A206032 -id:A206032 - cp's OEIS frontend

A280085 Partial sums of A206032 (Product_{d|n} sigma(d)).

1, 4, 8, 29, 35, 179, 187, 502, 554, 878, 890, 29114, 29128, 29704, 30280, 40045, 40063, 113071, 113091, 208347, 209371, 210667, 210691, 25612291, 25612477, 25614241, 25616321, 25842113, 25842143, 52715999, 52716031, 53331226, 53333530, 53336446, 53338750

Offset: 1

Jaroslav Krizek, Dec 26 2016

nonn

sigma(n) is the sum of the divisors of n (A000203).

Cf. A000203, A007429, A206032, A280077 (partial sums of A007429), A280086 (partial products of A206032).

[&+[&*[SumOfDivisors(d): d in Divisors(k)]: k in [1..n]]: n in [1..100]]

Accumulate@ Array[Product[DivisorSigma[1, d], {d, Divisors@ #}] &, 35] (* Michael De Vlieger, Dec 26 2016 *)

a(n) = Sum_{i=1..n} A206032(i).

A280086 Partial products of A206032 (Product_{d|n} sigma(d)).

Original entry on oeis.org

1, 3, 12, 252, 1512, 217728, 1741824, 548674560, 28531077120, 9244068986880, 110928827842560, 3130855237028413440, 43831973318397788160, 25247216631397125980160, 14542396779684744564572160, 142006504553621530673047142400, 2556117081965187552114848563200

Offset: 1

Jaroslav Krizek, Dec 26 2016

nonn

sigma(n) is the sum of the divisors of n (A000203).

Cf. A000203, A007429, A206032, A280078 (partial products of A007429), A280085 (partial sums of A206032).

[&*[&*[SumOfDivisors(d): d in Divisors(k)]: k in [1..n]]: n in [1..100]]

FoldList[Times[#1, #2] &, Array[Product[DivisorSigma[1, d], {d, Divisors@ #}] &, 17]] (* Michael De Vlieger, Dec 26 2016 *)

a(n) = Product_{i=1..n} A206032(i).

A266265 Product of products of divisors of divisors of n.

Original entry on oeis.org

1, 2, 3, 16, 5, 216, 7, 1024, 81, 1000, 11, 2985984, 13, 2744, 3375, 1048576, 17, 34012224, 19, 64000000, 9261, 10648, 23, 63403380965376, 625, 17576, 59049, 481890304, 29, 19683000000000, 31, 34359738368, 35937, 39304, 42875, 4738381338321616896, 37, 54872

Offset: 1

Jaroslav Krizek, Dec 25 2015

nonn

a(n) = Product_{d|n} A007955(d) where A007955(m) = product of divisors of m.
From G. C. Greubel, Dec 31 2015: (Start)
for n>=1: 10^3|a(10*n), 10^6|a(20*n), 10^9|a(30*n).
for n>=0: 10^6|a(60*n+50), 10^9|a(60*n+70). (End)

			For n = 6; with b(n) = A007955(n); a(6) = b(1)*b(2)*b(3)*b(6) = 1*2*3*36 = 216.

G. C. Greubel, Table of n, a(n) for n = 1..1200

Cf. A007955 (product of divisors of n), A175317 (sum of products of divisors of divisors of n), A206032 (product of sums of divisors of divisors of n).

[&*[&*[b: b in Divisors(d)]: d in Divisors(n)]: n in [1..100]]

A266265 := proc(n)
    mul( A007955(d),d=numtheory[divisors](n)) ;
end proc:
seq(A266265(n),n=1..10) ; # R. J. Mathar, Feb 13 2019

Table[Product[Times @@ Divisors@ d, {d, Divisors@ n}], {n, 38}] (* Michael De Vlieger, Dec 31 2015 *)

a(n) = {my(d = divisors(n)); prod(k=1, #d, dd = divisors(d[k]); prod(kk=1,#dd, dd[kk]));} \\ Michel Marcus, Dec 27 2015

a(p) = p for p = prime.

a(n) = Product_{d|n} d^tau(n/d). - Ridouane Oudra, Apr 09 2023

A175317 a(n) = Sum_{d|n} A007955(d) where A007955(m) = product of divisors of m.

