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 11 results. Next

A280075 Partial products of A211776 (Product_{d|n} tau(d)).

Original entry on oeis.org

1, 2, 4, 24, 48, 768, 1536, 36864, 221184, 3538944, 7077888, 2038431744, 4076863488, 65229815808, 1043677052928, 125241246351360, 250482492702720, 72138957898383360, 144277915796766720, 41552039749468815360, 664832635991501045760, 10637322175864016732160
Offset: 1

Views

Author

Jaroslav Krizek, Dec 25 2016

Keywords

Comments

tau(n) is the number of positive divisors of n (A000005).

Crossrefs

Cf. A000005, A175596 (partial products of A007425), A237349 (partial sums of A211776).

Programs

  • Magma
    [&*[&*[NumberOfDivisors(d): d in Divisors(k)]: k in [1..n]]: n in [1..100]]
  • Mathematica
    FoldList[Times, Table[Product[DivisorSigma[0, d], {d, Divisors@ n}], {n, 22}]] (* Michael De Vlieger, Dec 25 2016 *)

Formula

a(n) = prod_{i=1..n} A211776(i).

A280076 Numbers n such that Sum_{d|n} tau(d) = Product_{d|n} tau(d).

Original entry on oeis.org

1, 4, 9, 25, 49, 121, 169, 289, 361, 529, 841, 961, 1369, 1681, 1849, 2209, 2809, 3481, 3721, 4489, 5041, 5329, 6241, 6889, 7921, 9409, 10201, 10609, 11449, 11881, 12769, 16129, 17161, 18769, 19321, 22201, 22801, 24649, 26569, 27889, 29929, 32041, 32761, 36481
Offset: 1

Views

Author

Jaroslav Krizek, Dec 25 2016

Keywords

Comments

Union of 1 and A001248 (squares of primes).
Numbers n such that A007425(n) = A211776(n).
Numbers n such that A007425(n) = Sum_{d|n} tau(d) = A211776(n) = Product_{d|n} tau(d) = 6.
Also squares of noncomposite numbers (A008578).
Subsequence of A350343. - Lorenzo Sauras Altuzarra, Sep 18 2022

Examples

			9 is a term because Sum_{d|9} tau(d) = 1+2+3 = Product_{d|9} tau(d) = 1*2*3 = 6.

Links

Crossrefs

Cf. A001248, A007425, A008578, A211776, A331294, A350343.

Programs

  • Magma
    [n: n in [1..1000000] | &*[NumberOfDivisors(d): d in Divisors(n)]  eq &+[NumberOfDivisors(d): d in Divisors(n)]]
  • Mathematica
    Select[Range@ 37500, Total@ # == Times @@ # &@ Map[DivisorSigma[0, #] &, Divisors@ #] &] (* Michael De Vlieger, Dec 25 2016 *)
  • PARI
    isok(n) = my(d = divisors(n), nd = apply(numdiv, d)); vecsum(nd) == prod(k=1, #nd, nd[k]); \\ Michel Marcus, Jun 26 2017

Formula

A007425(a(n)) = A211776(a(n)) = 6.
Apparently, a(n) = A331294(n + 3) if n > 5. - Lorenzo Sauras Altuzarra, Sep 18 2022

A280077 Partial sums of A007429 (Sum_{d|n} sigma(d)).

Original entry on oeis.org

1, 5, 10, 21, 28, 48, 57, 83, 101, 129, 142, 197, 212, 248, 283, 340, 359, 431, 452, 529, 574, 626, 651, 781, 819, 879, 937, 1036, 1067, 1207, 1240, 1360, 1425, 1501, 1564, 1762, 1801, 1885, 1960, 2142, 2185, 2365, 2410, 2553, 2679, 2779, 2828, 3113, 3179
Offset: 1

Views

Author

Jaroslav Krizek, Dec 25 2016

Keywords

Comments

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

Links

Crossrefs

Cf. A000203, A237349 (partial sums of A211776), A280078 (partial products of A007429).

