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.

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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).

A307100 a(n) = Sum_{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, 5, 7, 29, 11, 155, 15, 285, 88, 415, 23, 10547, 27, 803, 917, 5405, 35, 35228, 39, 48439, 1785, 1963, 47, 2665011, 386, 2735, 3004, 132539, 59, 6481465, 63, 202013, 4385, 4663, 4925, 90744884, 75, 5819, 6117, 20528695, 83, 24896285, 87, 513091, 547748, 8515
Offset: 1

Views

Author

Jaroslav Krizek, Apr 25 2019

Keywords

Comments

n divides a(n) for n = 1, 21, 333592, ...

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) = 155.

Crossrefs

Cf. A000005, A007955, A307101.

Programs

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

Formula

a(p) = 2*(p+1) for p = primes (A000040).

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).

A307892 a(n) = lcm(tau(n), pod(n)) / n, where tau(k) = the number of divisors of k (A000005) and pod(n) = the product of divisors of k (A007955).

Original entry on oeis.org

1, 1, 2, 6, 2, 6, 2, 8, 3, 10, 2, 144, 2, 14, 60, 320, 2, 324, 2, 1200, 84, 22, 2, 13824, 15, 26, 108, 2352, 2, 27000, 2, 3072, 132, 34, 140, 279936, 2, 38, 156, 64000, 2, 74088, 2, 5808, 4050, 46, 2, 26542080, 21, 7500, 204, 8112, 2, 157464, 220, 175616, 228
Offset: 1

Views

Author

Jaroslav Krizek, May 03 2019

Keywords

Comments

n divides lcm(tau(n), pod(n)) for all n >= 1.

Examples

			For n=4: a(4) = lcm(tau(4), pod(4))/4 = lcm(3, 8)/4 = 24/4 = 6.

Links

Crossrefs

Cf. A000005, A007955, A324528.

Programs

  • Magma
    [LCM(NumberOfDivisors(n), &*[d: d in Divisors(n)]) / n: n in [1.. 10^5]];
  • Mathematica
    Table[(LCM[DivisorSigma[0,n],Times@@Divisors[n]])/n,{n,60}] (* Harvey P. Dale, Sep 22 2024 *)

Formula

a(n) = A324528(n) / n.

A322672 a(n) = Product_{d|n} (pod(d)/d) where pod(k) is the product of the divisors of k (A007955).

Original entry on oeis.org

1, 1, 1, 2, 1, 6, 1, 16, 3, 10, 1, 1728, 1, 14, 15, 1024, 1, 5832, 1, 8000, 21, 22, 1, 191102976, 5, 26, 81, 21952, 1, 24300000, 1, 1048576, 33, 34, 35, 470184984576, 1, 38, 39, 4096000000, 1, 130691232, 1, 85184, 91125, 46, 1, 64925062108545024, 7, 125000, 51
Offset: 1

Views

Author

Jaroslav Krizek, Dec 23 2018

Keywords

Examples

			For n = 6; a(6) = pod(1)/1 * pod(2)/2 * pod(3)/3 * pod(6)/6 = 1/1 * 2/2 * 3/3 * 36/6 = 6.

Crossrefs

Cf. A006881, A007955, A322671.

Programs

  • Magma
    [&*[&*[c: c in Divisors(d)] / d: d in Divisors(n)]: n in [1..100]];
  • Mathematica
    Array[Product[Apply[Times, Divisors@ d]/d, {d, Divisors@ #}] &, 51] (* Michael De Vlieger, Jan 19 2019 *)
  • PARI
    a(n) = my(x=1); fordiv(n, d, x*=vecprod(divisors(d))/d); x; \\ Michel Marcus, Dec 23 2018

Formula

a(n) = n for n = 1 and squarefree semiprimes (A006881).

A323706 a(n) = numerator of Sum_{d|n} tau(d)/pod(d) where tau(k) = the number of the divisors of k (A000005) and pod(k) = the product of the divisors of k (A007955).