Original entry on oeis.org

1, 3, 4, 11, 6, 42, 8, 75, 31, 108, 12, 1778, 14, 206, 234, 1099, 18, 5901, 20, 8116, 452, 498, 24, 333618, 131, 692, 760, 22166, 30, 810372, 32, 33867, 1104, 1176, 1238, 10085333, 38, 1466, 1538, 2568180, 42, 3112382, 44, 85690, 91386, 2142, 48, 255138610

Offset: 1

Jaroslav Krizek, Apr 01 2010

nonn

			For n = 4, with b(n) = A007955(n), a(4) = b(1) + b(2) + b(4) = 1 + 2 + 8 = 11.

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A007429, A007955, A206032, A266265.

Subsequences: A008864, A181388 \ {0}.

a[n_] := DivisorSum[n, #^(DivisorSigma[0, #]/2) &]; Array[a, 50] (* Amiram Eldar, Oct 23 2021 *)

a(n) = sumdiv(n, d, vecprod(divisors(d))); \\ Michel Marcus, Dec 09 2014 and Oct 23 2021

from math import isqrt
from sympy import divisor_count, divisors
def A175317(n): return sum(isqrt(d)**c if (c:=divisor_count(d)) & 1 else d**(c//2) for d in divisors(n,generator=True)) # Chai Wah Wu, Jun 24 2022

From Bernard Schott, Oct 26 2021: (Start)
a(1) = 1 (the only fixed point).
a(p) = p+1 for prime p only.
a(2^k) = A181388(k+1). (End)

Corrected by Jaroslav Krizek, Apr 02 2010

Edited and more terms from Michel Marcus, Dec 09 2014

A206031 a(n) = product of numbers k <= sigma(n) such that k = sigma(d) for any divisor d of n where sigma = A000203.

Original entry on oeis.org

1, 3, 4, 21, 6, 144, 8, 315, 52, 324, 12, 28224, 14, 576, 576, 9765, 18, 73008, 20, 95256, 1024, 1296, 24, 25401600, 186, 1764, 2080, 225792, 30, 26873856, 32, 615195, 2304, 2916, 2304, 1302170688, 38, 3600, 3136, 128595600, 42, 84934656, 44, 762048, 584064

Offset: 1

Jaroslav Krizek, Feb 03 2012

nonn

Sequence is not the same as A206032(n): a(66) = 35831808, A206032(66) = 429981696.

In sequence A206032 are multiplied all values of sigma(d) of all divisors d of numbers n, in sequence a(n) are multiplied only distinct values of sigma(d) of all divisors d of numbers n.

			For n=6 -> divisors d of 6: 1,2,3,6; corresponding values of sigma(d): 1,3,4,12; a(6) = Product of k = 1*3*4*12 = 144. For n=66 -> divisors d of 66: 1,2,3,6,11,22,33,66; corresponding values of sigma(d): 1,3,4,12,12,36,48,144; a(66) = Product of k = 1*3*4*12*36*48*144 = 35831808.

Antti Karttunen, Table of n, a(n) for n = 1..4096

Cf. A184388, A184395, A206032, A206033.

Table[Times @@ Union[DivisorSigma[1, Divisors[n]]], {n, 100}] (* T. D. Noe, Feb 10 2012 *)

a(n)=my(d=vecsort(apply(sigma,divisors(n)),,8));prod(i=2,#d,d[i]) \\ Charles R Greathouse IV, Feb 19 2013

a(p) = p+1, a(pq) = ((p+1)*(q+1))^2 for p, q = distinct primes.

A206033 a(1) =1; for n>=1: a(n) = product of numbers k <= sigma(n) such that k is not equal to sigma(d) for any divisor d of n where sigma = A000203.

