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.

A336722 a(n) = gcd(tau(n), sigma(n), pod(n)) where tau(k) is the number of divisors of k (A000005), sigma(k) is the sum of divisors of k (A000203) and pod(k) is the product of divisors of k (A007955).

Original entry on oeis.org

1, 1, 1, 1, 1, 4, 1, 1, 1, 2, 1, 2, 1, 4, 1, 1, 1, 3, 1, 2, 1, 4, 1, 4, 1, 2, 1, 2, 1, 8, 1, 1, 1, 2, 1, 1, 1, 4, 1, 2, 1, 8, 1, 2, 3, 4, 1, 2, 1, 1, 1, 2, 1, 8, 1, 8, 1, 2, 1, 12, 1, 4, 1, 1, 1, 8, 1, 2, 1, 8, 1, 3, 1, 2, 1, 2, 1, 8, 1, 2, 1, 2, 1, 4, 1, 4, 1, 4, 1, 6, 1, 2, 1, 4, 1, 12, 1, 1, 3, 1, 1, 8, 1, 2, 1
Offset: 1

Views

Author

Jaroslav Krizek, Aug 01 2020

Keywords

Comments

a(n) = tau(n) for numbers n: 1, 6, 14, 22, 30, 38, 42, 46, 54, 56, 60, 62, 66, 70, 78, 86, 94, 96, 102, ...

Examples

			a(6) = gcd(tau(6), sigma(6), pod(6)) = gcd(4, 12, 36) = 4.

Links

Crossrefs

Cf. A009205 (gcd(tau(n), sigma(n))), A306671 (gcd(tau(n), pod(n))), A306682 (gcd(sigma(n), pod(n))).
Cf. A000005 (tau(n)), A000203 (sigma(n)), A007955 (pod(n)), A336723 (lcm(tau(n), sigma(n), pod(n))).
Cf. A277521 (numbers k such that a(k) = tau(k) and simultaneously A336723(k) = pod(k)).

Programs

  • Magma
    [GCD([#Divisors(n), &+Divisors(n), &*Divisors(n)]): n in [1..100]];
  • Mathematica
    a[n_] := GCD @@ {(d = DivisorSigma[0,n]), DivisorSigma[1, n], n^(d/2)}; Array[a, 100] (* Amiram Eldar, Aug 01 2020 *)
  • PARI
    A007955(n) = if(issquare(n, &n), n^numdiv(n^2), n^(numdiv(n)/2)); \\ From A007955
A336722(n) = gcd(A007955(n), gcd(numdiv(n), sigma(n))); \\ Antti Karttunen, Aug 10 2020

Formula

a(p) = 1 for p = primes (A000040).
a(n) = gcd(A007955(n), A009205(n)). - Antti Karttunen, Aug 10 2020

Extensions

Data section extended up to a(105) by Antti Karttunen, Aug 10 2020

A308867 Sum of all the parts in the partitions of n into 6 parts.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 6, 7, 16, 27, 50, 77, 132, 182, 280, 390, 560, 748, 1044, 1349, 1800, 2310, 2992, 3749, 4776, 5875, 7332, 8937, 10948, 13166, 15960, 18972, 22688, 26763, 31654, 36995, 43416, 50320, 58520, 67431, 77800, 89052, 102144, 116186, 132396, 149895
Offset: 0

Views

Author

Wesley Ivan Hurt, Jun 29 2019

Keywords

Links

Crossrefs

Cf. A238340, A308868, A308869, A306670, A306671, A308872, A308873.

Programs

  • Mathematica
    Table[n*Sum[Sum[Sum[Sum[Sum[1, {i, j, Floor[(n - j - k - l - m)/2]}], {j, k, Floor[(n - k - l - m)/3]}], {k, l, Floor[(n - l - m)/4]}], {l, m, Floor[(n - m)/5]}], {m, Floor[n/6]}], {n, 0, 100}]
Table[Total[Flatten[IntegerPartitions[n,{6}]]],{n,0,50}] (* Harvey P. Dale, Oct 29 2024 *)

Formula

a(n) = n * Sum_{m=1..floor(n/6)} Sum_{l=m..floor((n-m)/5)} Sum_{k=l..floor((n-l-m)/4)} Sum_{j=k..floor((n-k-l-m)/3)} Sum_{i=j..floor((n-j-k-l-m)/2)} 1.
a(n) = n * A238340(n).
a(n) = A308868(n) + A308869(n) + A306670(n) + A306671(n) + A308872(n) + A308873(n).

