A191414 -id:A191414 - cp's OEIS frontend

A034448 usigma(n) = sum of unitary divisors of n (divisors d such that gcd(d, n/d)=1); also called UnitarySigma(n).

1, 3, 4, 5, 6, 12, 8, 9, 10, 18, 12, 20, 14, 24, 24, 17, 18, 30, 20, 30, 32, 36, 24, 36, 26, 42, 28, 40, 30, 72, 32, 33, 48, 54, 48, 50, 38, 60, 56, 54, 42, 96, 44, 60, 60, 72, 48, 68, 50, 78, 72, 70, 54, 84, 72, 72, 80, 90, 60, 120, 62, 96, 80, 65, 84, 144, 68, 90, 96, 144

Offset: 1

N. J. A. Sloane, Dec 11 1999

Row sums of the triangle in A077610. - Reinhard Zumkeller, Feb 12 2002

Multiplicative with a(p^e) = p^e+1 for e>0. - Franklin T. Adams-Watters, Sep 11 2005

			Unitary divisors of 12 are 1, 3, 4, 12. Or, 12=3*2^2 hence usigma(12)=(3+1)*(2^2+1)=20.

James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 147.

T. D. Noe, Table of n, a(n) for n = 1..10000
Octavio A. Agustín-Aquino, Prime injections and quasipolarities, Matematiche 69 (2014) 159-168
Steven R. Finch, Unitarism and Infinitarism, February 25, 2004. [Cached copy, with permission of the author]
Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 50.
Carl Pomerance and Hee-Sung Yang, Variant of a theorem of Erdős on the sum-of-proper-divisors function, Math. Comp., to appear (2014).
Tim Trudgian, The sum of the unitary divisor function, Publications de l'Institut Mathématique 2015 Vol. 97, Issue 111, pp. 175-180.
Eric Weisstein's World of Mathematics, Unitary Divisor Function
Wikipedia, Unitary divisor

Cf. A000203, A034444, A034460, A047994, A048250, A064000, A064609, A191414.
Cf. A063937 (squares > 1).
Cf. A188999, A301981, A301982.

a034448 = sum . a077610_row  -- Reinhard Zumkeller, Feb 12 2012
(Python 3.8+)
from math import prod
from sympy import factorint
def A034448(n): return prod(p**e+1 for p, e in factorint(n).items()) # Chai Wah Wu, Jun 20 2021

A034448 := proc(n) local ans, i:ans := 1: for i from 1 to nops(ifactors(n)[ 2 ]) do ans := ans*(1+ifactors(n)[ 2 ][ i ][ 1 ]^ifactors(n)[ 2 ] [ i ] [ 2 ]): od: RETURN(ans) end:
a := proc(n) local i; numtheory[divisors](n); select(d -> igcd(d,n/d)=1, %); add(i,i=%) end; # Peter Luschny, May 03 2009

usigma[n_] := Block[{d = Divisors[n]}, Plus @@ Select[d, GCD[ #, n/# ] == 1 &]]; Table[ usigma[n], {n, 71}] (* Robert G. Wilson v, Aug 28 2004 *)
Table[DivisorSum[n, # &, CoprimeQ[#, n/#] &], {n, 70}] (* Michael De Vlieger, Mar 01 2017 *)
usigma[n_] := If[n == 1, 1, Times @@ (1 + Power @@@ FactorInteger[n])]; Array[usigma, 100] (* faster since avoids generating divisors, Giovanni Resta, Apr 23 2017 *)

A034448(n)=sumdiv(n,d,if(gcd(d,n/d)==1,d)) \\ Rick L. Shepherd

A034448(n) = {my(f=factorint(n)); prod(k=1, #f[,2], f[k,1]^f[k,2]+1)} \\ Andrew Lelechenko, Apr 22 2014

a(n)=sumdivmult(n,d,if(gcd(d,n/d)==1,d)) \\ Charles R Greathouse IV, Sep 09 2014

If n = Product p_i^e_i, usigma(n) = Product (p_i^e_i + 1). - Vladeta Jovovic, Apr 19 2001
Dirichlet generating function: zeta(s)*zeta(s-1)/zeta(2s-1). - Franklin T. Adams-Watters, Sep 11 2005
Conjecture: a(n) = sigma(n^2/rad(n))/sigma(n/rad(n)), where sigma = A000203 and rad = A007947. - Velin Yanev, Aug 20 2017
This conjecture is easily verified since all the functions involved are multiplicative and proving it for prime powers is straightforward. - Juan José Alba González, Mar 19 2021
From Amiram Eldar, May 29 2020: (Start)
Sum_{d|n, gcd(d, n/d) = 1} a(d) * (-1)^omega(n/d) = n.
a(n) <= sigma(n) = A000203(n), with equality if and only if n is squarefree (A005117). (End)
Sum_{k=1..n} a(k) ~ Pi^2 * n^2 / (12*zeta(3)). - Vaclav Kotesovec, May 20 2021
a(n) = uphi(n^2)/uphi(n) = A191414(n)/uphi(n), where uphi(n) = A047994(n). - Amiram Eldar, Sep 21 2024

More terms from Erich Friedman

A350388 a(n) is the largest unitary divisor of n that is a square.

