cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 10 results.

A280116 Partial sums of A266265 (Product_{d|n} pod(d)).

Original entry on oeis.org

1, 3, 6, 22, 27, 243, 250, 1274, 1355, 2355, 2366, 2988350, 2988363, 2991107, 2994482, 4043058, 4043075, 38055299, 38055318, 102055318, 102064579, 102075227, 102075250, 63403483040626, 63403483041251, 63403483058827, 63403483117876, 63403965008180
Offset: 1

Views

Author

Jaroslav Krizek, Dec 26 2016

Keywords

Comments

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

Crossrefs

Cf. A007955, A175317, A266265, A280114 (partial sums of A175317), A280117 (partial products of A266265).

Programs

  • Magma
    [&+[&*[&*[b: b in Divisors(d)]: d in Divisors(k)]: k in [1..n]]: n in [1..1000]];
  • Mathematica
    Accumulate@ Array[Product[Times @@ Divisors@ d, {d, Divisors@ #}] &, 28] (* Michael De Vlieger, Dec 26 2016 *)

Formula

a(n) = Sum_{i=1..n} A266265(i).

A280117 Partial products of A266265 (Product_{d|n} pod(d)), where pod(n) is the product of the divisors of n (A007955).

Original entry on oeis.org

1, 2, 6, 96, 480, 103680, 725760, 743178240, 60197437440, 60197437440000, 662171811840000, 1977234435405250560000, 25704047660268257280000, 70531906779776097976320000, 238045185381744330670080000000, 249608468306847943276709806080000000
Offset: 1

Views

Author

Jaroslav Krizek, Dec 26 2016

Keywords

Crossrefs

Cf. A007955, A175317, A266265, A280115 (partial sums of A175317), A280116 (partial sums of A266265).

Programs

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

Formula

a(n) = Product_{i=1..n} A266265(i).

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

Views

Author

Jaroslav Krizek, Apr 01 2010

Keywords

Examples

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

Links

Crossrefs

Cf. A007429, A007955, A206032, A266265.
Subsequences: A008864, A181388 \ {0}.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, #^(DivisorSigma[0, #]/2) &]; Array[a, 50] (* Amiram Eldar, Oct 23 2021 *)
  • PARI
    a(n) = sumdiv(n, d, vecprod(divisors(d))); \\ Michel Marcus, Dec 09 2014 and Oct 23 2021
  • Python
    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

Formula

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)

Extensions

Corrected by Jaroslav Krizek, Apr 02 2010
Edited and more terms from Michel Marcus, Dec 09 2014

A323760 Numerator of Product_{d|n} (pod(d)/tau(d)) where pod(k) = the product of the divisors of k and tau(k) = the number of the divisors of k.

Original entry on oeis.org

1, 1, 3, 8, 5, 27, 7, 128, 27, 125, 11, 10368, 13, 343, 3375, 131072, 17, 118098, 19, 2000000, 9261, 1331, 23, 6879707136, 625, 2197, 19683, 15059072, 29, 38443359375, 31, 2147483648, 35937, 4913, 42875, 101559956668416, 37, 6859, 59319, 10240000000000, 41
Offset: 1

Views

Author

Jaroslav Krizek, Jan 26 2019

Keywords

Comments

Product_{d|n} (pod(d)/tau(d)) > 1 for all n > 2.

Examples

			For n=4; Product_{d|4} (pod(d)/tau(d)) = (pod(1)/tau(1))*(pod(2)/tau(2))*(pod(4)/tau(4)) = (1/1)*(2/2)*(8/3) = 8/3; a(4) = 8.

Crossrefs

Cf. A211776, A266265, A323761 (denominator).

Programs

  • Magma
    [Numerator(&*[&*[c: c in Divisors(d)] / NumberOfDivisors(d): d in Divisors(n)]): n in [1..100]]
  • Maple
    A323760 := proc(n)
    numer(A266265(n)/A211776(n)) ;
end proc:
seq(A323760(n),n=1..20) ; # R. J. Mathar, Feb 13 2019
  • PARI
    a(n) = my(p=1, vd); fordiv(n, d, vd = divisors(d); p *= vecprod(vd)/#vd); numerator(p); \\ Michel Marcus, Jan 27 2019

Formula

a(p) = p for primes p > 2.

A323761 Denominator of Product_{d|n} (pod(d)/tau(d)) where pod(k) = the product of the divisors of k and tau(k) = the number of the divisors of k.

