A034676 -id:A034676 - cp's OEIS frontend

A254520 Möbius transform of A034676.

1, 4, 9, 12, 25, 36, 49, 48, 72, 100, 121, 108, 169, 196, 225, 192, 289, 288, 361, 300, 441, 484, 529, 432, 600, 676, 648, 588, 841, 900, 961, 768, 1089, 1156, 1225, 864, 1369, 1444, 1521, 1200, 1681, 1764, 1849, 1452, 1800, 2116, 2209, 1728, 2352, 2400

Offset: 1

Álvar Ibeas, Jan 31 2015

The Dirichlet convolution of a(n) and sigma(n) is sigma(n^2).

Álvar Ibeas, Table of n, a(n) for n = 1..10000

Cf. A000290, A000203, A034676, A063659, A065764.

a(n) = n^2*sumdiv(n, d, if (issquare(d), moebius(sqrtint(d))/d)); \\ Michel Marcus, Feb 10 2015

a(n) = n^2 * Sum_{d^2 | n} (moebius(d) / d^2).
Multiplicative with a(p) = p^2; a(p^e) = p^(2e) - p^(2e-2), for e > 1.
Dirichlet g.f.: zeta(s-2) / zeta(2s-2).
Sum_{k=1..n} a(k) ~ 30 * n^3 / Pi^4. - Vaclav Kotesovec, Jan 11 2019
Sum_{k>=1} 1/a(k) = Product_{primes p} (1 + 1/p^2 + 1/(p^2 - 1)^2) = 1.681923034881403168503816690236967736500606659628336043348190538886262268... - Vaclav Kotesovec, Sep 20 2020
a(n) = n*A063659(n). - Ridouane Oudra, Jul 26 2025

A229996 For every positive integer m, let u(m) = (d(1),d(2),...,d(k)) be the unitary divisors of m. The sequence (a(n)) consists of successive numbers m which d(k)/d(1) + d(k-1)/d(2) + ... + d(k)/d(1) is an integer.

Original entry on oeis.org

1, 10, 65, 130, 260, 340, 1105, 1972, 2210, 4420, 8840, 9860, 15650, 20737, 32045, 41474, 44200, 51272, 55250, 64090, 75140, 82948, 103685, 128180, 207370, 207553, 221000, 256360, 352529, 414740, 415106, 512720, 532100, 705058, 759025, 813800, 829480, 830212

Offset: 1

Clark Kimberling, Oct 31 2013

The integer sums d(k)/d(1) + d(k-1)/d(2) + ... + d(k)/d(1) are given by A229999. - Clark Kimberling, Jun 16 2018

Also numbers m such that the sum of the squares of the unitary divisors of m is divisible by m (the unitary version of A046762). - Amiram Eldar, Jun 16 2018

			The first 10 sums: 1, 5/2, 10/3, 17/4, 26/5, 25/3, 50/7, 65/8, 82/9, 13, so that a(1) = 1 and a(10) = 13.

Amiram Eldar, Table of n, a(n) for n = 1..333 (terms below 10^10)

Cf. A034676, A046762, A229994, A077610, A229999.

z = 1000; r[n_] := Select[Divisors[n], GCD[#, n/#] == 1 &]; k[n_] := Length[r[n]];
t[n_] := Table[r[n][[k[n] + 1 - i]]/r[n][[k[1] + i - 1]], {i, 1, k[n]}];
s = Table[Plus @@ t[n], {n, 1, z}]; a[n_] := If[IntegerQ[s[[n]]], 1, 0]; u = Table[a[n], {n, 1, z}]; Flatten[Position[u, 1]]  (* A229996 *)
usigma2[n_] :=  If[n == 1, 1, Times @@ (1 + Power @@@ FactorInteger[n]^2)]; seqQ[n_] := Divisible[usigma2[n], n]; Select[Range[10^6], seqQ] (* Amiram Eldar, Jun 16 2018 *)

is(n) = {my(f = factor(n)); !(prod(i = 1, #f~, f[i,1]^(2*f[i,2]) + 1) % n);} \\ Amiram Eldar, Jun 16 2024

Definition corrected by Clark Kimberling, Jun 16 2018

A366535 The sum of unitary divisors of the exponentially odd numbers (A268335).