Original entry on oeis.org

1, 1, 1, 4, 1, 1, 1, 1, 9, 1, 1, 4, 1, 1, 1, 16, 1, 9, 1, 4, 1, 1, 1, 1, 25, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 36, 1, 1, 1, 1, 1, 1, 1, 4, 9, 1, 1, 16, 49, 25, 1, 4, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 9, 64, 1, 1, 1, 4, 1, 1, 1, 9, 1, 1, 25, 4, 1, 1, 1, 16, 81, 1, 1, 4

Offset: 1

Amiram Eldar, Dec 28 2021

First differs from A056623 at n = 32.

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

Cf. A000290, A001222, A008833, A056623, A071974, A076479, A191414, A268335, A335275, A350386, A350389.

f[p_, e_] := If[EvenQ[e], p^e, 1]; 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]%2, 1, f[i,1]^f[i,2]));} \\ Amiram Eldar, Oct 01 2023

Multiplicative with a(p^e) = p^e if e is even and 1 otherwise.
a(n) = n/A350389(n).
a(n) = A071974(n)^2.
a(n) = A008833(n) if and only if n is in A335275.
A001222(a(n)) = A350386(n).
a(n) = 1 if and only if n is an exponentially odd number (A268335).
a(n) = n if and only if n is a positive square (A000290 \ {0}).
Sum_{k=1..n} a(k) ~ c * n^(3/2), where c = (1/3) * Product_{p prime} (1 + sqrt(p)/(1 + p + p^2)) = 0.59317173657411718128... [updated Oct 16 2022]
Dirichlet g.f.: zeta(2*s-2) * zeta(2*s) * Product_{p prime} (1 + 1/p^s - 1/p^(2*s) - 1/p^(3*s-2)). - Amiram Eldar, Oct 01 2023
Sum_{d|n, gcd(d, n/d) == 1} A076479(d) * a(n/d) = A191414(sqrt(n)) if n is a square, and 0 otherwise. - Amiram Eldar, Jun 01 2025

A382661 The unitary Jordan totient function applied to the exponentially odd numbers (A268335).

Original entry on oeis.org

1, 3, 8, 24, 24, 48, 63, 72, 120, 168, 144, 192, 288, 360, 384, 360, 528, 504, 504, 728, 840, 576, 960, 1023, 960, 864, 1152, 1368, 1080, 1344, 1512, 1680, 1152, 1848, 1584, 2208, 2304, 2808, 2184, 2880, 3024, 2880, 2520, 3480, 3720, 2880, 4032, 2880, 4488, 4224

Offset: 1

Amiram Eldar, Apr 02 2025

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

Cf. A013664, A065463, A191414, A268335.

Similar sequences: A358346, A363825, A366438, A366439, A366534, A366535, A367417, A368711, A368979, A374456, A382660.

f[p_, e_] := p^(2*e)-1; uj2[1] = 1; uj2[n_] := Times @@ f @@@ FactorInteger[n]; expOddQ[n_] := AllTrue[FactorInteger[n][[;; , 2]], OddQ]; uj2 /@ Select[Range[100], expOddQ]

uj2(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i, 1]^(2*f[i, 2])-1);}
isexpodd(n) = {my(f = factor(n)); for(i=1, #f~, if(!(f[i, 2] % 2), return (0))); 1;}
list(lim) = apply(uj2, select(isexpodd, vector(lim, i, i)));

a(n) = A191414(A268335(n)).

Sum_{k=1..n} a(k) ~ c * n^3, where c = (zeta(6)/(3*d^3)) * Product_{p prime} (1 - 1/p^2 + 1/p^5 - 2/p^6 + 1/p^7) = 0.59726984314764530141..., and d = A065463 is the asymptotic density of the exponentially odd numbers.

A382663 The unitary Jordan totient function applied to the cubefree numbers (A004709).

Original entry on oeis.org

1, 3, 8, 15, 24, 24, 48, 80, 72, 120, 120, 168, 144, 192, 288, 240, 360, 360, 384, 360, 528, 624, 504, 720, 840, 576, 960, 960, 864, 1152, 1200, 1368, 1080, 1344, 1680, 1152, 1848, 1800, 1920, 1584, 2208, 2400, 1872, 2304, 2520, 2808, 2880, 2880, 2520, 3480, 2880

Offset: 1

Amiram Eldar, Apr 02 2025

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

Cf. A002117, A004709, A191414.

Similar sequences: A366440, A366536, A366537, A368712, A368779, A369889, A376365, A376366, A379717, A382419, A382662.

f[p_, e_] := p^(2*e)-1; uj2[1] = 1; uj2[n_] := Times @@ f @@@ FactorInteger[n]; cubeFreeQ[n_] := AllTrue[FactorInteger[n][[;; , 2]], # < 3 &]; uj2 /@ Select[Range[100], cubeFreeQ]

uj2(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i, 1]^(2*f[i, 2])-1); }
iscubefree(n) = {my(f = factor(n)); for(i=1, #f~, if(f[i, 2] > 2, return (0))); 1; }
list(lim) = apply(uj2, select(iscubefree, vector(lim, i, i)));

a(n) = A191414(A004709(n)).

