume.

Cf. A034448, A064609.
Cf. A057434, A061502, A072379.
Cf. A002117, A013661.

Accumulate[Table[DivisorSum[n, # &, CoprimeQ[#, n/#] &]^2, {n, 1, 50}]] (* Vaclav Kotesovec, Feb 04 2019 *)

A034448(n) = {my(f = factor(n)); prod(i=1, #f~, 1 + f[i, 1]^f[i, 2]);}
lista(nmax) = {my(s = 0); for(n = 1, nmax, s += A034448(n)^2; print1(s, ", "));} \\ Amiram Eldar, Jul 24 2024

a(n) = Sum_{k=1..n} usigma(k)^2 = Sum_{k=1..n} A034448(k)^2.
Asymptotic expression: a(n) = Sum_{k<=n} usigma(k)^2 = (zeta(2)*zeta(3)*alpha_1/3)*n^3 + O(n^2*log(n)^4), alpha_1 = Product_{p prime} (1+1/p^2-2/p^3-1/p^4-2/p^5+3/p^6), zeta(2) = A013661 and zeta(3) = A002117.
alpha_1 = 1.001619936509160661474009830789... . - Amiram Eldar, Jul 24 2024

A379513 Numerators of the partial sums of the reciprocals of the sum of unitary divisors function (A034448).

Original entry on oeis.org

1, 4, 19, 107, 39, 61, 259, 817, 853, 97, 301, 307, 2209, 187, 2279, 39583, 121129, 122557, 124699, 126127, 509863, 171541, 173921, 526523, 6930479, 6983519, 7063079, 7118771, 7193027, 802663, 405199, 13495327, 1131701, 30726097, 123670153, 622026437, 11910394103

Offset: 1

Amiram Eldar, Dec 23 2024

			Fractions begin with 1, 4/3, 19/12, 107/60, 39/20, 61/30, 259/120, 817/360, 853/360, 97/40, 301/120, 307/120, ...

Amiram Eldar, Table of n, a(n) for n = 1..1000
Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018. See p. 51.
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.
László Tóth, Alternating Sums Concerning Multiplicative Arithmetic Functions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1. See section 4.9, pp. 28-29.
Rimer Zurita, Generalized Alternating Sums of Multiplicative Arithmetic Functions, Journal of Integer Sequences, Vol. 23 (2020), Article 20.10.4. See section 4.3, pp. 12-15.

Cf. A034448, A064609, A370898, A379514 (denominators), A379515.

Cf. A001620, A308041.

usigma[n_] := Times @@ (1 + Power @@@ FactorInteger[n]); usigma[1] = 1; Numerator[Accumulate[Table[1/usigma[n], {n, 1, 50}]]]

usigma(n) = {my(f = factor(n)); prod(i = 1, #f~, 1 + f[i, 1]^f[i, 2]);}
list(nmax) = {my(s = 0); for(k = 1, nmax, s += 1 / usigma(k); print1(numerator(s), ", "))};

a(n) = numerator(Sum_{k=1..n} 1/A034448(k)).

a(n)/A379514(n) = B * log(n) + D + O(log(n)^(5/3) * log(log(n))^(4/3) / n), where B = A308041, D = B * (gamma + A1 - A2), gamma = A001620, A1 = Sum_{p prime} ((p*C(p)*log(p)/(p-1)) * Sum_{k>=1} (k/(p^k*(p^(k+1)+1)))), A2 = Sum_{p prime} ((C(p)*log(p)/p^2) * Sum_{k>=0} (1/(p^k*(p^(k+1)+1)))), and C(p) = 1 - (p/(p-1)) * Sum_{k>=1} (1/(p^k*(p^(k+1)+1))) (Sita Ramaiah and Suryanarayana, 1980).

A379514 Denominators of the partial sums of the reciprocals of the sum of unitary divisors function (A034448).

Original entry on oeis.org

1, 3, 12, 60, 20, 30, 120, 360, 360, 40, 120, 120, 840, 70, 840, 14280, 42840, 42840, 42840, 42840, 171360, 57120, 57120, 171360, 2227680, 2227680, 2227680, 2227680, 2227680, 247520, 123760, 4084080, 340340, 9189180, 36756720, 183783600, 3491888400, 3491888400

Offset: 1

Amiram Eldar, Dec 23 2024

Amiram Eldar, Table of n, a(n) for n = 1..1000
Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018. See p. 51.
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.
László Tóth, Alternating Sums Concerning Multiplicative Arithmetic Functions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1. See section 4.9, pp. 28-29.
Rimer Zurita, Generalized Alternating Sums of Multiplicative Arithmetic Functions, Journal of Integer Sequences, Vol. 23 (2020), Article 20.10.4. See section 4.3, pp. 12-15.