Original entry on oeis.org

1, 3, 4, 6, 12, 8, 9, 18, 12, 14, 24, 24, 18, 20, 32, 36, 24, 36, 42, 28, 30, 72, 32, 33, 48, 54, 48, 38, 60, 56, 54, 42, 96, 44, 72, 48, 72, 54, 84, 72, 72, 80, 90, 60, 62, 96, 84, 144, 68, 96, 144, 72, 74, 114, 96, 168, 80, 126, 84, 108, 132, 120, 108, 90, 112

Offset: 1

Amiram Eldar, Oct 12 2023

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

Cf. A034448, A065463, A077610, A268335, A366439, A366534.

Similar sequences: A034676, A366537, A366539.

s[n_] := Module[{f = FactorInteger[n], e}, e = f[[;;, 2]]; If[AllTrue[e, OddQ], Times @@ (1 + Power @@@ f), Nothing]]; s[1] = 1; Array[s, 100]

lista(max) = for(k = 1, max, my(f = factor(k), e = f[, 2], isexpodd = 1); for(i = 1, #e, if(!(e[i] % 2), isexpodd = 0; break)); if(isexpodd, print1(prod(i = 1, #e, 1 + f[i, 1]^e[i]), ", ")));

a(n) = A034448(A268335(n)).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = (zeta(4)/d^2) * Product_{p prime} (1 - 2/p^4 + 1/p^5) = 1.92835521961603199612..., d = A065463 is the asymptotic density of the exponentially odd numbers.
The asymptotic mean of the unitary abundancy index of the exponentially odd numbers: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A268335(k) = c * d = 1.35841479521454692063... .

A286880 Square array A(n,k), n>=0, k>=1, read by antidiagonals, where row n is the sum of n-th powers of unitary divisors of k (divisors d such that gcd(d, k/d) = 1).

Original entry on oeis.org

1, 2, 1, 2, 3, 1, 2, 4, 5, 1, 2, 5, 10, 9, 1, 4, 6, 17, 28, 17, 1, 2, 12, 26, 65, 82, 33, 1, 2, 8, 50, 126, 257, 244, 65, 1, 2, 9, 50, 252, 626, 1025, 730, 129, 1, 4, 10, 65, 344, 1394, 3126, 4097, 2188, 257, 1, 2, 18, 82, 513, 2402, 8052, 15626, 16385, 6562, 513, 1, 4, 12, 130, 730, 4097, 16808, 47450, 78126, 65537, 19684, 1025, 1

Offset: 0

Ilya Gutkovskiy, Aug 02 2017

For row r > 0, Sum_{k=1..n} A(r,k) ~ zeta(r+1) * n^(r+1) / ((r+1) * zeta(r+2)). - Vaclav Kotesovec, May 20 2021

			Square array begins:
1,   2,    2,     2,     2,     4,  ...
1,   3,    4,     5,     6,    12,  ...
1,   5,   10,    17,    26,    50,  ...
1,   9,   28,    65,   126,   252,  ...
1,  17,   82,   257,   626,  1394,  ...
1,  33,  244,  1025,  3126,  8052,  ...

Eric Weisstein's World of Mathematics, Unitary Divisor
Eric Weisstein's World of Mathematics, Unitary Divisor Function
Index entries for sequences related to sums of divisors

Rows n=0-8 give: A034444, A034448, A034676, A034677, A034678, A034679, A034680, A034681, A034682.

Cf. A077610, A109974, A285425.

Dirichlet g.f. of row n: zeta(s)*zeta(s-n)/zeta(2*s-n).

A366537 The sum of unitary divisors of the cubefree numbers (A004709).

