A319677 -id:A319677 - cp's OEIS frontend

A331177 Number of values of k, 1 <= k <= n, with A319677(k) = A319677(n), where A319677(n) = n/gcd(n, uphi(n)), and uphi is unitary totient function (A047994).

1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 3, 3, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 2, 3, 1, 1, 4, 1, 1, 1, 2, 1, 2, 1, 2, 1, 4, 1, 2, 3, 2, 3, 2, 1, 4, 1, 2, 1, 1, 1, 2, 1, 3, 1, 2, 1, 3, 1, 2, 3, 1, 1, 5, 1, 3, 1, 2, 1, 5, 1, 2, 1, 1, 1, 2, 1, 1, 3, 2, 1, 1, 1, 2, 1, 4, 1, 2, 1, 1, 3

Offset: 1

Antti Karttunen, Jan 11 2020

nonn

Ordinal transform of A319677.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A047994, A319677.

Cf. also A330739, A331175.

up_to = 65537;
ordinal_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), pt); for(i=1, length(invec), if(mapisdefined(om,invec[i]), pt = mapget(om, invec[i]), pt = 0); outvec[i] = (1+pt); mapput(om,invec[i],(1+pt))); outvec; };
A047994(n) = { my(f=factor(n)); prod(i=1, #f~, (f[i, 1]^f[i, 2])-1); };
A319677(n) = n/gcd(n, A047994(n));
v331177 = ordinal_transform(vector(up_to, n, A319677(n)));
A331177(n) = v331177[n];

A327837 Decimal expansion of the asymptotic mean of the number of exponential divisors function (A049419).

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Sep 27 2019

			1.602317102305418052349626315621161003776939495785572...

Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 52 (constant Z3).
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 (eq. 2.37 and 3.18, pp. 1346 and 1354).
Abdelhakim Smati and Jie Wu, On the exponential divisor function, Publications de l'Institut Mathématique, Vol. 61 (1997), pp. 21-32.
László Tóth, An order result for the exponential divisor function, Publ. Math. Debrecen, Vol. 71, No. 1-2 (2007), pp. 165-171, arXiv preprint,, arXiv:0708.3552 [math.NT], 2007.
László Tóth, Alternating sums concerning multiplicative arithmetic functions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1 (section 4.10, p. 30).
Jie Wu, Problème de diviseurs exponentiels et entiers exponentiellement sans facteur carré, Journal de théorie des nombres de Bordeaux, Vol. 7, No. 1, (1995), pp. 133-141.

Cf. A000005, A047994, A049419, A145353, A361013, A361967, A379517, A379518.

Cf. A059956 (constant for unitary divisors), A306071 (bi-unitary), A327576 (infinitary).

$MaxExtraPrecision = 1500; m = 1500; em = 500; f[x_] := 1 + Log[1 + Sum[x^e * (DivisorSigma[0, e] - DivisorSigma[0, e - 1]), {e, 2, em}]]; c = Rest[ CoefficientList[Series[f[x], {x, 0, m}], x] * Range[0, m] ]; RealDigits[ Exp[NSum[Indexed[c, k] * PrimeZetaP[k]/k, {k, 2, m}, NSumTerms -> m, WorkingPrecision -> m]], 10, 100][[1]]

Equals lim_{k->oo} A145353(k)/k.
Equals Product_{p prime} (1 + Sum_{e >= 2} p^(-e) * (d(e) - d(e-1))), where d(e) is the number of divisors of e (A000005).
Equals Product_{p prime} (1 - 1/p) * (2 - (log(p-1) + QPolyGamma(0, 1, 1/p)) / log(p)). - Vaclav Kotesovec, Feb 27 2023
From Amiram Eldar, Dec 24 2024: (Start)
Equals lim_{m->oo} (1/m) * Sum_{k=1..m} k/uphi(k) = lim_{m->oo} (1/m) * Sum_{k=1..m} A319677(k)/A319676(k), where uphi(k) is the unitary totient function (A047994).
Equals lim_{m->oo} (1/log(m)) * Sum_{k=1..m} 1/uphi(k) = lim_{m->oo} (1/log(m)) * A379517(m)/A379518(m).
Equals lim_{m->oo} (1/m) * Sum_{k=1..m} A361967(k).
Equals Product_{p prime} ((1-1/p) * (1 + Sum_{k>=1} 1/(p^k-1))).
Equals Product_{p prime} (1 + (1-1/p) * Sum_{k>=1} 1/(p^k*(p^k-1))). (End)

More digits from Vaclav Kotesovec, Jun 13 2021

A332883 If n = Product (p_j^k_j) then a(n) = denominator of Product (1 + 1/p_j^k_j).