Cf. A034448, A064609, A370898, A379513 (numerators), A379516.

usigma[n_] := Times @@ (1 + Power @@@ FactorInteger[n]); usigma[1] = 1; Denominator[Accumulate[Table[1/usigma[n], {n, 1, 50}]]]

usigma(n) = {my(f = factor(n)); prod(i = 1, #f~, 1 + f[i, 1]^f[i, 2]);}
list(nmax) = {my(s = 0); for(k = 1, nmax, s += 1 / usigma(k); print1(denominator(s), ", "))};

a(n) = denominator(Sum_{k=1..n} 1/A034448(k)).

A327566 Partial sums of the infinitary divisors sum function: a(n) = Sum_{k=1..n} isigma(k), where isigma is A049417.

Original entry on oeis.org

1, 4, 8, 13, 19, 31, 39, 54, 64, 82, 94, 114, 128, 152, 176, 193, 211, 241, 261, 291, 323, 359, 383, 443, 469, 511, 551, 591, 621, 693, 725, 776, 824, 878, 926, 976, 1014, 1074, 1130, 1220, 1262, 1358, 1402, 1462, 1522, 1594, 1642, 1710, 1760, 1838, 1910, 1980

Offset: 1

Amiram Eldar, Sep 17 2019

nonn

Differs from A307159 at n >= 16.

Steven R. Finch, Mathematical Constants II, Cambridge University Press, 2018, section 1.7.5, pp. 53-54.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Graeme L. Cohen and Peter Hagis, Jr., Arithmetic functions associated with infinitary divisors of an integer, International Journal of Mathematics and Mathematical Sciences, Vol. 16, No. 2 (1993), pp. 373-383.

Cf. A049417 (isigma), A327574.

Cf. A024916 (all divisors), A064609 (unitary), A307042 (exponential), A307159 (bi-unitary).

f[p_, e_] := p^(2^(-1 + Position[Reverse @ IntegerDigits[e, 2], ?(# == 1 &)])); isigma[1] = 1; isigma[n] := Times @@ (Flatten @ (f @@@ FactorInteger[n]) + 1); Accumulate[Array[isigma, 52]]

a(n) ~ c * n^2, where c = 0.730718... (A327574).

A379515 Numerators of the partial alternating sums of the reciprocals of the sum of unitary divisors function (A034448).

Original entry on oeis.org

1, 2, 11, 43, 53, 4, 37, 293, 329, 103, 113, 107, 809, 129, 809, 12913, 41119, 39691, 41833, 8081, 33395, 32443, 33871, 10973, 148361, 48275, 7149, 34861, 108119, 319937, 164941, 1761311, 112361, 662011, 5405483, 26502319, 516671461, 508357441, 3620857237, 3556192637

Offset: 1

Amiram Eldar, Dec 23 2024

			Fractions begin with 1, 2/3, 11/12, 43/60, 53/60, 4/5, 37/40, 293/360, 329/360, 103/120, 113/120, 107/120, ...

Amiram Eldar, Table of n, a(n) for n = 1..1000
Olivier Bordellès and Benoit Cloitre, An Alternating Sum Involving the Reciprocal of Certain Multiplicative Functions, Journal of Integer Sequences, Vol. 16 (2013), Article 13.6.3. See p. 4, eq. (vi).
László Tóth, Alternating Sums Concerning Multiplicative Arithmetic Functions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1. See section 4.9, pp. 28-29.

Cf. A034448, A064609, A308041, A323482, A370898, A379513, A379516 (denominators).

usigma[n_] := Times @@ (1 + Power @@@ FactorInteger[n]); usigma[1] = 1; Numerator[Accumulate[Table[(-1)^(n+1)/usigma[n], {n, 1, 50}]]]

usigma(n) = {my(f = factor(n)); prod(i = 1, #f~, 1 + f[i, 1]^f[i, 2]);}
list(nmax) = {my(s = 0); for(k = 1, nmax, s += (-1)^(k+1) / usigma(k); print1(numerator(s), ", "))};

a(n) = numerator(Sum_{k=1..n} (-1)^(k+1)/A034448(k)).

a(n)/A379516(n) = E * log(n) + F + O(log(n)^(5/3) * log(log(n))^(4/3) / n^u), where u > 0, E = A308041 * (2/(A323482 + 1/2) - 1) = 0.10259754363391420806..., and F is a constant.

cp's OEIS Frontend

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

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A034448 usigma(n) = sum of unitary divisors of n (divisors d such that gcd(d, n/d)=1); also called UnitarySigma(n).

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A307159 Partial sums of the bi-unitary divisors sum function: Sum_{k=1..n} bsigma(k), where bsigma is A188999.

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A370898 Partial alternating sums of the sum of unitary divisors function (A034448).

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A307042 Partial sums of the exponential divisors sum function: Sum_{k=1..n} esigma(k), where esigma is A051377.

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A074789 Partial sums of usigma(n)^2: square of the sum of unitary divisors of n.

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A379513 Numerators of the partial sums of the reciprocals of the sum of unitary divisors function (A034448).

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