Sum_{k=1..n} a(k) ~ c * n^3, where c = (zeta(3)^3/3) * Product_{p prime} (1 - 2/p^3 + 1/p^4 - 1/p^6 + 1/p^7) = 0.42656661743049439763... .

A297365 Numbers k such that uphi(k)usigma(k) = uphi(k+1)usigma(k+1), where uphi is the unitary totient function (A047994) and usigma the sum of unitary divisors (A034448).

Original entry on oeis.org

5, 11, 19, 71, 247, 271, 991, 2232, 6200, 8271, 10295, 16744, 18496, 18576, 25704, 26656, 102175, 122607, 166624, 225939, 301103, 747967, 7237384, 7302592, 15760224, 21770800, 28121184, 72967087, 98617024, 104577848, 173859007, 253496176, 335610184, 371191600

Offset: 1

Amiram Eldar, Dec 29 2017

nonn

Equivalently, numbers k such that A191414(k) = A191414(k+1). - Amiram Eldar, Nov 09 2023

			11 is in the sequence since uphi(11) * usigma(11) = 10 * 12 = uphi(12) * usigma(12) = 6 * 20 = 120.

Amiram Eldar, Table of n, a(n) for n = 1..47 (terms below 10^11)

The unitary version of A244439.

Cf. A034448, A047994, A191414.

usigma[n_] := If[n == 1, 1, Times @@ (1 + Power @@@ FactorInteger[n])];
uphi[n_] := (Times @@ (Table[#[[1]]^#[[2]] - 1, {1}] & /@ FactorInteger[n]))[[1]]; u[n_] := uphi[n]*usigma[n]; aQ[n_] := u[n] == u[n + 1]; Select[Range[10^6], aQ]

A191414(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i, 1]^(2*f[i, 2])-1); }
lista(kmax) = {my(a1 = 1, a2); for(k = 2, kmax, a2 = A191414(k); if(a1 == a2, print1(k-1, ", ")); a1 = a2); } \\ Amiram Eldar, Nov 09 2023

A348011 a(n) = phi(n^2) * Sum_{d|n} 2^omega(d) / d.

Original entry on oeis.org

1, 4, 10, 20, 28, 40, 54, 88, 102, 112, 130, 200, 180, 216, 280, 368, 304, 408, 378, 560, 540, 520, 550, 880, 740, 720, 954, 1080, 868, 1120, 990, 1504, 1300, 1216, 1512, 2040, 1404, 1512, 1800, 2464, 1720, 2160, 1890, 2600, 2856, 2200, 2254, 3680, 2730, 2960

Offset: 1

Ilya Gutkovskiy, Sep 24 2021

Cf. A000010, A001221, A002618, A007434, A034444, A047994, A060648, A062355, A069097, A191414, A341772.

Table[EulerPhi[n^2] DivisorSum[n, 2^PrimeNu[#]/# &], {n, 50}]
f[p_, e_] := p^(e - 1) ((p + 1) p^e - 2); a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 50]

a(n) = eulerphi(n^2)*sumdiv(n, d, 2^omega(d)/d); \\ Michel Marcus, Sep 24 2021

Multiplicative with a(p^e) = p^(e-1) * ((p + 1) * p^e - 2).
a(n) = Sum_{k=1..n, gcd(n,k) = 1} gcd(n,k-1)^2.
a(n) = Sum_{k=1..n} uphi(gcd(n,k)^2).
a(n) = Sum_{d|n} phi(n/d) * uphi(d^2).
Sum_{k=1..n} a(k) ~ c * n^3, where c = (Pi^2/18) * Product_{p prime} (1 - 2/p^3 + 1/p^4) = 0.4083249979... . - Amiram Eldar, Nov 05 2022

cp's OEIS Frontend

A034448 usigma(n) = sum of unitary divisors of n (divisors d such that gcd(d, n/d)=1); also called UnitarySigma(n).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

PARI

PARI

Formula

Extensions

A350388 a(n) is the largest unitary divisor of n that is a square.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A382661 The unitary Jordan totient function applied to the exponentially odd numbers (A268335).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A382663 The unitary Jordan totient function applied to the cubefree numbers (A004709).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A297365 Numbers k such that uphi(k)*usigma(k) = uphi(k+1)*usigma(k+1), where uphi is the unitary totient function (A047994) and usigma the sum of unitary divisors (A034448).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A348011 a(n) = phi(n^2) * Sum_{d|n} 2^omega(d) / d.

Views

Author

Keywords

Crossrefs

Programs

Mathematica

PARI

Formula

A297365 Numbers k such that uphi(k)usigma(k) = uphi(k+1)usigma(k+1), where uphi is the unitary totient function (A047994) and usigma the sum of unitary divisors (A034448).