Programs

  • Magma
    [&+[&+[SumOfDivisors(d): d in Divisors(k)]: k in [1..n]]: n in [1..100]]
  • PARI
    a(n) = sum(k=1, n, sumdiv(k, d, sigma(d))); \\ Michel Marcus, May 29 2018
  • PARI
    my(N=50, x='x+O('x^N)); Vec(sum(k=1, N, sigma(k)*x^k/(1-x^k))/(1-x)) \\ Seiichi Manyama, Jul 24 2022

Formula

a(n) = Sum_{i=1..n} A007429(i).
a(n) = Sum_{k=1..n} A000203(k) * floor(n/k). - Daniel Suteu, May 28 2018
a(n) = Sum_{k=1..n} A000005(k)/2 * floor(n/k) * floor(1+n/k). - Daniel Suteu, May 28 2018
a(n) ~ Pi^4 * n^2 / 72. - Vaclav Kotesovec, Nov 06 2018
G.f.: (1/(1-x)) * Sum_{k>=1} sigma(k) * x^k/(1 - x^k). - Seiichi Manyama, Jul 24 2022

A237349 a(n) = Sum_{i=1..n} ( Product_{k|i} d(k) ), where d(n) = A000005(n).

Original entry on oeis.org

1, 3, 5, 11, 13, 29, 31, 55, 61, 77, 79, 367, 369, 385, 401, 521, 523, 811, 813, 1101, 1117, 1133, 1135, 10351, 10357, 10373, 10397, 10685, 10687, 14783, 14785, 15505, 15521, 15537, 15553, 62209, 62211, 62227, 62243, 71459, 71461, 75557, 75559, 75847, 76135
Offset: 1

Views

Author

Wesley Ivan Hurt, Feb 06 2014

Keywords

Comments

Sum of all the products formed by multiplying together the number of divisors of each divisor of the numbers from 1 to n.
Partial sums of A211776. [Joerg Arndt, Feb 11 2014]

Examples

			a(3) = 5. Sum_{i=1..3} ( Product_{k|i} d(k) ) =
( Product_{k|1} d(k) ) + ( Product_{k|2} d(k) ) + ( Product_{k|3} d(k) ) = ( d(1) ) + ( d(1) * d(2) ) + ( d(1) * d(3) ) = 1 + (1)(2) + (1)(2) = 5.

Crossrefs

Cf. A000005, A211776.

Programs

  • Maple
    with(numtheory); A237349:=n->add(mul(tau(k)^(1-ceil(i/k)+floor(i/k)), k=1..i), i=1..n); seq(A237349(n), n=1..50);
  • Mathematica
    Table[Sum[Product[DivisorSigma[0, k]^(1-Ceiling[i/k]+Floor[i/k]), {k, i}], {i, n}], {n, 50}]

Formula

a(n) = Sum_{i=1..n} ( Product_{k|i} A000005(k) ).

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.

A280078 Partial products of A007429 (Sum_{d|n} sigma(d)).

Original entry on oeis.org

1, 4, 20, 220, 1540, 30800, 277200, 7207200, 129729600, 3632428800, 47221574400, 2597186592000, 38957798880000, 1402480759680000, 49086826588800000, 2797949115561600000, 53161033195670400000, 3827594390088268800000, 80379482191853644800000
Offset: 1

Views

Author

Jaroslav Krizek, Dec 25 2016

Keywords

Comments

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

Crossrefs

Cf. A000203, A280075 (partial products of A211776), A280077 (partial sums of A007429).

Programs

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

Formula

a(n) = Product_{i=1..n} A007429(i).

A334470 a(n) = Product_{d|n} (A253139(n) / tau(d)) where A253139(n) = lcm_{d|n} tau(d).

