A062758 -id:A062758 - cp's OEIS frontend

A174939 a(n) = Sum_{k<=n} A007955(k) * A007955(k) = Sum_{k<=n} A007955(k)^2, where A007955(m) = product of divisors of m.

1, 5, 14, 78, 103, 1399, 1448, 5544, 6273, 16273, 16394, 3002378, 3002547, 3040963, 3091588, 4140164, 4140453, 38152677, 38153038, 102153038, 102347519, 102581775, 102582304, 110177896480, 110177912105, 110178369081, 110178900522, 110660790826, 110660791667

Offset: 1

Jaroslav Krizek, Apr 02 2010

nonn

			For n = 4, A007955(n) = b(n): a(4) = b(1)^2 + b(2)^2 + b(3)^2 + b(4)^2 = 1^2 + 2^2 + 3^2 + 8^2 = 78.

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Cf. A007955, A062758.

Accumulate@ Array[#^DivisorSigma[0, #] &, 29] (* Michael De Vlieger, May 03 2022 *)

a(n) = sum(k=1, n, k^numdiv(k)); \\ Michel Marcus, May 03 2022

from sympy import divisor_count
from itertools import count, islice
def agen():
    an = 1
    for k in count(2):
        yield an
        an += k**divisor_count(k)
print(list(islice(agen(), 29))) # Michael S. Branicky, May 03 2022

a(n) = Sum_{k=1..n} A062758(k). - Michel Marcus, May 03 2022

a(27) and beyond from Michael S. Branicky, May 03 2022

A236287 a(n) = sigma(n)^tau(n), where tau(n) = A000005(n) = the number of divisors of n and sigma(n) = A000203(n) = the sum of divisors of n.

Original entry on oeis.org

1, 9, 16, 343, 36, 20736, 64, 50625, 2197, 104976, 144, 481890304, 196, 331776, 331776, 28629151, 324, 3518743761, 400, 5489031744, 1048576, 1679616, 576, 167961600000000, 29791, 3111696, 2560000, 30840979456, 900, 722204136308736, 1024, 62523502209, 5308416

Offset: 1

Jaroslav Krizek, Jan 23 2014

nonn

			a(4) = sigma(4)^tau(4) = 7^3 = 343.

Jaroslav Krizek, Table of n, a(n) for n = 1..100

Cf. A000005 (tau(n)), A000203 (sigma(n)), A062758 (n^tau(n)), A217872 (sigma(n)^n), A236285 (tau(n)^sigma(n)), A236286.

Table[DivisorSigma[1, n]^DivisorSigma[0, n], {n, 1000}]

s=[]; for(n=1, 40, s=concat(s, sigma(n, 1)^sigma(n, 0))); s \\ Colin Barker, Jan 24 2014

a(n) = A000203(n)^A000005(n).

A290480 Product of proper unitary divisors of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 6, 1, 1, 1, 10, 1, 12, 1, 14, 15, 1, 1, 18, 1, 20, 21, 22, 1, 24, 1, 26, 1, 28, 1, 27000, 1, 1, 33, 34, 35, 36, 1, 38, 39, 40, 1, 74088, 1, 44, 45, 46, 1, 48, 1, 50, 51, 52, 1, 54, 55, 56, 57, 58, 1, 216000, 1, 62, 63, 1, 65, 287496, 1, 68, 69, 343000, 1, 72, 1, 74, 75, 76, 77, 474552, 1, 80

Offset: 1

Ilya Gutkovskiy, Aug 03 2017

nonn

			a(12) = 12 because 12 has 6 divisors {1, 2, 3, 4, 6, 12} among which 3 are proper unitary {1, 3, 4} and 1*3*4 = 12.

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

Cf. A001221, A007774 (fixed points), A007955, A007956, A034460, A061537, A062758, A063919, A078599, A087652, A126192, A136655, A183091.

with(numtheory):
a:= n-> mul(d, d=select(x-> igcd(x, n/x)=1, divisors(n) minus {n})):
seq(a(n), n=1..80);  # Alois P. Heinz, Aug 03 2017

Table[Product[d, {d, Select[Divisors[n], GCD[#, n/#] == 1 &]}]/n, {n, 80}]
Table[n^(2^(PrimeNu[n] - 1) - 1), {n, 80}]

A290480(n) = if(1==n,n,n^(2^(omega(n)-1)-1)); \\ Antti Karttunen, Aug 06 2018

from sympy import divisors, gcd, prod
def a(n): return prod(d for d in divisors(n) if gcd(d, n//d) == 1)//n
print([a(n) for n in range(1, 51)]) # Indranil Ghosh, Aug 04 2017

a(n) = A061537(n)/n.
a(n) = n^(2^(omega(n)-1)-1), where omega() is the number of distinct primes dividing n (A001221).
a(n) = 1 if n is a prime power.

A069141 a(n) = n^2*(n+1)!/(n^tau(n)) where tau(n) is the number of divisors of n.