Original entry on oeis.org

1, 2, 6, 240, 120, 3326400, 5040, 4151347200, 119750400, 19760412672000, 39916800, 10802449851605508096000000, 6227020800, 1077167364120207360000, 1077167364120207360000, 842072570832352567099392000000, 355687428096000

Offset: 1

Jaroslav Krizek, Feb 03 2012

nonn

In sequence A206032 are multiplied all values of sigma(d) of all divisors d of numbers n, in sequence A206031 are multiplied only distinct values of sigma(d) of all divisors d of numbers n and in sequence a(n) are multiplied numbers k (1<=k<=sigma(n)) such that sigma(d) = k has no solution for neither divisor d of number n.

			For n=6 -> divisors d of 6: 1,2,3,6; corresponding values of sigma(d): 1,3,4,12; a(6) = Product of k = 2*5*6*7*8*9*10*11 = 3326400.

Cf. A184388, A206031, A206032.

Table[Times @@ Complement[Range[DivisorSigma[1, n]], DivisorSigma[1, Divisors[n]]], {n, 100}] (* T. D. Noe, Feb 10 2012 *)

A280114 Partial sums of A175317 (Sum_{d|n} pod(d)).

Original entry on oeis.org

1, 4, 8, 19, 25, 67, 75, 150, 181, 289, 301, 2079, 2093, 2299, 2533, 3632, 3650, 9551, 9571, 17687, 18139, 18637, 18661, 352279, 352410, 353102, 353862, 376028, 376058, 1186430, 1186462, 1220329, 1221433, 1222609, 1223847, 11309180, 11309218, 11310684

Offset: 1

Jaroslav Krizek, Dec 26 2016

nonn

pod(n) is the product of the divisors of n (A007955).

Cf. A007955, A175317, A206032, A280085 (partial sums of A206032), A280115 (partial products of A175317).

[&+[&+[&*[b: b in Divisors(d)]: d in Divisors(k)]: k in [1..n]]: n in [1..1000]];

Accumulate@ Array[Sum[Times @@ Divisors@ d, {d, Divisors@ #}] &, 38] (* Michael De Vlieger, Dec 26 2016 *)

a(n) = Sum_{i=1..n} A175317(i).

A280115 Partial products of A175317 (Sum_{d|n} pod(d)).

Original entry on oeis.org

1, 3, 12, 132, 792, 33264, 266112, 19958400, 618710400, 66820723200, 801848678400, 1425686950195200, 19959617302732800, 4111681164362956800, 962133392460931891200, 1057384598314564148428800, 19032922769662154671718400, 112313277263776374717810278400

Offset: 1

Jaroslav Krizek, Dec 26 2016

nonn

pod(n) is the product of the divisors of n (A007955).

Cf. A007955, A175317, A206032, A280086 (partial products of A206032), A280114 (partial sums of A175317).

[&*[&+[&*[b: b in Divisors(d)]: d in Divisors(k)]: k in [1..n]]: n in [1..1000]]

FoldList[Times[#1, #2] &, Array[Sum[Times @@ Divisors@ d, {d, Divisors@ #}] &, 18]] (* Michael De Vlieger, Dec 26 2016 *)

a(n) = Product_{i=1..n} A175317(i).

A290478 Triangle read by rows in which row n lists the sum of the divisors of each divisor of n.

Original entry on oeis.org

1, 1, 3, 1, 4, 1, 3, 7, 1, 6, 1, 3, 4, 12, 1, 8, 1, 3, 7, 15, 1, 4, 13, 1, 3, 6, 18, 1, 12, 1, 3, 4, 7, 12, 28, 1, 14, 1, 3, 8, 24, 1, 4, 6, 24, 1, 3, 7, 15, 31, 1, 18, 1, 3, 4, 12, 13, 39, 1, 20, 1, 3, 7, 6, 18, 42, 1, 4, 8, 32, 1, 3, 12, 36, 1, 24, 1, 3, 4, 7

Offset: 1

Michel Lagneau, Aug 03 2017

Or, in the triangle A027750(n), replace each element with the sum of its divisors.
The row whose index x is a prime power p^m (p prime and m >= 0) is equal to (1, sigma(p), sigma(p^2), ..., sigma(p^(m-1))).
We observe the following properties of row n when n is the product of k distinct primes, k = 1,2,...:
when n = prime(m), row n = (1, prime(m)+1);
when n is the product of two distinct primes p < q, row n = (1, p+1, q+1,(p+1)(q+1));
when n is the product of three distinct primes p < q < r, row n = (1, p+1, q+1, r+1, (p+1)(q+1), (p+1)(r+1), (q+1)(r+1), sigma(p*q*r));