Original entry on oeis.org

1, 1, 2, 3, 2, 2, 2, 3, 2, 2, 2, 1, 2, 2, 16, 15, 2, 1, 2, 9, 16, 2, 2, 1, 6, 2, 8, 9, 2, 8, 2, 45, 16, 2, 16, 1, 2, 2, 16, 9, 2, 8, 2, 9, 32, 2, 2, 25, 6, 9, 16, 9, 2, 1, 16, 9, 16, 2, 2, 1, 2, 2, 32, 315, 16, 8, 2, 9, 16, 8, 2, 1, 2, 2, 32, 9, 16, 8, 2, 9
Offset: 1

Views

Author

Jaroslav Krizek, Jan 27 2019

Keywords

Comments

Product_{d|n} (pod(d)/tau(d)) > 1 for all n > 2.

Examples

			For n=4; Product_{d|4} (pod(d)/tau(d)) = (pod(1)/tau(1))*(pod(2)/tau(2))*(pod(4)/tau(4)) = (1/1)*(2/2)*(8/3) = 8/3; a(4) = 3.

Links

Crossrefs

Cf. A211776, A266265, A323760 (numerator), A323762.

Programs

  • Magma
    [Denominator(&*[&*[c: c in Divisors(d)] / NumberOfDivisors(d): d in Divisors(n)]): n in [1..100]]
  • Maple
    A323761 := proc(n)
    denom(A266265(n)/A211776(n)) ;
end proc:
seq(A323761(n),n=1..20) ; # R. J. Mathar, Feb 13 2019
  • PARI
    a(n) = my(p=1, vd); fordiv(n, d, vd = divisors(d); p *= vecprod(vd)/#vd); denominator(p); \\ Michel Marcus, Jan 27 2019

Formula

a(p) = 2 for prime p > 2.
a(n) = 1 for numbers in A323762.

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

Views

Author

Jaroslav Krizek, Apr 25 2019

Keywords

Comments

n divides a(n) for all n.

Examples

			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.

Links

Crossrefs

Cf. A000203, A007955, A325029.

Programs

  • Magma
    [&*[&+ [c: c in Divisors(d)] * &*[c: c in Divisors(d)]: d in Divisors(n)]: n in [1..100]]
  • Mathematica
    Table[Times@@(DivisorSigma[1,#]Times@@Divisors[#]&/@Divisors[n]),{n,30}] (* Harvey P. Dale, Dec 10 2024 *)
  • PARI
    a(n) = my(d=divisors(n)); prod(k=1, #d, my(dd=divisors(d[k])); vecsum(dd)*vecprod(dd)); \\ Michel Marcus, Apr 25 2019

Formula

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).

A220849 a(n) = Product_{d|n} Product_{d_x|n , d_x <= d} d_x.

Original entry on oeis.org

1, 2, 3, 16, 5, 432, 7, 1024, 81, 2000, 11, 71663616, 13, 5488, 10125, 1048576, 17, 816293376, 19, 2048000000, 27783, 21296, 23, 219122084616339456, 625, 35152, 59049, 15420489728, 29, 2550916800000000000, 31, 34359738368, 107811, 78608, 214375
Offset: 1

Views

Author

Jaroslav Krizek, Dec 22 2012

Keywords

Comments

Conjecture: sequence is injective (all terms of this sequence occur only once).
The subsequence of fixed points consists of 1 together with the primes (A008578). - Bernard Schott, Oct 26 2021

Examples

			The divisors of 6 are 1, 2, 3, 6. a(n) = 1*(1*2)*(1*2*3)*(1*2*3*6) = 1*2*6*36 = 432.

Links

Crossrefs

Cf. A000292, A006881, A007955, A008578 (fixed points), A064945, A266265.

Programs

  • Mathematica
    a[n_] := Module[{d = Divisors[n], nd}, nd = Length[d]; Product[d[[i]]^(nd - i + 1), {i, 1, nd}]]; Array[a, 35] (* Amiram Eldar, Oct 23 2021 *)
  • PARI
    a(n) = my(d=divisors(n)); prod(k=1, #d, vecprod(select(x->(x<=d[k]), d))); \\ Michel Marcus, Oct 23 2021

Formula

a(p) = p for prime p.
From Bernard Schott, Oct 29 2021: (Start)
For p prime and k >= 0, a(p^k) = p^A000292(k).
For n = p*q, p < q primes (A006881), a(n) = p*n^3. (End)

Extensions

a(24) corrected by Seiichi Manyama, Oct 23 2021

A307101 a(n) = Product_{d|n} (tau(d)*pod(d)) where tau(k) = the number of divisors of k (A000005) and pod(k) = the product of the divisors of k (A007955).

Original entry on oeis.org

1, 4, 6, 96, 10, 3456, 14, 24576, 486, 16000, 22, 859963392, 26, 43904, 54000, 125829120, 34, 9795520512, 38, 18432000000, 148176, 170368, 46, 584325558976905216, 3750, 281216, 1417176, 138784407552, 58, 80621568000000000, 62, 24739011624960, 574992, 628864
Offset: 1

Views

Author

Jaroslav Krizek, Apr 25 2019

Keywords

Comments

n divides a(n) for all n.