Original entry on oeis.org

1, 3, 4, 5, 6, 12, 8, 10, 18, 12, 20, 14, 24, 24, 18, 30, 20, 30, 32, 36, 24, 26, 42, 40, 30, 72, 32, 48, 54, 48, 50, 38, 60, 56, 42, 96, 44, 60, 60, 72, 48, 50, 78, 72, 70, 54, 72, 80, 90, 60, 120, 62, 96, 80, 84, 144, 68, 90, 96, 144, 72, 74, 114, 104, 100, 96

Offset: 1

Amiram Eldar, Oct 12 2023

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

Cf. A002117, A004709, A005117, A034448, A062822, A077610, A358040, A366440, A366536.

Similar sequences: A034676, A366535, A366539.

s[n_] := Module[{f = FactorInteger[n], e}, e = f[[;;, 2]]; If[AllTrue[e, # < 3 &], Times @@ (1 + Power @@@ f), Nothing]]; s[1] = 1; Array[s, 100]

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

from sympy.ntheory.factor_ import udivisor_sigma
from sympy import mobius, integer_nthroot
def A366537(n):
    def f(x): return n+x-sum(mobius(k)*(x//k**3) for k in range(1, integer_nthroot(x,3)[0]+1))
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    return udivisor_sigma(m) # Chai Wah Wu, Aug 05 2024

a(n) = A034448(A004709(n)).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = zeta(3)^2 * Product_{p prime} (1 + 1/p^2 - 2/p^3 + 1/p^4 - 1/p^5) = 1.665430860774244601005... .
The asymptotic mean of the unitary abundancy index of the cubefree numbers: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A004709(k) = c / zeta(3) = 1.38548421160152785073... .

A366539 The sum of unitary divisors of the exponentially 2^n-numbers (A138302).

Original entry on oeis.org

1, 3, 4, 5, 6, 12, 8, 10, 18, 12, 20, 14, 24, 24, 17, 18, 30, 20, 30, 32, 36, 24, 26, 42, 40, 30, 72, 32, 48, 54, 48, 50, 38, 60, 56, 42, 96, 44, 60, 60, 72, 48, 68, 50, 78, 72, 70, 54, 72, 80, 90, 60, 120, 62, 96, 80, 84, 144, 68, 90, 96, 144, 72, 74, 114, 104

Offset: 1

Amiram Eldar, Oct 12 2023

Also the sum 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. A034448, A049417, A077609, A077610, A138302, A271727, A366538.

Similar sequences: A034676, A366535, A366537.

s[n_] := Module[{f = FactorInteger[n], e}, e = f[[;;, 2]]; If[AllTrue[e, # == 2^IntegerExponent[#, 2] &], Times @@ (1 + Power @@@ f), Nothing]]; s[1] = 1; Array[s, 100]

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

a(n) = A034448(A138302(n)).
a(n) = A049417(A138302(n)).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = (1/d^2) * Product_{p prime} f(1/p) = 1.58107339851877782285..., d = A271727 is the asymptotic density of A138302, and f(x) = 1 + x^2 + 2 * Sum_{k>=2} (x^(2^k)-x^(2^k+1)).
The asymptotic mean of the unitary abundancy index of A138302: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A138302(k) = c * d = 1.37948208055913856387... .

A374538 a(n) is the sum of the squares of the unitary divisors of n that are exponentially odd numbers (A268335).

Original entry on oeis.org

1, 5, 10, 1, 26, 50, 50, 65, 1, 130, 122, 10, 170, 250, 260, 1, 290, 5, 362, 26, 500, 610, 530, 650, 1, 850, 730, 50, 842, 1300, 962, 1025, 1220, 1450, 1300, 1, 1370, 1810, 1700, 1690, 1682, 2500, 1850, 122, 26, 2650, 2210, 10, 1, 5, 2900, 170, 2810, 3650, 3172

Offset: 1

Amiram Eldar, Jul 11 2024

The number of unitary divisors of n that are exponentially odd is A055076(n) and their sum is A358346(n).

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

Cf. A000290, A005117, A034676, A055076, A268335, A350389, A358346, A374537.

