A034448 -id:A034448 - cp's OEIS frontend

A178450 Dirichlet inverse of A034448 (unitary sigma).

1, -3, -4, 4, -6, 12, -8, -6, 6, 18, -12, -16, -14, 24, 24, 8, -18, -18, -20, -24, 32, 36, -24, 24, 10, 42, -12, -32, -30, -72, -32, -12, 48, 54, 48, 24, -38, 60, 56, 36, -42, -96, -44, -48, -36, 72, -48, -32, 14, -30, 72, -56, -54, 36, 72, 48, 80, 90, -60, 96, -62, 96, -48, 16, 84, -144, -68, -72, 96, -144

Offset: 1

R. J. Mathar, Dec 22 2010

Antti Karttunen, Table of n, a(n) for n = 1..20000 (first 1000 terms from Andrew Howroyd)

Cf. A034448, A378434.

import Math.NumberTheory.Primes
a n = product . map (\(p, e) -> if even e then 2*unPrime p^(e`div`2) else -(unPrime p+1)*unPrime p^(e`div`2)) $ factorise n -- Sebastian Karlsson, Dec 04 2021

usigma[n_] := If[n==1, 1, Times @@ (1 + Power @@@ FactorInteger[n])];
a[n_] := a[n] = If[n==1, 1, -Sum[usigma[n/d] a[d], {d, Most@Divisors[n]}]];
Array[a, 70] (* Jean-François Alcover, Feb 16 2020 *)
f[p_, e_] := If[OddQ[e], -(p+1)*p^((e-1)/2), 2*p^(e/2)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 24 2023 *)

seq(n)={dirdiv(vector(n, n, n==1), vector(n, n, sumdivmult(n, d, if(gcd(d, n/d)==1, d))))} \\ Andrew Howroyd, Aug 05 2018

A178450(n) = { my(f=factor(n)); prod(i=1, #f~, if(!(f[i,2]%2), 2*(f[i, 1]^(f[i, 2]/2)), -(1+f[i,1])*(f[i, 1]^((f[i, 2]-1)/2)))); }; \\ (After the multiplicative formula) - Antti Karttunen, Nov 26 2024

Dirichlet g.f.: zeta(2s-1)/(zeta(s)*zeta(s-1)). - R. J. Mathar, Apr 14 2011

Multiplicative with a(p^e) = 2*p^(e/2) if e is even, -(p+1)*p^((e-1)/2) if e is odd. - Sebastian Karlsson, Dec 04 2021

A301981 Euler transform of A034448.

Original entry on oeis.org

1, 1, 4, 8, 19, 37, 84, 154, 313, 581, 1109, 2001, 3696, 6518, 11637, 20215, 35173, 60007, 102404, 171960, 288286, 477586, 788527, 1289539, 2101394, 3396594, 5469267, 8747285, 13934572, 22068218, 34815513, 54640049, 85434022, 132964684, 206193983, 318414629

Offset: 0

Vaclav Kotesovec, Mar 30 2018

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A001615, A034448, A156303, A301594, A301982.

nmax = 40; A034448 = Flatten[{1, Table[Times @@ (1 + Power @@@ FactorInteger[k]), {k, 2, nmax+1}]}]; CoefficientList[Series[Exp[Sum[Sum[A034448[[k]] * x^(j*k) / j, {k, 1, Floor[nmax/j] + 1}], {j, 1, nmax}]], {x, 0, nmax}], x]

G.f.: Product_{k>=1} 1/(1-x^k)^A034448(k).

Conjecture: a(n) ~ exp((3*Pi*n)^(2/3)/2 - 1/2) * A^6 / (2 * 3^(5/6) * Pi^(1/3) * n^(5/6)), where A is the Glaisher-Kinkelin constant A074962.

A301982 Expansion of Product_{k>=1} (1 + x^k)^A034448(k).

