A077610 -id:A077610 - cp's OEIS frontend

A222085 Sum of the least divisors of n whose LCM is equal to n.

1, 3, 4, 7, 6, 6, 8, 15, 13, 8, 12, 10, 14, 10, 9, 31, 18, 21, 20, 12, 11, 14, 24, 24, 31, 16, 40, 14, 30, 11, 32, 63, 15, 20, 13, 25, 38, 22, 17, 20, 42, 19, 44, 18, 18, 26, 48, 52, 57, 43, 21, 20, 54, 66, 17, 22, 23, 32, 60, 15, 62, 34, 20, 127, 19, 23, 68

Offset: 1

Paolo P. Lava, Feb 11 2013

nonn

			The divisors of 20 are 1, 2, 4, 5, 10, 20 while the least divisors of 20 whose LCM is equal to 20 are 1, 2, 4, 5. Then a(20) = 1+2+4+5 = 12.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Paolo P. Lava)

Cf. A000005, A000961, A001358, A003418, A005179, A024619, A034444, A077610, A222084.

with(numtheory);
A222085:=proc(q)
local a,b,c,j,n,v; print(1);
for n from 2 to q do a:=ifactors(n)[2]; b:=nops(a); c:=0;
  for j from 1 to b do if a[j][1]^a[j][2]>c then c:=a[j][1]^a[j][2]; fi; od;
  a:=op(sort([op(divisors(n))])); b:=nops(divisors(n)); v:=0;
  for j from 1 to b do v:=v+a[j]; if a[j]=c then break; fi; od; print(v);
od; end:
A222085(100000000);

s[n_] := Module[{sum=0, L=1}, Do[sum+=d; L = LCM[L, d]; If[L == n, Break[]], {d, Divisors[n]}]; sum]; Array[s, 67] (* Amiram Eldar, Nov 05 2019 *)

a(n)=my(s,L=1);fordiv(n,d,s+=d;L=lcm(L,d);if(L==n,return(s))) \\ Charles R Greathouse IV, Feb 14 2013

A305995 Rectangular array read by downward antidiagonals; row n consists of the numbers m such that n is the denominator of d(k)/d(1) + d(k-1)/d(2) + ... + d(k)/d(1), where d(1),d(2),...,d(k) are the unitary divisors of m.

Original entry on oeis.org

1, 10, 2, 65, 68, 3, 130, 520, 6, 4, 260, 1768, 15, 40, 5, 340, 2600, 30, 104, 50, 12, 1105, 6760, 60, 1040, 1700, 120, 7, 1972, 17680, 150, 20560, 3250, 312, 14, 8, 2210, 62600, 195, 35360, 7825, 600, 35, 2080, 9, 4420, 165896, 204, 85280, 27625, 3120, 70, 4112, 18, 20

Offset: 1

Clark Kimberling, Jun 16 2018

Every positive integer occurs exactly once, so that as a sequence, this is a permutation of the positive integers. The numbers in row n are divisible by n; see A305996 for the quotients.

			Northwest corner:
   1    10    65   130    260     340    1105
   2    68   520  1768   2600    6760   17680
   3     6    15    30     60     150     195
   4    40   104  1040  20560   35360   85280
   5    50  1700  3250   7825   27625   31300
  12   120   312   600   3120   61680  106080
   7    14    35    70    140     175     350
   8  2080  4112  6560  32800   38048   52000
   9    18    90   369    585     612     738

Cf. A077610, A229994, A229996, A229997, A229998, A229998, A305996.

t[n_] := Table[r[n][[k[n] + 1 - i]]/r[n][[k[1] + i - 1]], {i, 1, k[n]}];
s = Table[Total[t[n]], {n, 1, z}]; a[n_] := If[IntegerQ[s[[n]]], 1, 0];
d = Denominator[s];
row[n_] := Flatten[Position[d, n]]
TableForm[Table[row[n], {n, 1, 10}]]  (* A305995 array *)
r1[n_, k_] := row[n][[k]]; zz = 10;
Flatten[Table[r1[n - k + 1, k], {n, zz}, {k, n, 1, -1}]]  (* A305995 sequence *)

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

Original entry on oeis.org

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

A328328 Unitary admirable numbers: numbers k such that there is a proper unitary divisor d of k such that usigma(k) - 2d = 2k, where usigma is the sum of unitary divisors function (A034448).

