A177754 -id:A177754 - cp's OEIS frontend

A306950 Numbers m that divide A177754(m) = Sum_{k=1..m} uphi(k), where uphi is the unitary totient function (A047994).

1, 2, 32, 36, 39, 50, 62, 147, 169, 190, 203, 467, 1035, 1075, 2174, 2475, 27047, 28097, 91087, 181175, 215795, 539654, 580160, 668988, 868879, 2611450, 14359486, 118119399, 1030191204, 1109928219, 2362155122

Offset: 1

Amiram Eldar, Mar 17 2019

The unitary version of A048290.

			32 is in the sequence since A177754(32) = 384 = 32 * 12 is divisible by 32.

Amiram Eldar, Table of n, a(n), A177754(a(n))/a(n) for n = 1..31.

Cf. A047994, A048290, A064611, A177754.

uphi[1] = 1; uphi[n_] := Product[{p, e} = pe; p^e-1, {pe, FactorInteger[n]}]; seq={}; s = 0; Do[s = s + uphi[n]; If[Divisible[s,n], AppendTo[seq, n]], {n, 1, 10^6}]; seq (* after Jean-François Alcover at A047994 *)

A370899 Partial alternating sums of the unitary totient function (A047994).

Original entry on oeis.org

1, 0, 2, -1, 3, 1, 7, 0, 8, 4, 14, 8, 20, 14, 22, 7, 23, 15, 33, 21, 33, 23, 45, 31, 55, 43, 69, 51, 79, 71, 101, 70, 90, 74, 98, 74, 110, 92, 116, 88, 128, 116, 158, 128, 160, 138, 184, 154, 202, 178, 210, 174, 226, 200, 240, 198, 234, 206, 264, 240, 300, 270

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. A047994, A065463, A177754.

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

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

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

a(n) = Sum_{k=1..n} (-1)^(k+1) * A047994(k).

a(n) = c * n^2 + O(n * log(n)^(5/3) * log(log(n))^(4/3)), where c = A065463 / 10 = 0.07044422... (Tóth, 2017).

A306070 Partial sums of A116550: Sum_{k=1..n} bphi(k) where bphi(k) is the bi-unitary analog of Euler's totient function.

Original entry on oeis.org

1, 2, 4, 7, 11, 14, 20, 27, 35, 41, 51, 59, 71, 80, 89, 104, 120, 132, 150, 164, 178, 193, 215, 232, 256, 274, 300, 321, 349, 364, 394, 425, 448, 472, 497, 526, 562, 589, 617, 648, 688, 709, 751, 786, 820, 853, 899, 935, 983, 1019, 1056, 1098, 1150, 1189

Offset: 1

Amiram Eldar, Jun 19 2018

nonn

The bi-unitary version of A002088 and A177754.

Amiram Eldar, Table of n, a(n) for n = 1..10000
László Tóth, On the bi-unitary analogues of Euler's arithmetical function and the gcd-sum function, Journal of Integer Sequences, Vol. 12 (2009), Article 09.5.2.

Cf. A002088, A177754, A116550, A306071.

bphi[1] = 1; bphi[n_] := With[{pp = Power @@@ FactorInteger[n]}, Count[Range[n], m_ /; Intersection[pp, Power @@@ FactorInteger[m]] == {}]]; Accumulate[Table[bphi[n], {n, 1, 100}]] (* after Jean-François Alcover at A116550 *)
phi[x_, n_] := DivisorSum[n, MoebiusMu[#]*Floor[x/#] &]; bphi[n_] := DivisorSum[n, (-1)^PrimeNu[#]*phi[n/#, #] &, CoprimeQ[#, n/#] &]; Accumulate[Array[bphi, 100]] (* Amiram Eldar, Jun 30 2025 *)

udivs(n) = {my(d = divisors(n)); select(x->(gcd(x, n/x)==1), d); }
gcud(n, m) = vecmax(setintersect(udivs(n), udivs(m)));
bphi(n) = if (n==1, 1, sum(k=1, n-1, gcud(n, k) == 1));
a(n) = sum(k=1, n, bphi(k)); \\ Michel Marcus, Jun 20 2018

a(n) = A*n^2/2 + O(n*log(n)^2), where A = A306071.

A379517 Numerators of the partial sums of the reciprocals of the unitary totient function (A047994).