Original entry on oeis.org

1, 1, 3, 7, 12, 26, 52, 92, 170, 310, 541, 945, 1636, 2760, 4639, 7743, 12725, 20795, 33730, 54184, 86547, 137351, 216387, 339069, 528394, 818664, 1262245, 1936835, 2957432, 4496330, 6807191, 10262275, 15411203, 23056461, 34366753, 51047199, 75567600

Offset: 0

Vaclav Kotesovec, Mar 30 2018

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A001615, A034448, A156303, A301594, A301981.

nmax = 30; A034448 = Flatten[{1, Table[Times @@ (1 + Power @@@ FactorInteger[k]),{k, 2, nmax+1}]}]; CoefficientList[Series[Exp[Sum[-(-1)^j/j * Sum[A034448[[k]] * x^(j*k), {k, 1, Floor[nmax/j] + 1}], {j, 1, nmax}]], {x, 0, nmax}], x]

Conjecture: a(n) ~ exp(3 * Pi^(2/3) * n^(2/3) / 2^(5/3)) / (2^(4/3) * sqrt(3) * Pi^(1/6) * n^(2/3)).

A308041 Decimal expansion of lim_{m->oo} (1/log(m))*Sum_{k=1..m} 1/usigma(k), where usigma(k) is the sum of unitary divisors of k (A034448).

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, May 10 2019

			0.76871836244648519867273433245535052523425574041190...

Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 51 (constant Y3).
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. 3.8-3.9, p. 1352-1353).
László Tóth, Alternating sums concerning multiplicative arithmetic functions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1 (section 4.9, p. 29).

Cf. A034448, A063974, A308039 (corresponding limit with sigma).

$MaxExtraPrecision = 1000; m = 1000; f[p_] := 1 - (p - 1)/p*Sum[1/p^k/(p^k + 1), {k, 1, m}]; c = Rest[CoefficientList[Series[Log[f[1/x]], {x, 0, m}], x]*Range[0, m]];RealDigits[f[2]*Exp[NSum[Indexed[c, k]*(PrimeZetaP[k] - 1/2^k)/k, {k, 2, m}, NSumTerms -> m, WorkingPrecision -> m]], 10, 100][[1]]

From Amiram Eldar, Dec 23 2024: (Start)
Equals Product_{p prime} ((1-1/p) * (1 + Sum_{k>=1} 1/(p^k+1))).
Equals lim_{m->oo} (1/m) * Sum_{k=1..m} A063974(k). (End)

More digits from Vaclav Kotesovec, Jun 13 2021

A323159 Greatest common divisor of product (1+(p^e)) and product (1+p), where p ranges over prime factors of n, with e corresponding exponent; a(n) = gcd(A034448(n), A048250(n)).

Original entry on oeis.org

1, 3, 4, 1, 6, 12, 8, 3, 2, 18, 12, 4, 14, 24, 24, 1, 18, 6, 20, 6, 32, 36, 24, 12, 2, 42, 4, 8, 30, 72, 32, 3, 48, 54, 48, 2, 38, 60, 56, 18, 42, 96, 44, 12, 12, 72, 48, 4, 2, 6, 72, 14, 54, 12, 72, 24, 80, 90, 60, 24, 62, 96, 16, 1, 84, 144, 68, 18, 96, 144, 72, 6, 74, 114, 8, 20, 96, 168, 80, 6, 2, 126, 84, 32, 108, 132, 120, 36, 90, 36

Offset: 1

Antti Karttunen, Jan 09 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A034448, A048250, A323160, A323163.

A034448(n) = { my(f=factorint(n)); prod(k=1, #f~, 1+(f[k, 1]^f[k, 2])); }; \\ After code in A034448
A048250(n) = factorback(apply(p -> p+1,factor(n)[,1]));
A323159(n) = gcd(A034448(n), A048250(n));

a(n) = gcd(A034448(n), A048250(n)).

A323160 a(n) = gcd(n, A323159(n)) = gcd(n, A034448(n), A048250(n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 6, 1, 1, 1, 2, 1, 4, 1, 2, 3, 1, 1, 6, 1, 2, 1, 2, 1, 12, 1, 2, 1, 4, 1, 6, 1, 1, 3, 2, 1, 2, 1, 2, 1, 2, 1, 6, 1, 4, 3, 2, 1, 4, 1, 2, 3, 2, 1, 6, 1, 8, 1, 2, 1, 12, 1, 2, 1, 1, 1, 6, 1, 2, 3, 2, 1, 6, 1, 2, 1, 4, 1, 6, 1, 2, 1, 2, 1, 4, 1, 2, 3, 4, 1, 18, 7, 4, 1, 2, 5, 12, 1, 2, 3, 2, 1, 6, 1, 2, 3

Offset: 1

Antti Karttunen, Jan 09 2019

nonn

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

Cf. A034448, A048250, A323159, A323163, A323166.