Original entry on oeis.org

2, 6, 24, 30, 720, 140, 40320, 5670, 403200, 399168, 479001600, 300300, 87178291200, 6671808000, 92990177280, 86837751000, 6402373705728000, 1158789632000, 2432902008176640000, 319318388573184, 2548754484756480000, 53413257724968960000, 620448401733239439360000

Offset: 1

Benoit Cloitre, Apr 08 2002

The expression n^2*(n+1)!/(n^tau(n)) is always an integer. n^tau(n) is also the product of square divisors of n (cf. A062758).

Amiram Eldar, Table of n, a(n) for n = 1..448

Cf. A000005, A062758.

a[n_] := (n+1)! / n^(DivisorSigma[0,n] - 2); Array[a, 23] (* Amiram Eldar, Aug 03 2024 *)

a(n) = (n+1)! / n^(numdiv(n)-2); \\ Amiram Eldar, Aug 03 2024

a(22)-a(23) from Amiram Eldar, Aug 03 2024

A165797 a(n) = n^( sigma(n) - tau(n) ).

Original entry on oeis.org

1, 2, 9, 256, 625, 1679616, 117649, 8589934592, 3486784401, 100000000000000, 25937424601, 552061438912436417593344, 23298085122481, 83668255425284801560576, 332525673007965087890625

Offset: 1

Jaroslav Krizek, Sep 27 2009

nonn

The power of n with exponent given by the difference between its sum of divisors and its count of divisors.

			a(4) = 4^(sigma(4)-tau(4)) = 4^(7-3) = 4^4 = 256.

Jaroslav Krizek, Table of n, a(n) for n = 1..50

Table[n^[ DivisorSigma[1, n] - DivisorSigma[0, n]], {n, 50}]

a(n) = n^(A000203(n)-A000005(n)) = n^A000203(n) / n^A000005(n) = n^A065608(n).
a(n) = A100879(n) / A062758(n).
a(p) = p^(p-1) for p = prime.

Slightly edited by R. J. Mathar, Sep 29 2009

A236288 a(n) = sigma(n)^n / sigma(n)^tau(n), where tau(n) = A000005(n) = the number of divisors of n and sigma(n) = A000203(n) = the sum of divisors of n.

Original entry on oeis.org

1, 1, 4, 7, 216, 144, 32768, 50625, 4826809, 34012224, 5159780352, 481890304, 4049565169664, 63403380965376, 1521681143169024, 25408476896404831, 6746640616477458432, 12381557655576425121, 13107200000000000000000, 53148384174432398229504, 38685626227668133590597632

Offset: 1

Jaroslav Krizek, Jan 23 2014

nonn

Conjecture: number 1 is the only number n such that sigma(n)^(n - tau(n)) = sigma(n+1)^(n + 1 - tau(n+1)).

Conjecture: number 1 is the only number n such that sigma(n)^(n - tau(n)) = sigma(k)^(k - tau(k)) has solution for distinct numbers n and k.

			a(4) = sigma(4)^(4 - tau(4)) = 7^(4 - 3) = 7.

Jaroslav Krizek, Table of n, a(n) for n = 1..50

Cf. A000005 (tau(n)), A000203 (sigma(n)), A062758 (n^tau(n)), A217872 (sigma(n)^n), A236285 (tau(n)^sigma(n)), A236287 (sigma(n)^tau(n)).

Table[DivisorSigma[1, n]^[n - DivisorSigma[0, n]], {n, 50}]

s=[]; for(n=1, 30, s=concat(s, sigma(n, 1)^(n-sigma(n, 0)))); s \\ Colin Barker, Jan 24 2014

a(n) = sigma(n)^(n - tau(n)).

a(n) = A217872(n) / A236287(n) = A000203(n)^n / A000203(n)^A000005(n) = A000203(n)^A049820(n).

A277169 Product of squares of proper divisors of n.

Original entry on oeis.org

1, 1, 1, 4, 1, 36, 1, 64, 9, 100, 1, 20736, 1, 196, 225, 4096, 1, 104976, 1, 160000, 441, 484, 1, 191102976, 25, 676, 729, 614656, 1, 729000000, 1, 1048576, 1089, 1156, 1225, 78364164096, 1, 1444, 1521, 4096000000, 1, 5489031744, 1, 3748096, 4100625, 2116, 1, 28179280429056, 49, 6250000

Offset: 1

Ilya Gutkovskiy, Oct 19 2016

			a(6) = 36 because 6 has 3 proper divisors {1,2,3} and 1^2*2^2*3^2 = 36.

Robert Israel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Divisor Product
Eric Weisstein's World of Mathematics, Proper divisors

Cf. A000290, A001248, A007422, A007955, A007956, A008578, A062758.

seq(n^(numtheory:-tau(n)-2), n=1..50); # Robert Israel, Nov 13 2016

Table[n^(DivisorSigma[0, n] - 2), {n, 1, 50}]

a(n) = n^(sigma_0(n)-2).
a(n) = n^A000005(n)/A000290(n).
a(n) = A000290(A007956(n))/A000290(n).
a(n) = A000290(A007955(n)/n)/A000290(n).
a(n) = A062758(n)/A000290(n).
a(n) = 1 if n is prime or n = 1 (A008578).
a(n) = n if n is square of prime (A001248).
a(n) = n^2 if n is multiplicatively perfect number (A007422).