Original entry on oeis.org

1, 2, 5, 17, 37, 43, 15, 109, 225, 239, 1223, 3809, 1293, 4019, 1031, 209, 1693, 1735, 5261, 5345, 5429, 27649, 306659, 310619, 312929, 317549, 4155857, 4195897, 603091, 615961, 619393, 19304143, 19463731, 1228951, 9898103, 4982299, 1251116, 2524397, 10164083

Offset: 1

Amiram Eldar, Dec 24 2024

			Fractions begin with 1, 2, 5/2, 17/6, 37/12, 43/12, 15/4, 109/28, 225/56, 239/56, 1223/280, 3809/840, ...

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. 52.
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 4.10, pp. 30-31.
Rimer Zurita, Generalized Alternating Sums of Multiplicative Arithmetic Functions, Journal of Integer Sequences, Vol. 23 (2020), Article 20.10.4. See section 4.5, pp. 16-17.

Cf. A001620, A047994, A177754, A370899, A327837, A379518 (denominators), A379519.

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

uphi(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 / uphi(k); print1(numerator(s), ", "))};

a(n) = numerator(Sum_{k=1..n} 1/A047994(k)).

a(n)/A379518(n) = L * log(n) + M + O(log(n)^(5/3)/n), where L = A327837, M = L * (gamma - B + A1 + A2), gamma = A001620, B = Sum_{p prime} (1-1/p) * log(p) * Sum_{k>=1} k/(p^k*(p^k-1)) / A(p), A1 = Sum_{p prime} log(p)/(p^2*(p-1)*A(p)), A2 = Sum_{p prime} ((A*(p)(p)*log(p)/p^2), A(p) = 1 + (1-1/p) * Sum_{k>=1} 1/(p^k*(p^k-1)), and A*(p) = Sum_{k>=1} 1/(p^k*p^(k+1)-1)*A(p)) (Sita Ramaiah and Suryanarayana, 1980).

A379518 Denominators of the partial sums of the reciprocals of the unitary totient function (A047994).

Original entry on oeis.org

1, 1, 2, 6, 12, 12, 4, 28, 56, 56, 280, 840, 280, 840, 210, 42, 336, 336, 1008, 1008, 1008, 5040, 55440, 55440, 55440, 55440, 720720, 720720, 102960, 102960, 102960, 3191760, 3191760, 199485, 1595880, 797940, 199485, 398970, 1595880, 11171160, 1117116, 279279

Offset: 1

Amiram Eldar, Dec 24 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. 52.
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 4.10, pp. 30-31.
Rimer Zurita, Generalized Alternating Sums of Multiplicative Arithmetic Functions, Journal of Integer Sequences, Vol. 23 (2020), Article 20.10.4. See section 4.5, pp. 16-17.

Cf. A047994, A177754, A370899, A379517 (numerators), A379520.

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

uphi(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 / uphi(k); print1(denominator(s), ", "))};

a(n) = denominator(Sum_{k=1..n} 1/A047994(k)).

A379519 Numerators of the partial alternating sums of the reciprocals of the unitary totient function (A047994).

Original entry on oeis.org

1, 0, 1, 1, 5, -1, 1, -5, 11, -31, -71, -211, -47, -281, -22, -29, -359, -569, -1427, -1847, -1427, -1931, -18721, -22681, -20371, -24991, -297163, -37467, -34607, -44617, -125843, -4141373, -3769001, -2117233, -327013, -2117233, -6041389, -6662009, -774568, -3297757

Offset: 1

Amiram Eldar, Dec 24 2024

			Fractions begin with 1, 0, 1/2, 1/6, 5/12, -1/12, 1/12, -5/84, 11/168, -31/168, -71/840, -211/840, ...

Amiram Eldar, Table of n, a(n) for n = 1..1000
László Tóth, Alternating Sums Concerning Multiplicative Arithmetic Functions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1. See section 4.10, pp. 30-31.

Cf. A047994, A065442, A177754, A327837, A370899, A379517, A379520 (denominators).

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

uphi(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) / uphi(k); print1(numerator(s), ", "))};

a(n) = numerator(Sum_{k=1..n} (-1)^(k+1)/A047994(k)).

a(n)/A379520(n) = T * log(n) + U + O(log(n)^(5/3) / n^u), where u > 0, T = A327837 * (2/(A065442 + 1) - 1), and U is a constant.

A379520 Denominators of the partial alternating sums of the reciprocals of the unitary totient function (A047994).