Examples

			a(6) = (tau(1)*pod(1)) * (tau(2)*pod(2)) * (tau(3)*pod(3)) * (tau(6)*pod(6)) = (1*1) * (2*2) * (2*3) * (4*36) = 3456.

Crossrefs

Cf. A000005, A007955, A307100.

Programs

  • Magma
    [&*[# [c: c in Divisors(d)] * &*[c: c in Divisors(d)]: d in Divisors(n)]: n in [1..100]]
  • PARI
    a(n) = my(d=divisors(n)); prod(k=1, #d, my(dd=divisors(d[k])); #dd*vecprod(dd)); \\ Michel Marcus, Apr 25 2019

Formula

a(n) = Product_{d|n} tau(d) * Product_{d|n} pod(d) = A211776(n) * A266265(n).
a(p) = 2p for p = primes (A000040).

A324981 a(n) = Product_{d|n} (d*pod(d)) where pod(k) = the product of the divisors of k (A007955).

Original entry on oeis.org

1, 4, 9, 128, 25, 7776, 49, 65536, 2187, 100000, 121, 5159780352, 169, 537824, 759375, 1073741824, 289, 198359290368, 361, 512000000000, 4084101, 5153632, 529, 21035720123168587776, 78125, 11881376, 43046721, 10578455953408, 841, 15943230000000000000, 961
Offset: 1

Views

Author

Jaroslav Krizek, Mar 22 2019

Keywords

Examples

			a(6) = 1*pod(1) * 2*pod(2) * 3*pod(3) * 6*pod(6) = (1*1) * (2*2) * (3*3) * (6*36) = 7776.

Crossrefs

Cf. A007955, A174933 (Sum_{d|n} (d*pod(d))), A266265.

Programs

  • Magma
    [&*[d * &*[c: c in Divisors(d)]: d in Divisors(n)]: n in [1..100]]
  • Mathematica
    Array[Times @@ Map[# Apply[Times, Divisors@ #] &, Divisors@ #] &, 31] (* Michael De Vlieger, Mar 24 2019 *)
  • PARI
    a(n) = my(d=divisors(n), p=1); fordiv(n, d, p*=d*vecprod(divisors(d))); p;  \\ Michel Marcus, Mar 22 2019

Formula

a(n) = (Product_{d|n} d) * (Product_{d|n} pod(d)) = A007955(n) * A266265(n).
a(p) = p^2 for p = primes (A000040).

A334489 a(n) = Product_{d|n} (pod(n)/pod(d)) where pod(n) = A007955(n), the product of divisors of n.

Original entry on oeis.org

1, 2, 3, 32, 5, 7776, 7, 16384, 243, 100000, 11, 8916100448256, 13, 537824, 759375, 1073741824, 17, 1156831381426176, 19, 4096000000000000, 4084101, 5153632, 23, 2315513501476187716057433112576, 3125, 11881376, 4782969, 232218265089212416, 29
Offset: 1

Views

Author

Jaroslav Krizek, May 03 2020

Keywords

Examples

			For n = 6; divisors d of 6: {1, 2, 3, 6}; pod(d): {1, 2, 3, 36}; lcm_{d|6} pod(d) = pod(6) = 36; a(6) = 36/1 * 36/2 * 36/3 * 36/36 = 7776.

Crossrefs

Cf. Similar sequences for functions lcm_{d|n} tau(d) and lcm_{d|n} sigma(d): A334470, A334471.
Cf. A000005, A000040, A007955, A266265.

Programs

  • Magma
    [&*[ LCM([&*Divisors(d): d in Divisors(n)]) / &*Divisors(d): d in Divisors(n)]: n in [1..100]]
  • Mathematica
    pod[n_] := Times @@ Divisors[n]; a[n_] := pod[n]^Length[(d = Divisors[n])]/Times @@ (pod /@ d); Array[a, 30] (* Amiram Eldar, May 03 2020 *)
  • PARI
    pod(n) = vecprod(divisors(n));
a(n) = my(d=divisors(n), podn = pod(n)); prod(k=1, #d, podn/pod(d[k])); \\ Michel Marcus, May 03-11 2020

Formula

a(p) = p for p = primes (A000040).
a(n) = ((lcm_{d|n} pod(d))^tau(n)) / Product_{d|n} (pod(d)) = A007955(n)^A000005(n)/A266265(n).
a(n) = n^c(n) where c(n) only depends on the prime signature of n. - David A. Corneth, May 05 2020
Showing 1-10 of 10 results.

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