Original entry on oeis.org

1, 2, 5, 19, 7, 25, 9, 39, 16, 61, 13, 101, 15, 113, 469, 2501, 19, 2809, 21, 11263, 865, 265, 25, 133489, 178, 365, 1300, 29431, 31, 327601, 33, 40019, 2017, 613, 2069, 3659761, 39, 761, 2773, 921041, 43, 1203049, 45, 109255, 66692, 1105, 49, 410700293, 444
Offset: 1

Views

Author

Jaroslav Krizek, Jan 24 2019

Keywords

Comments

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

Examples

			For n=4; Sum_{d|4} tau(d)/pod(d) = tau(1)/pod(1) + tau(2)/pod(2) + tau(4)/pod(4) = 1/1 + 2/2 + 3/8 = 19/8; a(4) = 19.

Crossrefs

Cf. A000005, A007955, A323707 (denominator).

Programs

  • Magma
    [Numerator(&+[NumberOfDivisors(d) / &*[c: c in Divisors(d)]: d in Divisors(n)]): n in [1..100]]
  • PARI
    a(n) = numerator(sumdiv(n, d, numdiv(d)/vecprod(divisors(d)))); \\ Michel Marcus, Jan 25 2019

Formula

a(p) = p + 2 for prime p > 2.

A323707 a(n) = denominator of Sum_{d|n} tau(d)/pod(d) where tau(k) = the number of the divisors of k (A000005) and pod(k) = the product of the divisors of k (A007955).

Original entry on oeis.org

1, 1, 3, 8, 5, 9, 7, 16, 9, 25, 11, 32, 13, 49, 225, 1024, 17, 972, 19, 4000, 441, 121, 23, 41472, 125, 169, 729, 10976, 29, 101250, 31, 16384, 1089, 289, 1225, 1119744, 37, 361, 1521, 320000, 41, 388962, 43, 42592, 30375, 529, 47, 127401984, 343, 62500, 2601
Offset: 1

Views

Author

Jaroslav Krizek, Jan 26 2019

Keywords

Comments

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

Examples

			For n=4; Sum_{d|4} tau(d)/pod(d) = tau(1)/pod(1) + tau(2)/pod(2) + tau(4)/pod(4) = 1/1 + 2/2 + 3/8 = 19/8; a(4) = 8.

Crossrefs

Cf. A000005, A007955, A323706 (numerator).

Programs

  • Magma
    [Denominator(&+[NumberOfDivisors(d) / &*[c: c in Divisors(d)]: d in Divisors(n)]): n in [1..100]]
  • Mathematica
    Array[Denominator@ DivisorSum[#, DivisorSigma[0, #]/Apply[Times, Divisors@ #] &] &, 51] (* Michael De Vlieger, Jan 27 2019 *)
  • PARI
    a(n) = denominator(sumdiv(n, d, numdiv(d)/vecprod(divisors(d)))); \\ Michel Marcus, Jan 26 2019

Formula

a(n) = 1 for n = 1, 2, ... (no other n <= 5*10^6).
a(p) = p for prime p > 2.

A323762 Numbers m such that Product_{d|m} (pod(d)/tau(d)) is an integer h where pod(k) = the product of the divisors of k (A007955) and tau(k) = the number of the divisors of k (A000005).

Original entry on oeis.org

1, 2, 12, 18, 24, 36, 54, 60, 72, 84, 90, 108, 120, 126, 132, 150, 156, 168, 180, 198, 204, 216, 228, 234, 240, 252, 264, 270, 276, 294, 300, 306, 312, 342, 348, 360, 372, 378, 396, 408, 414, 420, 444, 450, 456, 468, 480, 492, 504, 516, 522, 540, 552, 558, 564
Offset: 1

Views

Author

Jaroslav Krizek, Jan 27 2019

Keywords

Comments

Corresponding values of integers h: 1, 1, 10368, 118098, 6879707136, 101559956668416, ...
Product_{d|n} (pod(d)/tau(d)) > 1 for all n > 2.

Examples

			12 is a term because Product_{d|12} (pod(d)/tau(d)) = (pod(1)/tau(1))*(pod(2)/tau(2))*(pod(3)/tau(3))*(pod(4)/tau(4))*(pod(6)/tau(6))*(pod(12)/tau(12)) = (1/1)*(2/2)*(3/2)*(8/3)*(36/4)*(1728/6) = 10368 (integer).