			Row 6 is (a(11), a(12), a(13), a(14)) = (1, 3, 4, 12) because sigma(A027750(11))= sigma(1) = 1, sigma(A027750(12))= sigma(2) = 3, sigma(A027750(13))= sigma(3) = 4 and sigma(A027750(14)) = sigma(6) = 12.
Triangle begins:
  1;
  1,  3;
  1,  4;
  1,  3,  7;
  1,  6;
  1,  3,  4, 12;
  1,  8;
  1,  3,  7, 15;
  1,  4, 13;
  1,  3,  6, 18;
  ...

Michel Marcus, Table of n, a(n) for n = 1..10006 (rows 1 to 1358, flattened).

Cf. A000203, A027750, A290532.

Cf. A007429 (row sums), A206032 (row products).

[[SumOfDivisors(d): d in Divisors(n)]: n in [1..20]]; // Vincenzo Librandi, Sep 08 2017

with(numtheory):nn:=100:
for n from 1 to nn do:
  d1:=divisors(n):n1:=nops(d1):
   for i from 1 to n1 do:
     s:=sigma(d1[i]):
     printf(`%d, `,s):
   od:
od:

Array[DivisorSigma[1, Divisors@ #] &, 24] // Flatten (* Michael De Vlieger, Aug 07 2017 *)

row(n) = apply(sigma, divisors(n)); \\ Michel Marcus, Dec 27 2021

a(n) = sigma(A027750(n)).

A325030 a(n) = Product_{d|n} (sigma(d)*pod(d)) where sigma(k) = the sum of the divisors of k (A000203) and pod(k) = the product of the divisors of k (A007955).

Original entry on oeis.org

1, 6, 12, 336, 30, 31104, 56, 322560, 4212, 324000, 132, 84276412416, 182, 1580544, 1944000, 10239344640, 306, 2483164449792, 380, 6096384000000, 9483264, 13799808, 552, 1610547321930095001600, 116250, 31004064, 122821920, 108806975520768, 870

Offset: 1

Jaroslav Krizek, Apr 25 2019

nonn

n divides a(n) for all n.

			a(6) = (sigma(1)*pod(1)) * (sigma(2)*pod(2)) * (sigma(3)*pod(3)) * (sigma(6)*pod(6)) = (1*1) * (3*2) * (4*3) * (12*36) = 31104.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A000203, A007955, A325029.

[&*[&+ [c: c in Divisors(d)] * &*[c: c in Divisors(d)]: d in Divisors(n)]: n in [1..100]]

Table[Times@@(DivisorSigma[1,#]Times@@Divisors[#]&/@Divisors[n]),{n,30}] (* Harvey P. Dale, Dec 10 2024 *)

a(n) = my(d=divisors(n)); prod(k=1, #d, my(dd=divisors(d[k])); vecsum(dd)*vecprod(dd)); \\ Michel Marcus, Apr 25 2019

a(n) = Product_{d|n} sigma(d) * Product_{d|n} pod(d) = A206032(n) * A266265(n).

a(p) = p*(p+1) for p = primes (A000040).

cp's OEIS Frontend

A280085 Partial sums of A206032 (Product_{d|n} sigma(d)).

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A280086 Partial products of A206032 (Product_{d|n} sigma(d)).

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A266265 Product of products of divisors of divisors of n.

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A175317 a(n) = Sum_{d|n} A007955(d) where A007955(m) = product of divisors of m.

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A206031 a(n) = product of numbers k <= sigma(n) such that k = sigma(d) for any divisor d of n where sigma = A000203.

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A206033 a(1) =1; for n>=1: a(n) = product of numbers k <= sigma(n) such that k is not equal to sigma(d) for any divisor d of n where sigma = A000203.

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A280114 Partial sums of A175317 (Sum_{d|n} pod(d)).

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A280115 Partial products of A175317 (Sum_{d|n} pod(d)).