Original entry on oeis.org

1, 1, 2, 6, 12, 12, 12, 84, 168, 168, 840, 840, 280, 840, 105, 105, 1680, 1680, 5040, 5040, 5040, 5040, 55440, 55440, 55440, 55440, 720720, 80080, 80080, 80080, 240240, 7447440, 7447440, 3723720, 620620, 3723720, 11171160, 11171160, 1396395, 5585580, 2234232, 2234232

Offset: 1

Amiram Eldar, Dec 24 2024

Amiram Eldar, Table of n, a(n) for n = 1..1000
László Tóth, Alternating Sums Concerning Multiplicative Arithmetic Functions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1. See section 4.10, pp. 30-31.

Cf. A047994, A177754, A370899, A379518, A379519 (numerators).

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

uphi(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) / uphi(k); print1(denominator(s), ", "))};

a(n) = denominator(Sum_{k=1..n} (-1)^(k+1)/A047994(k)).

A327572 Partial sums of an infinitary analog of Euler's phi function: a(n) = Sum_{k=1..n} iphi(k), where iphi is A091732.

Original entry on oeis.org

1, 2, 4, 7, 11, 13, 19, 22, 30, 34, 44, 50, 62, 68, 76, 91, 107, 115, 133, 145, 157, 167, 189, 195, 219, 231, 247, 265, 293, 301, 331, 346, 366, 382, 406, 430, 466, 484, 508, 520, 560, 572, 614, 644, 676, 698, 744, 774, 822, 846, 878, 914, 966, 982, 1022, 1040

Offset: 1

Amiram Eldar, Sep 17 2019

nonn

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. A091732 (iphi), A327575.

Cf. A002088 (sums of phi), A177754 (unitary), A306070 (bi-unitary).

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

a(n) ~ c * n^2, where c = 0.328935... (A327575).

A321613 Partial products of the unitary totient function (A047994): a(n) = Product_{k=1..n} uphi(k).

Original entry on oeis.org

1, 1, 2, 6, 24, 48, 288, 2016, 16128, 64512, 645120, 3870720, 46448640, 278691840, 2229534720, 33443020800, 535088332800, 4280706662400, 77052719923200, 924632639078400, 11095591668940800, 110955916689408000, 2441030167166976000, 34174422340337664000

Offset: 1

Amiram Eldar, Dec 19 2018

nonn

a(n) is also the determinant of the symmetric n X n matrix M defined by M(i,j) = ugcd(i,j) for 1 <= i,j <= n, where ugcd(i,j) in the greatest common unitary divisor of i and j (A165430).

The unitary version of A001088.

			a(4) = uphi(1) * uphi(2) * uphi(3) * uphi(4) = 1 * 1 * 2 * 3 = 6.

Amiram Eldar, Table of n, a(n) for n = 1..481
H. Jager, The unitary analogues of some identities for certain arithmetical functions, Nederl. Akad. Wetensch. Proc. Ser. A, Vol. 64 (1961), pp. 508-515.

Cf. A001088, A047994, A165430, A177754.

uphi[1] = 1; uphi[n_] := Times @@ (-1 + Power @@@ FactorInteger[n]); FoldList[ Times, uphi /@ Range[50]]

uphi(n) = my(f=factor(n)~); prod(i=1, #f, f[1, i]^f[2, i]-1); \\ A047994
a(n) = prod(k=1, n, uphi(k)); \\ Michel Marcus, Dec 19 2018

cp's OEIS Frontend

A306950 Numbers m that divide A177754(m) = Sum_{k=1..m} uphi(k), where uphi is the unitary totient function (A047994).

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A370899 Partial alternating sums of the unitary totient function (A047994).

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A306070 Partial sums of A116550: Sum_{k=1..n} bphi(k) where bphi(k) is the bi-unitary analog of Euler's totient function.

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A379517 Numerators of the partial sums of the reciprocals of the unitary totient function (A047994).

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A379518 Denominators of the partial sums of the reciprocals of the unitary totient function (A047994).

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A379519 Numerators of the partial alternating sums of the reciprocals of the unitary totient function (A047994).

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A379520 Denominators of the partial alternating sums of the reciprocals of the unitary totient function (A047994).

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A327572 Partial sums of an infinitary analog of Euler's phi function: a(n) = Sum_{k=1..n} iphi(k), where iphi is A091732.

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