Crossrefs

Cf. A000005, A007955, A323760, A323761.

Programs

  • Magma
    [n: n in [1..1000] | Denominator(&*[&*[c: c in Divisors(d)] / NumberOfDivisors(d): d in Divisors(n)]) eq 1]
  • PARI
    isok(n) = my(p=1, vd); fordiv(n, d, vd = divisors(d); p *= vecprod(vd)/#vd); denominator(p) == 1; \\ Michel Marcus, Jan 30 2019

Formula

A323761(a(n)) = 1.

A324363 a(n) = numerator of Sum_{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, 5, 7, 27, 11, 25, 15, 231, 76, 97, 23, 2185, 27, 369, 91, 3727, 35, 9049, 39, 19041, 1565, 887, 47, 48775, 306, 615, 2092, 65, 59, 63601, 63, 119327, 1259, 1042, 4143, 55891387, 75, 2595, 5243, 1278633, 83, 713689, 87, 96711, 125216, 3785, 95, 339061279
Offset: 1

Views

Author

Jaroslav Krizek, Feb 23 2019

Keywords

Comments

Sum_{d|n} sigma(d)/pod(d) > 1 for all n > 1.

Examples

			For n=4; Sum_{d|4} sigma(d)/pod(d) = sigma(1)/pod(1) + sigma(2)/pod(2) + sigma(4)/pod(4) = 1/1 + 3/2 + 7/8 = 27/8; a(4) = 27.

Crossrefs

Cf. A000040, A000203, A007955, A324364 (denominators).

Programs

  • Magma
    [Numerator(&+[SumOfDivisors(d) / &*[c: c in Divisors(d)]: d in Divisors(n)]): n in [1..100]]
  • Mathematica
    Array[Numerator@ DivisorSum[#, Total[#]/(Times @@ #) &@ Divisors@ # &] &, 48] (* Michael De Vlieger, Feb 24 2019 *)
  • PARI
    a(n) = numerator(sumdiv(n, d, sigma(d)/vecprod(divisors(d)))); \\ Michel Marcus, Feb 23 2019

Formula

a(p) = 2p+1 for p = primes (A000040).

A324364 a(n) = denominator of Sum_{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, 2, 3, 8, 5, 6, 7, 64, 27, 25, 11, 432, 13, 98, 25, 1024, 17, 1944, 19, 4000, 441, 242, 23, 9216, 125, 169, 729, 14, 29, 11250, 31, 32768, 363, 289, 1225, 10077696, 37, 722, 1521, 256000, 41, 129654, 43, 21296, 30375, 1058, 47, 63700992, 343, 125000, 867
Offset: 1

Views

Author

Jaroslav Krizek, Feb 23 2019

Keywords

Comments

Sum_{d|n} sigma(d)/pod(d) > 1 for all n > 1.

Examples

			For n=4; Sum_{d|4} sigma(d)/pod(d) = sigma(1)/pod(1) + sigma(2)/pod(2) + sigma(4)/pod(4) = 1/1 + 3/2 + 7/8 = 27/8; a(4) = 8.

Crossrefs

Cf. A000040, A000203, A007955, A008578, A324363 (numerators).

Programs

  • Magma
    [Denominator(&+[SumOfDivisors(d) / &*[c: c in Divisors(d)]: d in Divisors(n)]): n in [1..100]]
  • Mathematica
    Array[Denominator@ DivisorSum[#, Total[#]/(Times @@ #) &@ Divisors@ # &] &, 51] (* Michael De Vlieger, Feb 24 2019 *)
  • PARI
    a(n) = denominator(sumdiv(n, d, sigma(d)/vecprod(divisors(d)))); \\ Michel Marcus, Feb 23 2019

Formula

a(p) = p for p = primes (A000040).
a(n) = n for n = 6 or when n is a noncomposite (in A008578).
a(n) = 1 for n = 1, ... (no other n <= 10^5).
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