Original entry on oeis.org

30, 42, 66, 70, 78, 102, 114, 138, 150, 174, 186, 222, 246, 258, 282, 294, 318, 354, 366, 402, 420, 426, 438, 474, 498, 534, 582, 606, 618, 630, 642, 654, 660, 678, 726, 750, 762, 780, 786, 822, 834, 840, 894, 906, 942, 978, 990, 1002, 1014, 1020, 1038, 1074, 1086

Offset: 1

Amiram Eldar, Oct 12 2019

nonn

Differs from A302574(n) at n >= 30.
Equivalently, numbers that equal to the sum of their proper unitary divisors, with one of them taken with a minus sign.
The unitary version of A111592.
The squarefree terms are also admirable numbers (A111592). The nonsquarefree terms are 150, 294, 420, 630, 660, 726, 750, 780, 840, 990, ...
The unitary abundant numbers (A034683) that are not unitary admirable numbers are: 210, 330, 390, 462, 510, 546, 570, 690, 714, 770, 798, 858, 870, 910, 924, 930, 966, ...

			150 is in the sequence since 150 = 1 + 2 + 3 - 6 + 25 + 50 + 75 is the sum of its proper unitary divisors with one of them, 6, taken with a minus sign.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A034448, A077610, A111592, A302574.

Subsequence of A034683 and A290466.

usigma[1] = 1; usigma[n_] := Times @@ (1 + Power @@@ FactorInteger[n]); aQ[n_] := (ab = usigma[n] - 2n) > 0 && EvenQ[ab] && ab/2 < n && Divisible[n, ab/2] && CoprimeQ[2*n/ab, ab/2]; Select[Range[1086], aQ]

A348001 Number of distinct values obtained when the unitary totient function (A047994) is applied to the unitary divisors of n.

Original entry on oeis.org

1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 4, 2, 2, 4, 2, 2, 2, 2, 4, 4, 2, 2, 4, 2, 2, 2, 4, 2, 4, 2, 2, 4, 2, 4, 4, 2, 2, 4, 4, 2, 4, 2, 4, 4, 2, 2, 4, 2, 2, 4, 4, 2, 2, 4, 4, 4, 2, 2, 8, 2, 2, 4, 2, 4, 4, 2, 4, 4, 4, 2, 4, 2, 2, 4, 4, 4, 4, 2, 4, 2, 2, 2, 7, 4, 2, 4

Offset: 1

Amiram Eldar, Sep 23 2021

nonn

			n = 6 has four unitary divisors: 1, 2, 3 and 6. Applying A047994 to these gives 1, 1, 2 and 2, with just 2 distinct values, thus a(6) = 2.
n = 12 has four unitary divisors: 1, 3, 4 and 12. Applying A047994 to these gives 4 distinct values, 1, 2, 3 and 6, thus a(12) = 4.

Amiram Eldar, Table of n, a(n) for n = 1..10000

The unitary version of A319696.

Cf. A047994, A077610, A278568.

f[p_, e_] := p^e - 1; uphi[1] = 1; uphi[n_] := Times @@ f @@@ FactorInteger[n]; a[n_] := Length @ Union[uphi /@ Select[Divisors[n], CoprimeQ[#, n/#] &]]; Array[a,100]

a(2^e) = 2 for e > 1.
a(p^e) = 2 for an odd prime p and e > 0.
a(n) >= omega(n), with equality if and only if n is in A278568.

A385198 The number of integers k from 1 to n such that the greatest divisor of k that is a unitary divisor of n is a prime power (A246655).

