A087893 -id:A087893 - cp's OEIS frontend

A309307 Number of unitary divisors of n (excluding 1).

0, 1, 1, 1, 1, 3, 1, 1, 1, 3, 1, 3, 1, 3, 3, 1, 1, 3, 1, 3, 3, 3, 1, 3, 1, 3, 1, 3, 1, 7, 1, 1, 3, 3, 3, 3, 1, 3, 3, 3, 1, 7, 1, 3, 3, 3, 1, 3, 1, 3, 3, 3, 1, 3, 3, 3, 3, 3, 1, 7, 1, 3, 3, 1, 3, 7, 1, 3, 3, 7, 1, 3, 1, 3, 3, 3, 3, 7, 1, 3, 1, 3, 1, 7, 3, 3, 3, 3, 1, 7, 3, 3, 3, 3, 3, 3, 1, 3, 3, 3

Offset: 1

Ilya Gutkovskiy, Jul 21 2019

nonn

Also the number of squarefree divisors > 1.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Unitary Divisor

Cf. A000225, A001221, A001222, A032741, A034444, A070824, A077610, A087893.

nmax = 100; CoefficientList[Series[Sum[MoebiusMu[k]^2 x^k/(1 - x^k), {k, 2, nmax}], {x, 0, nmax}], x] // Rest
Table[2^PrimeNu[n] - 1, {n, 1, 100}]

G.f.: Sum_{k>=2} mu(k)^2*x^k/(1 - x^k).
Dirichlet g.f.: zeta(s)*(zeta(s)/zeta(2*s) - 1).
a(n) = 2^omega(n) - 1.
a(n) = A000225(A001221(n)) = A034444(n) - 1.
Sum_{k=1..n} a(k) ~ 6*n*(log(n) + 2*gamma - 1 - Pi^2/6 - 12*zeta'(2)/Pi^2) / Pi^2, where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Oct 16 2019
a(n) = -1 + Sum_{d|n} mu(d)^2. - Wesley Ivan Hurt, Feb 04 2022

A335269 Numbers for which the harmonic mean of the nontrivial unitary divisors is an integer.

Original entry on oeis.org

228, 345, 1645, 2120, 4025, 4386, 4977, 7725, 8041, 13026, 23881, 24157, 24336, 51925, 88473, 115957, 150161, 169893, 229177, 255041, 278721, 322592, 342637, 377201, 490725, 538625, 656937, 1497517, 1566981, 2132021, 3256261, 3847001, 4646101, 5054221, 5524897

Offset: 1

Amiram Eldar, May 29 2020

nonn

A number m is a term if the set {d|m ; d > 1, d < m, gcd(d, m/d) = 1} is nonempty and the harmonic mean its members is an integer.
The corresponding harmonic means are 8, 9, 15, 16, 25, 12, 21, 15, 33, 12, ...
Equivalently, numbers m such that omega(m) > 1 and (usigma(m)-m-1) | m*(2^omega(m)-2), where usigma is the sum of unitary divisors (A034448), and 2^omega(m)-2 = A034444(m)-2 = A087893 (m) is the number of the nontrivial unitary divisors of m.
The squarefree terms of A247078 are also terms of this sequence.

			228 is a term since the harmonic mean of its nontrivial unitary divisors, {3, 4, 12, 19, 57, 76} is 8 which is an integer.

Amiram Eldar, Table of n, a(n) for n = 1..150

The unitary version of A247078.

Cf. A006086, A034444, A034448, A077610, A087893, A335268, A335270.

usigma[1] = 1; usigma[n_] := Times @@ (1 + Power @@@ FactorInteger[n]); Select[Range[10^6], (omega = PrimeNu[#]) > 1 && Divisible[#*(2^omega - 2), usigma[#] - # - 1] &]

A066341 Sum of distinct terms in n-th row of Fermat's triangle.

Original entry on oeis.org

1, 1, 1, 1, 8, 1, 1, 1, 12, 1, 14, 1, 16, 17, 1, 1, 20, 1, 22, 23, 24, 1, 26, 1, 28, 1, 30, 1, 94, 1, 1, 35, 36, 37, 38, 1, 40, 41, 42, 1, 130, 1, 46, 47, 48, 1, 50, 1, 52, 53, 54, 1, 56, 57, 58, 59, 60, 1, 184, 1, 64, 65, 1, 67, 202, 1, 70, 71, 214, 1, 74, 1, 76, 77, 78, 79, 238, 1

Offset: 2

Wouter Meeussen, Jan 01 2002

			Fermat's triangle (A066340) is {1}, {1, 1}, {1, 0, 1}, {1, 1, 1, 1}, {1, 4, 3, 4, 1}, ... and the distinct terms in each row are {1}, {1}, {0, 1}, {1}, {1, 3, 4}, ... with sums 1, 1, 1, 1, 8, ...

Antti Karttunen, Table of n, a(n) for n = 2..13737 (terms 2..2000 from Muniru A Asiru)

Cf. A066340, A001221, A007875, A087893.

List(List(List([2..80],n->List([1..n-1],m->PowerMod(m,Phi(n),n))),Set),Sum); # Muniru A Asiru, Aug 06 2018

Plus@@@(Union/@Table[ (PowerMod[ #, EulerPhi[ k ], k ])&/@ Range[ k-1 ], {k, 2, 256} ]) or equivalently Table[ w=Length[ FactorInteger[ k ]];(2^(w-1)-1)*k+2^(w-1), {k, 2, 256} ]

A066341(n) = { my(ph = eulerphi(n),m=Map(),t,s=0); for(k=1,n-1,t = ((k^ph)%n); if(!mapisdefined(m, t), s += t; mapput(m,t,t))); (s); }; \\ Antti Karttunen, Aug 06 2018

Conjectures from Ridouane Oudra, Apr 05 2025: (Start)
a(n) = (n+1)*2^(omega(n)-1) - n, where omega(n) = A001221(n).
a(n) = (n+1)*A007875(n) - n.
a(n) = (n/2)*A087893(n) + A007875(n). (End)

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A309307 Number of unitary divisors of n (excluding 1).

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A335269 Numbers for which the harmonic mean of the nontrivial unitary divisors is an integer.

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A066341 Sum of distinct terms in n-th row of Fermat's triangle.

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