Original entry on oeis.org

1, 2, 3, 4, 5, 1, 7, 8, 9, 5, 11, 3, 13, 7, 5, 16, 17, 3, 19, 2, 21, 11, 23, 2, 25, 13, 27, 7, 29, 5, 31, 32, 11, 17, 35, 18, 37, 19, 39, 20, 41, 7, 43, 11, 3, 23, 47, 12, 49, 25, 17, 26, 53, 9, 55, 7, 57, 29, 59, 1, 61, 31, 63, 64, 65, 11, 67, 34, 23, 35

Offset: 1

Ilya Gutkovskiy, Feb 28 2020

Denominator of sum of reciprocals of unitary divisors of n.

			1, 3/2, 4/3, 5/4, 6/5, 2, 8/7, 9/8, 10/9, 9/5, 12/11, 5/3, 14/13, 12/7, 8/5, 17/16, ...

Antti Karttunen, Table of n, a(n) for n = 1..20000
Eric Weisstein's World of Mathematics, Unitary Divisor

Cf. A007947, A017666, A034448, A077610, A319677, A323166, A327158 (positions of 1's), A332881, A332882 (numerators).

a:= n-> denom(mul(1+i[1]^i[2], i=ifactors(n)[2])/n):
seq(a(n), n=1..80);  # Alois P. Heinz, Feb 28 2020

Table[If[n == 1, 1, Times @@ (1 + 1/#[[1]]^#[[2]] & /@ FactorInteger[n])], {n, 1, 70}] // Denominator
Table[Sum[If[GCD[d, n/d] == 1,  1/d, 0], {d, Divisors[n]}], {n, 1, 70}] // Denominator

a(n) = denominator(sumdiv(n, d, if (gcd(d, n/d)==1, 1/d))); \\ Michel Marcus, Feb 28 2020

a(n) = denominator of Sum_{d|n, gcd(d, n/d) = 1} 1/d.
a(n) = denominator of usigma(n)/n.
a(p) = p, where p is prime.
a(n) = n / A323166(n). - Antti Karttunen, Nov 13 2021

A319676 Numerator of A047994(n)/n where A047994 is the unitary totient function.

Original entry on oeis.org

1, 1, 2, 3, 4, 1, 6, 7, 8, 2, 10, 1, 12, 3, 8, 15, 16, 4, 18, 3, 4, 5, 22, 7, 24, 6, 26, 9, 28, 4, 30, 31, 20, 8, 24, 2, 36, 9, 8, 7, 40, 2, 42, 15, 32, 11, 46, 5, 48, 12, 32, 9, 52, 13, 8, 3, 12, 14, 58, 2, 60, 15, 16, 63, 48, 10, 66, 12, 44, 12, 70, 7, 72, 18, 16

Offset: 1

Michel Marcus, Sep 26 2018

Cf. A047994, A030163, A065463, A305678, A319481, A319677 (denominators).

Cf. A002110, A038110, A006093.

uphi[n_] := Product[{p, e} = pe; p^e - 1, {pe, FactorInteger[n]}];
a[n_] := If[n == 1, 1, Numerator[uphi[n]/n]];
Array[a, 100] (* Jean-François Alcover, Jan 10 2022 *)

a(n)=my(f=factor(n)~); numerator(prod(i=1, #f, f[1, i]^f[2, i]-1)/n);

a(p) = p-1, for p prime (see A006093).
a(A002110(n)) = A038110(n).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A319677(k) = Product_{p prime} (1 - 1/(p*(p+1))) = 0.7044422... (A065463). - Amiram Eldar, Nov 21 2022

cp's OEIS Frontend

A331177 Number of values of k, 1 <= k <= n, with A319677(k) = A319677(n), where A319677(n) = n/gcd(n, uphi(n)), and uphi is unitary totient function (A047994).

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A327837 Decimal expansion of the asymptotic mean of the number of exponential divisors function (A049419).

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A332883 If n = Product (p_j^k_j) then a(n) = denominator of Product (1 + 1/p_j^k_j).

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A319676 Numerator of A047994(n)/n where A047994 is the unitary totient function.

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