A034448(n) = { my(f=factorint(n)); prod(k=1, #f~, 1+(f[k, 1]^f[k, 2])); }; \\ After code in A034448
A048250(n) = factorback(apply(p -> p+1,factor(n)[,1]));
A323160(n) = gcd(n, gcd(A034448(n), A048250(n)));

a(n) = gcd(n, A323159(n)) = gcd(A048250(n), A323166(n)).

a(n) = gcd(n, A034448(n), A048250(n)).

A327160 Number of positive integers that are reachable from n with some combination of transitions x -> usigma(x)-x and x -> gcd(x,usigma(x)), where usigma is the sum of unitary divisors of n (A034448).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Aug 25 2019

nonn

Question: Is this sequence well defined for every n ? If A318882 is not well defined in whole N, then neither this can be.

			From n = 30 we can reach any of the following strictly positive numbers: 30 (e.g., with an empty sequence of transitions), 42 (as A034460(30) = 42), 54 (as A034460(42) = 54; note that A034460(54) = 30 again) and 6 as A323166(30) = A323166(42) = A323166(54) = 6 = A323166(6) = A034460(6), thus a(30) = 4.

Antti Karttunen, Table of n, a(n) for n = 1..20000
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000

Cf. A034448, A034460, A318882, A323166.

Cf. also A326198, A327161.

A034448(n) = { my(f=factorint(n)); prod(k=1, #f~, 1+(f[k, 1]^f[k, 2])); }; \\ After code in A034448
A327160aux(n,xs) = if(vecsearch(xs,n),xs, xs = setunion([n],xs); if(1==n,xs, my(a=A034448(n)-n, b=gcd(A034448(n),n)); xs = A327160aux(a,xs); if((a==b),xs, A327160aux(b,xs))));
A327160(n) = length(A327160aux(n,Set([])));

A327161 Number of positive integers that are reachable from n with some combination of transitions x -> usigma(x)-x and x -> gcd(x,phi(x)), where usigma is the sum of unitary divisors of n (A034448), and phi is Euler totient function (A000010).

Original entry on oeis.org

1, 2, 2, 3, 2, 3, 2, 4, 3, 5, 2, 5, 2, 6, 4, 5, 2, 7, 2, 6, 4, 7, 2, 6, 3, 6, 4, 6, 2, 10, 2, 6, 5, 7, 3, 8, 2, 8, 4, 7, 2, 10, 2, 6, 5, 7, 2, 8, 3, 8, 5, 8, 2, 10, 4, 6, 4, 7, 2, 4, 2, 8, 5, 7, 3, 6, 2, 8, 5, 9, 2, 9, 2, 8, 4, 7, 3, 5, 2, 9, 5, 7, 2, 8, 3, 8, 6, 7, 2, 4, 5, 7, 5, 9, 4, 11, 2, 11, 5, 13, 2, 10, 2, 8, 7

Offset: 1

Antti Karttunen, Aug 25 2019

nonn

Question: Is this sequence well-defined for every n > 0? If A318882 is not well-defined for all positive integers, then neither can this be.

			a(30) = 10 as the graph obtained from vertex-relations x -> A034460(x) and x -> A009195(x) spans the following ten numbers [1, 2, 4, 6, 8, 12, 18, 30, 42, 54], which is illustrated below:
.
  30 -> 42 -> 54 (-> 30 ...)
   |     |     |
   2 <-- 6 <- 18
   |  \        |
   1 <-- 4 <- 12
            \  |
             <-8

Antti Karttunen, Table of n, a(n) for n = 1..20000
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000

Cf. A000010, A009195, A034448, A034460, A318882, A326195.