Original entry on oeis.org

1, 2, 2, 36, 2, 16, 2, 864, 36, 16, 2, 10368, 2, 16, 16, 6480000, 2, 10368, 2, 10368, 16, 16, 2, 11943936, 36, 16, 864, 10368, 2, 4096, 2, 64800000, 16, 16, 16, 2176782336, 2, 16, 16, 11943936, 2, 4096, 2, 10368, 10368, 16, 2, 1343692800000000, 36, 10368, 16
Offset: 1

Views

Author

Jaroslav Krizek, May 01 2020

Keywords

Examples

			For n = 6; divisors d of 6: {1, 2, 3, 6}; tau(d): {1, 2, 2, 4}; lcm_{d|6} tau(d) = 4; a(6) = 4/1 * 4/2 * 4/2 * 4/4 = 16.

Crossrefs

Cf. A334471 (similar sequence with sigma(d)).
Cf. A000005, A000040, A211776, A253139.

Programs

  • Magma
    [&*[ LCM([#Divisors(d): d in Divisors(n)]) / #Divisors(d): d in Divisors(n)]: n in [1..100]]
  • Mathematica
    a[n_] := (LCM @@ (s = DivisorSigma[0, Divisors[n]]))^Length[s] / Times @@ s; Array[a, 51] (* Amiram Eldar, May 02 2020 *)
  • PARI
    a(n) = {my(d=divisors(n), lcmd = lcm(vector(#d, k, numdiv(d[k])))); vecprod(vector(#d, k, lcmd/numdiv(d[k])));} \\ Michel Marcus, May 02 2020

Formula

a(n) = ((lcm_{d|n} tau(d))^tau(n)) / Product_{d|n} tau(d).
a(n) = A253139(n)^A000005(n) / A211776(n).
a(p) = 2 for p = primes (A000040).
a(n) = 2^(k*2^(k-1)) if n is a product of k distinct primes.

A306705 a(n) = Product_{d|n} d*tau(d), where tau(k) = the number of the divisors of k (A000005).

Original entry on oeis.org

1, 4, 6, 48, 10, 576, 14, 1536, 162, 1600, 22, 497664, 26, 3136, 3600, 122880, 34, 1679616, 38, 2304000, 7056, 7744, 46, 3057647616, 750, 10816, 17496, 6322176, 58, 3317760000, 62, 23592960, 17424, 18496, 19600, 470184984576, 74, 23104, 24336, 23592960000, 82
Offset: 1

Views

Author

Jaroslav Krizek, Mar 05 2019

Keywords

Examples

			a(6) = 1*tau(1) * 2*tau(2) * 3*tau(3) * 6*tau(6) = (1*1) * (2*2) * (3*2) * (6*4) = 576.

Links

Crossrefs

Cf. A000005, A060640 (Sum_{d|n} d*tau(d)), A007955, A211776.

Programs

  • Magma
    [&*[d * NumberOfDivisors(d): d in Divisors(n)]: n in [1..100]]
  • Maple
    f:= proc(n) uses numtheory; local d;
  mul(d*tau(d),d = divisors(n))
end proc:
map(f, [$1..100]); # Robert Israel, Mar 24 2019
  • Mathematica
    Table[n^(DivisorSigma[0, n]/2) * Product[DivisorSigma[0, k], {k, Divisors[n]}], {n, 1, 60}] (* Vaclav Kotesovec, Mar 10 2019 *)
  • PARI
    a(n) = my(res = 1); fordiv(n, d, res *= d*numdiv(d)); res; \\ Michel Marcus, Mar 06 2019

Formula

a(p) = 2p for p = primes (A000040).
a(n) = (Product_{d|n} tau(d)) * (Product_{d|n} d) = A211776(n) * A007955(n).
From Robert Israel, Mar 24 2019: (Start)
a(p^k) = (k+1)! * p^(k*(k+1)/2) for primes p.
a(p*q) = 16*p^2*q^2 if p and q are distinct primes. (End)

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).
Showing 1-10 of 11 results. Next

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