Cf. A002117, A013661, A183699.

f[p_, e_] := 1 + If[OddQ[e], p^(2*e), 0]; a[n_] := Times @@ f @@@ FactorInteger[n]; a[1] = 1; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, 1 + if(f[i, 2]%2,  f[i, 1]^(2*f[i, 2]), 0));}

a(n) = A034676(A350389(n)).
a(n) >= 1 with equality if and only if n is a square (A000290).
a(n) <= A374537(n) with equality if and only if n is squarefree (A005117).
Multiplicative with a(p^e) = p^(2*e) + 1 if e is odd, and 1 otherwise.
Dirichlet g.f.: zeta(s) * zeta(2*s-4) * Product_{p prime} (1 + 1/p^(s-2) - 1/p^(2*s-4) - 1/p^(2*s-2)).
Sum_{k=1..n} a(k) ~ c * n^3 / 3, where c = zeta(2) * zeta(3) * Product_{p prime} (1 - 2/p^2 + 1/p^3 - 1/p^4 + 1/p^5) = 0.79482441214759383925... .

A130106 A051731 * diagonalized matrix of A063659.

Original entry on oeis.org

1, 1, 2, 1, 0, 3, 1, 2, 0, 3, 1, 0, 0, 0, 5, 1, 2, 3, 0, 0, 6, 1, 0, 0, 0, 0, 0, 7, 1, 2, 0, 3, 0, 0, 0, 6, 1, 0, 3, 0, 0, 0, 0, 0, 8, 1, 2, 0, 0, 5, 0, 0, 0, 0, 10, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 11, 1, 2, 3, 3, 0, 6, 0, 0, 0, 0, 0, 9, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 13, 1, 2, 0, 0, 0, 0, 7, 0, 0, 0, 0, 0, 0, 14

Offset: 1

Gary W. Adamson, May 07 2007

Right border = A063659, (1, 2, 3, 3, 5, 6, 7, 6, 8, 10, ...), the Moebius transform of A001615: (1, 3, 4, 6, 6, 12, 8, 12, 12, ...).
A130106 * (1, 2, 3, ...) = A034676: (1, 5, 10, 17, 26, 50, 50, ...).
A034676^(-1) * (1,2,3,...) = 1/1, 1/2, 2/3, 2/3, 4/5, 2/6, 6/7, 4/6, 6/8, 4/10, ...; where the numerators = phi(n), A000010: (1, 1, 2, 2, 4, 2, 6, 4, ...); and the denominators = A063659, the right border of the triangle: (1, 2, 3, 3, 5, 6, 7, 8, 10, ...).

			First few rows of the triangle:
  1;
  1, 2;
  1, 0, 3;
  1, 2, 0, 3;
  1, 0, 0, 0, 5;
  1, 2, 3, 0, 0, 6;
  1, 0, 0, 0, 0, 0, 7,
  1, 2, 0, 3, 0, 0, 0, 6;
  1, 0, 3, 0, 0, 0, 0, 0, 8;
  ...

Cf. A063659, A001615 (row sums), A051731, A000010.

m = 14;
A051731 = Table[If[Mod[n, k] == 0, 1, 0], {n, m}, {k, m}];
A063659 = Table[Sum[MoebiusMu[GCD[n, k]]^2, {k, n}], {n, m}] // DiagonalMatrix;
M = A051731.A063659;
Table[M[[n, k]], {n, m}, {k, n}] // Flatten (* Jean-François Alcover, Jan 18 2020 *)

Inverse Moebius transform of an infinite lower triangular matrix with A063659, (1, 2, 3, 3, 5, 6, 7, 6, 8, 10, ...) in the main diagonal and the rest zeros.

More terms from Jean-François Alcover, Jan 18 2020

A374539 The sum of the squares of the infinitary divisors of n.

Original entry on oeis.org

1, 5, 10, 17, 26, 50, 50, 85, 82, 130, 122, 170, 170, 250, 260, 257, 290, 410, 362, 442, 500, 610, 530, 850, 626, 850, 820, 850, 842, 1300, 962, 1285, 1220, 1450, 1300, 1394, 1370, 1810, 1700, 2210, 1682, 2500, 1850, 2074, 2132, 2650, 2210, 2570, 2402, 3130, 2900

Offset: 1

Amiram Eldar, Jul 11 2024

Also the sum of the infinitary divisors of n^2.