Original entry on oeis.org

0, 1, 1, 1, 1, 3, 1, 1, 1, 5, 1, 5, 1, 7, 6, 1, 1, 9, 1, 7, 8, 11, 1, 9, 1, 13, 1, 9, 1, 14, 1, 1, 12, 17, 10, 11, 1, 19, 14, 11, 1, 20, 1, 13, 12, 23, 1, 17, 1, 25, 18, 15, 1, 27, 14, 13, 20, 29, 1, 26, 1, 31, 14, 1, 16, 32, 1, 19, 24, 34, 1, 15, 1, 37, 26, 21

Offset: 1

Amiram Eldar, Jun 21 2025

			For n = 6, the greatest divisor of k that is a unitary divisor of 6 for k = 1 to 6 is 1, 2, 3, 2, 1 and 6, respectively. 3 of the values are prime powers, and therefore a(6) = 3.

Amiram Eldar, Table of n, a(n) for n = 1..10000

The unitary analog of A116512.
Cf. A047994, A065463, A069513, A077610, A246655, A384047.
The number of integers k from 1 to n such that the greatest divisor of k that is a unitary divisor of n is: A047994 (1), A384048 (squarefree), A384049 (cubefree), A384050 (powerful), A384051 (cubefull), A384052 (square), A384053 (cube), A384054 (exponentially odd), A384055 (odd), A384056 (power of 2), A384057 (3-smooth), A384058 (5-rough), A385195 (1 or 2), A385196 (prime), A385197 (noncomposite), this sequence (prime power), A385199 (1 or prime power).

f[p_, e_] := p^e - 1; a[1] = 0; a[n_] := Module[{fct = FactorInteger[n]}, (Times @@ f @@@ fct)*(Total[1/f @@@ fct])]; Array[a, 100]

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

The unitary convolution of A047994 (the unitary totient phi) with A069513 (the characteristic function of prime powers): a(n) = Sum_{d | n, gcd(d, n/d) == 1} A047994(d) * A069513(n/d).
a(n) = uphi(n) * Sum_{p^e || n} (1/(p^e-1)), where uphi = A047994, and p^e || n denotes that the prime power p^e unitarily divides n (i.e., p^e divides n but p^(e+1) does not divide n).
a(n) = A385199(n) - A047994(n).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = c1 * c2 = 0.26256423811374124133..., c1 = Product_{p prime}(1 - 1/(p*(p+1))) = A065463, and c2 = Sum_{p prime}(1/(p^2+p-1)) = 0.37272644617447080939... .

A334019 Sum of unitary divisors of n that are smaller than sqrt(n).

Original entry on oeis.org

0, 1, 1, 1, 1, 3, 1, 1, 1, 3, 1, 4, 1, 3, 4, 1, 1, 3, 1, 5, 4, 3, 1, 4, 1, 3, 1, 5, 1, 11, 1, 1, 4, 3, 6, 5, 1, 3, 4, 6, 1, 12, 1, 5, 6, 3, 1, 4, 1, 3, 4, 5, 1, 3, 6, 8, 4, 3, 1, 13, 1, 3, 8, 1, 6, 12, 1, 5, 4, 15, 1, 9, 1, 3, 4, 5, 8, 12, 1, 6, 1, 3, 1, 15, 6

Offset: 1

Amiram Eldar, Apr 12 2020

nonn

			The unitary divisors of 12 are {1, 3, 4, 12}, 1 and 3 are smaller than sqrt(12) and their sum is 1 + 3 = 4, hence a(12) = 4.

Jinyuan Wang, Table of n, a(n) for n = 1..10000

Cf. A034448, A070039, A077610, A334023.

a[n_] := DivisorSum[n, # &, #^2 < n && CoprimeQ[#, n/#] &]; Array[a, 100]

a(n) = sumdiv(n, d, if(gcd(d, n/d)==1 && dJinyuan Wang, Apr 12 2020

a(n) = A070039(n) for squarefree numbers (A005117) or squares of primes (A001248).

A366074 The number of "Fermi-Dirac primes" (A050376) that are unitary divisors of n.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Sep 28 2023

First differs from A293439 at n = 128.

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

Cf. A050376, A064547, A077610, A077761, A209229, A366073.

f[p_, e_] := If[e == 2^IntegerExponent[e, 2], 1, 0]; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = vecsum(apply(x -> (x == 1 << valuation(x, 2)), factor(n)[, 2]));

Additive with a(p^e) = A209229(e).