Cf. also A326196, A326198, A327160.

A034460(n) = (sumdivmult(n, d, if(gcd(d, n/d)==1, d))-n); \\ From A034460
A327161aux(n,xs) = if(vecsearch(xs,n),xs, xs = setunion([n],xs); if(1==n,xs, my(a=A034460(n), b=gcd(eulerphi(n),n)); xs = A327161aux(a,xs); if((a==b),xs, A327161aux(b,xs))));
A327161(n) = length(A327161aux(n,Set([])));

a(n) >= max(A318882(n), 1+A326195(n)).

A327164 Number of iterations of x -> gcd(usigma(x),x) needed to reach a fixed point, where usigma is the sum of unitary divisors of n (A034448).

Original entry on oeis.org

0, 1, 1, 1, 1, 0, 1, 1, 1, 2, 1, 2, 1, 2, 2, 1, 1, 1, 1, 3, 1, 2, 1, 3, 1, 2, 1, 2, 1, 1, 1, 1, 2, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 3, 2, 1, 2, 1, 2, 2, 2, 1, 1, 1, 2, 1, 2, 1, 0, 1, 2, 1, 1, 1, 1, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 2, 2, 1, 0, 2, 2, 1, 2, 2, 3, 1, 2, 2, 3, 1, 1, 1, 2, 2

Offset: 1

Antti Karttunen, Aug 28 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000

Cf. A034448, A323166, A327158 (positions of zeros).

Cf. also A326194.

A034448(n) = { my(f=factorint(n)); prod(k=1, #f~, 1+(f[k, 1]^f[k, 2])); }; \\ After code in A034448
A323166(n) = gcd(n, A034448(n));
A327164(n) = { my(u=A323166(n)); if(u==n,0,1+A327164(u)); }

A329858 Numbers k such that k and usigma(k) have the same set of prime divisors, where usigma(k) is the sum of unitary divisors of k (A034448).

Original entry on oeis.org

1, 6, 24, 60, 90, 180, 360, 378, 816, 1056, 1512, 3780, 9100, 10500, 12240, 13230, 15750, 15840, 26460, 31500, 36750, 40950, 46494, 51408, 52920, 63000, 63700, 66528, 73500, 87360, 94500, 95550, 110250, 145600, 145920, 147000, 163800, 181632, 185976, 220500

Offset: 1

Amiram Eldar, Nov 22 2019

nonn

			6 is in the sequence since 6 = 2 * 3 and usigma(6) = 12 = 2^2 * 3 both have the same set of prime divisors, {2, 3}.

Amiram Eldar, Table of n, a(n) for n = 1..435 (terms below 5*10^10)

The unitary version of A027598.

Cf. A007947, A034448.

rad[n_] := Times @@ (First@# & /@ FactorInteger@ n); usigma[1]=1; usigma[n_] := Times @@ (1 + Power @@@ FactorInteger[n]); Select[Range[2*10^5], rad[#] == rad[usigma[#]] &]

cp's OEIS Frontend

A178450 Dirichlet inverse of A034448 (unitary sigma).

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A301981 Euler transform of A034448.

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A301982 Expansion of Product_{k>=1} (1 + x^k)^A034448(k).

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A308041 Decimal expansion of lim_{m->oo} (1/log(m))*Sum_{k=1..m} 1/usigma(k), where usigma(k) is the sum of unitary divisors of k (A034448).

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A323159 Greatest common divisor of product (1+(p^e)) and product (1+p), where p ranges over prime factors of n, with e corresponding exponent; a(n) = gcd(A034448(n), A048250(n)).

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A323160 a(n) = gcd(n, A323159(n)) = gcd(n, A034448(n), A048250(n)).

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A327160 Number of positive integers that are reachable from n with some combination of transitions x -> usigma(x)-x and x -> gcd(x,usigma(x)), where usigma is the sum of unitary divisors of n (A034448).

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A327161 Number of positive integers that are reachable from n with some combination of transitions x -> usigma(x)-x and x -> gcd(x,phi(x)), where usigma is the sum of unitary divisors of n (A034448), and phi is Euler totient function (A000010).

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