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

Cf. A049417, A050376, A077609.

Similar sequences: A001157, A034676, A050999, A067558, A076577, A351265, A351307, A351647.

f[p_, e_] := p^(2^Position[Reverse@IntegerDigits[e, 2], ?(# == 1 &)]); a[1] = 1; a[n] := Times @@ (Flatten@(f @@@ FactorInteger[n]) + 1); Array[a, 100]

a(n) = {my(f = factor(n), b); prod(i = 1, #f~, b = binary(2 * f[i, 2]); prod(k=1, #b, if(b[k], 1+f[i, 1]^(2^(#b-k)), 1)));}

from math import prod
from sympy import factorint
def A374539(n): return prod(p**(1<Chai Wah Wu, Jul 11 2024

a(n) = A049417(n^2).
a(n) <= A001157(n), with equality if and only if n is in A036537.
Multiplicative with a(p^e) = Product{k>=1, e_k=1} (p^(2^(k+1)) + 1), where e = Sum_{k} e_k * 2^k is the binary representation of e, i.e., e_k is bit k of e.
Sum_{k=1..n} a(k) ~ c * n^3 / 3, where c = Product_{P} (1 + 1/(P^2*(P+1))) = 1.14142906130350119631..., and P are numbers of the form p^(2^k) where p is prime and k >= 0 (A050376).

A384452 a(n) is the sum of squares of the unitary divisors of n!.

Original entry on oeis.org

1, 5, 50, 650, 16900, 547924, 27396200, 1746641000, 139773881000, 13460683752200, 1642203417768400, 236441876606410000, 40195119023089700000, 7723888546922636420000, 1735183690969722609168800, 444206919394766468845892000, 128820006624482275965308680000, 41737604550102658693597600532800

Offset: 1

Darío Clavijo, Jun 02 2025

nonn

Project Euler, Problem 429: Sum of Squares of Unitary Divisors.
Wikipedia, Legendre's formula.

Cf. A000142, A034676, A064028, A077610.

f[p_, e_] := p^(2*e)+1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n!]; Array[a, 18] (* Amiram Eldar, Jun 02 2025 *)

row(n) = {my(d = divisors(n)); select(x->(gcd(x, n/x)==1), d); } \\ A077610
a(n) = norml2(row(n!)); \\ Michel Marcus, Jun 02 2025

from sympy import nextprime
def f(n,p):
  if n==0: return 0
  return f(n//p,p) + n//p
def a(n):
  s,p = 1, 2
  while p<=n:
    s *= p**(f(n,p)<<1)+1
    p = nextprime(p)
  return s
print([a(n) for n in range(1, 19)])

a(n) = Sum_{d|n!} (d^2 if gcd(d,n!//d) = 1).
a(n) = Product_{p <= n, p prime} (p^(2*f(n,p)))+1 with f(n,p) = f(floor(n/p)) + floor(n/p) and f(0,p) = 0 where f(n,p) is equivalent to the Legendre formula.
a(n) = A034676(n!).

cp's OEIS Frontend

A254520 Möbius transform of A034676.

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A229996 For every positive integer m, let u(m) = (d(1),d(2),...,d(k)) be the unitary divisors of m. The sequence (a(n)) consists of successive numbers m which d(k)/d(1) + d(k-1)/d(2) + ... + d(k)/d(1) is an integer.

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A366535 The sum of unitary divisors of the exponentially odd numbers (A268335).

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A286880 Square array A(n,k), n>=0, k>=1, read by antidiagonals, where row n is the sum of n-th powers of unitary divisors of k (divisors d such that gcd(d, k/d) = 1).

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A366537 The sum of unitary divisors of the cubefree numbers (A004709).

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A366539 The sum of unitary divisors of the exponentially 2^n-numbers (A138302).

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A374538 a(n) is the sum of the squares of the unitary divisors of n that are exponentially odd numbers (A268335).

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A130106 A051731 * diagonalized matrix of A063659.

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