Sum_{k=1..n} a(k) ~ n * (log(log(n)) + B + C), where B is Mertens's constant (A077761) and C = -P(2) + Sum_{k>=1} (P(2^k) - P(2^k+1)) = -0.13145993422430119364..., where P(s) is the prime zeta function.

A366538 The number of unitary divisors of the exponentially 2^n-numbers (A138302).

Original entry on oeis.org

1, 2, 2, 2, 2, 4, 2, 2, 4, 2, 4, 2, 4, 4, 2, 2, 4, 2, 4, 4, 4, 2, 2, 4, 4, 2, 8, 2, 4, 4, 4, 4, 2, 4, 4, 2, 8, 2, 4, 4, 4, 2, 4, 2, 4, 4, 4, 2, 4, 4, 4, 2, 8, 2, 4, 4, 4, 8, 2, 4, 4, 8, 2, 2, 4, 4, 4, 4, 8, 2, 4, 2, 4, 2, 8, 4, 4, 4, 2, 8, 4, 4, 4, 4, 4, 2, 4

Offset: 1

Amiram Eldar, Oct 12 2023

Also, the number of infinitary divisors of the terms of A138302, since A138302 is also the sequence of numbers whose sets of unitary divisors (A077610) and infinitary divisors (A077609) coincide.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A034444, A037445, A077609, A077610, A138302, A366539.

Similar sequences: A366534, A366536.

f[n_] := Module[{e = FactorInteger[n][[;;, 2]]}, If[AllTrue[e, # == 2^IntegerExponent[#, 2] &], 2^Length[e], Nothing]]; f[1] = 1; Array[f, 150]

lista(max) = for(k = 1, max, my(e = factor(k)[, 2], is = 1); for(i = 1, #e, if(e[i] >> valuation(e[i], 2) > 1, is = 0; break)); if(is, print1(2^#e, ", ")));

a(n) = A034444(A138302(n)).

a(n) = A037445(A138302(n)).

A368167 The largest unitary divisor of n that is a cubefull exponentially odd number (A335988).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 27, 1, 1, 1, 1, 32, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 27, 1, 8, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Amiram Eldar, Dec 14 2023

First differs from A056191 and A366126 at n = 32, and from A367513 at n = 64.
Also, the largest exponentially odd unitary divisor of the powerful part on n.
Also, the powerful part of the largest exponentially odd unitary divisor of n.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A057521, A077610, A335275, A335988, A350389, A368168, A368169.

Cf. A056191, A366126, A367513.

f[p_, e_] := If[e == 1 || EvenQ[e], 1, p^e]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i, 2] == 1 || !(f[i, 2]%2), 1, f[i, 1]^f[i, 2]));}

Multiplicative with a(p^e) = p^e if e is odd that is larger than 1, and 1 otherwise.
a(n) = A350389(A057521(n)).
a(n) = A057521(A350389(n)).
a(n) >= 1, with equality if and only if n is in A335275.
a(n) <= n, with equality if and only if n is in A335988.

cp's OEIS Frontend

A222085 Sum of the least divisors of n whose LCM is equal to n.

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A305995 Rectangular array read by downward antidiagonals; row n consists of the numbers m such that n is the denominator of d(k)/d(1) + d(k-1)/d(2) + ... + d(k)/d(1), where d(1),d(2),...,d(k) are the unitary divisors of m.

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

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A328328 Unitary admirable numbers: numbers k such that there is a proper unitary divisor d of k such that usigma(k) - 2d = 2k, where usigma is the sum of unitary divisors function (A034448).

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A348001 Number of distinct values obtained when the unitary totient function (A047994) is applied to the unitary divisors of n.

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A385198 The number of integers k from 1 to n such that the greatest divisor of k that is a unitary divisor of n is a prime power (A246655).

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A334019 Sum of unitary divisors of n that are smaller than sqrt(n).

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A366074 The number of "Fermi-Dirac primes" (A050376) that are unitary divisors of n.

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