A290479 Product of nonprime squarefree divisors of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 6, 1, 1, 1, 10, 1, 6, 1, 14, 15, 1, 1, 6, 1, 10, 21, 22, 1, 6, 1, 26, 1, 14, 1, 27000, 1, 1, 33, 34, 35, 6, 1, 38, 39, 10, 1, 74088, 1, 22, 15, 46, 1, 6, 1, 10, 51, 26, 1, 6, 55, 14, 57, 58, 1, 27000, 1, 62, 21, 1, 65, 287496, 1, 34, 69, 343000, 1, 6, 1, 74, 15, 38, 77, 474552, 1, 10

Offset: 1

Ilya Gutkovskiy, Aug 03 2017

nonn

			a(30) = 27000 because 30 has 8 divisors {1, 2, 3, 5, 6, 10, 15, 30} among which 5 are nonprime squarefree {1, 6, 10, 15, 30} and 1*6*10*15*30 = 27000.

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

Cf. A000005, A000469, A006881 (fixed points), A007947, A007955, A007956, A061537, A062758, A078599, A087652, A126192, A136655, A183091, A284118.

Table[Product[d, {d, Select[Divisors[n], !PrimeQ[#] && SquareFreeQ[#] &]}], {n, 80}]
Table[Last[Select[Divisors[n], SquareFreeQ]]^(DivisorSigma[0, Last[Select[Divisors[n], SquareFreeQ]]]/2 - 1), {n, 80}]

A290479(n) = if(1==n, n, my(r=factorback(factorint(n)[, 1])); (r^((numdiv(r)/2)-1))); \\ Antti Karttunen, Aug 06 2018

a(n) = A078599(n)/A007947(n).
a(n) = rad(n)^(d(rad(n))/2-1), where d() is the number of divisors of n (A000005) and rad() is the squarefree kernel of n (A007947).
a(n) = 1 if n is a prime power.

A332646 Numbers m with a divisor d such that d^tau(d) = m.

Original entry on oeis.org

1, 4, 9, 25, 49, 64, 121, 169, 289, 361, 529, 729, 841, 961, 1296, 1369, 1681, 1849, 2209, 2809, 3481, 3721, 4096, 4489, 5041, 5329, 6241, 6889, 7921, 9409, 10000, 10201, 10609, 11449, 11881, 12769, 15625, 16129, 17161, 18769, 19321, 22201, 22801, 24649, 26569

Offset: 1

Jaroslav Krizek, Feb 18 2020

nonn

Possible values for function n^tau(n) (A062758).

Supersequence of A189991 (numbers with prime factorization p^4*q^4; d = pq), A001248 (numbers with prime factorization p^2; d = p), A030516 (numbers with prime factorization p^6; d = p^2) and A280076.

			64 is a term because 4^3 = 64; 4 divides 64; tau(4) = 3.

Cf. A000005, A001248, A030516, A062758, A189991, A280076.

[n: n in [1..100000] | #[d: d in Divisors(n) | d^NumberOfDivisors(d) eq n] ge 1];

divPowerQ[n_] := AnyTrue[Divisors[n], #^DivisorSigma[0, #] == n &]; Select[Range[27000], divPowerQ] (* Amiram Eldar, Feb 18 2020 *)

isok(m) = fordiv(m, d, if (d^numdiv(d) == m, return (1))); \\ Michel Marcus, Feb 18 2020

A069152 a(n) = (n-1)!-n^tau(n)/n^2.

Original entry on oeis.org

0, 1, 2, 23, 84, 719, 4976, 40311, 362780, 3628799, 39896064, 479001599, 6227020604, 87178290975, 1307674363904, 20922789887999, 355687427991024, 6402373705727999, 121645100408672000, 2432902008176639559

Offset: 2

Benoit Cloitre, Apr 08 2002

n and a(n) have the same parity.

Cf. A062758.

Table[(n - 1)! - n^DivisorSigma[0, n]/n^2, {n, 2, 25}] (* Wesley Ivan Hurt, Jan 20 2024 *)

cp's OEIS Frontend

A174939 a(n) = Sum_{k<=n} A007955(k) * A007955(k) = Sum_{k<=n} A007955(k)^2, where A007955(m) = product of divisors of m.

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A236287 a(n) = sigma(n)^tau(n), where tau(n) = A000005(n) = the number of divisors of n and sigma(n) = A000203(n) = the sum of divisors of n.

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A290480 Product of proper unitary divisors of n.

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A069141 a(n) = n^2*(n+1)!/(n^tau(n)) where tau(n) is the number of divisors of n.

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A165797 a(n) = n^( sigma(n) - tau(n) ).

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A236288 a(n) = sigma(n)^n / sigma(n)^tau(n), where tau(n) = A000005(n) = the number of divisors of n and sigma(n) = A000203(n) = the sum of divisors of n.

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A277169 Product of squares of proper divisors of n.

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