A308868 Sum of the smallest parts in the partitions of n into 6 parts.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 5, 7, 12, 15, 22, 29, 40, 51, 70, 86, 112, 139, 176, 214, 269, 321, 394, 470, 567, 668, 801, 933, 1103, 1281, 1498, 1725, 2007, 2293, 2643, 3010, 3443, 3897, 4439, 4995, 5652, 6341, 7135, 7967, 8933, 9930, 11079, 12283, 13645
Offset: 0

Views

Author

Wesley Ivan Hurt, Jun 29 2019

Keywords

Links

Crossrefs

Cf. A238340, A308867, A308869, A306670, A306671, A308872, A308873.

Programs

  • Mathematica
    Table[Sum[Sum[Sum[Sum[Sum[m, {i, j, Floor[(n - j - k - l - m)/2]}], {j, k, Floor[(n - k - l - m)/3]}], {k, l, Floor[(n - l - m)/4]}], {l, m, Floor[(n - m)/5]}], {m, Floor[n/6]}], {n, 0, 50}]

Formula

a(n) = n * Sum_{m=1..floor(n/6)} Sum_{l=m..floor((n-m)/5)} Sum_{k=l..floor((n-l-m)/4)} Sum_{j=k..floor((n-k-l-m)/3)} Sum_{i=j..floor((n-j-k-l-m)/2)} m.
a(n) = A308867(n) - A308869(n) - A306670(n) - A306671(n) - A308872(n) - A308873(n).

A308869 Sum of the fifth largest parts in the partitions of n into 6 parts.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 5, 8, 13, 17, 25, 34, 48, 63, 86, 109, 143, 182, 232, 288, 363, 442, 547, 662, 804, 961, 1157, 1368, 1626, 1909, 2245, 2613, 3054, 3525, 4082, 4688, 5388, 6150, 7031, 7974, 9059, 10231, 11560, 12991, 14614, 16346, 18300, 20400
Offset: 0

Views

Author

Wesley Ivan Hurt, Jun 29 2019

Keywords

Links

Crossrefs

Cf. A238340, A308867, A308868, A306670, A306671, A308872, A308873.

Programs

  • Mathematica
    Table[Sum[Sum[Sum[Sum[Sum[l, {i, j, Floor[(n - j - k - l - m)/2]}], {j, k, Floor[(n - k - l - m)/3]}], {k, l, Floor[(n - l - m)/4]}], {l, m, Floor[(n - m)/5]}], {m, Floor[n/6]}], {n, 0, 50}]

Formula

a(n) = Sum_{m=1..floor(n/6)} Sum_{l=m..floor((n-m)/5)} Sum_{k=l..floor((n-l-m)/4)} Sum_{j=k..floor((n-k-l-m)/3)} Sum_{i=j..floor((n-j-k-l-m)/2)} l.
a(n) = A308867(n) - A308868(n) - A306670(n) - A306671(n) - A308872(n) - A308873(n).

A308872 Sum of the second largest parts in the partitions of n into 6 parts.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 1, 3, 5, 10, 15, 27, 37, 59, 82, 120, 160, 227, 293, 396, 508, 664, 832, 1068, 1314, 1650, 2012, 2477, 2980, 3628, 4314, 5178, 6111, 7250, 8477, 9975, 11566, 13483, 15543, 17970, 20577, 23646, 26907, 30712, 34785, 39469, 44472, 50217
Offset: 0

Views

Author

Wesley Ivan Hurt, Jun 29 2019

Keywords

Links

Crossrefs

Cf. A238340, A308867, A308868, A308869, A306670, A306671, A308873.

Programs

  • Mathematica
    Table[Sum[Sum[Sum[Sum[Sum[i, {i, j, Floor[(n - j - k - l - m)/2]}], {j, k, Floor[(n - k - l - m)/3]}], {k, l, Floor[(n - l - m)/4]}], {l, m, Floor[(n - m)/5]}], {m, Floor[n/6]}], {n, 0, 50}]

Formula

a(n) = Sum_{m=1..floor(n/6)} Sum_{l=m..floor((n-m)/5)} Sum_{k=l..floor((n-l-m)/4)} Sum_{j=k..floor((n-k-l-m)/3)} Sum_{i=j..floor((n-j-k-l-m)/2)} i.
a(n) = A308867(n) - A308868(n) - A308869(n) - A306670(n) - A306671(n) - A308873(n).

A308873 Sum of the largest parts in the partitions of n into 6 parts.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 2, 5, 9, 17, 27, 46, 67, 103, 146, 210, 285, 396, 520, 694, 896, 1162, 1466, 1865, 2310, 2881, 3525, 4321, 5215, 6317, 7535, 9011, 10653, 12603, 14761, 17316, 20113, 23390, 26990, 31146, 35698, 40939, 46632, 53139, 60221, 68236, 76931
Offset: 0

Views

Author

Wesley Ivan Hurt, Jun 29 2019

Keywords

Links

Crossrefs

Cf. A238340, A308867, A308868, A308869, A306670, A306671, A308872.

Programs

  • Mathematica
    Table[Sum[Sum[Sum[Sum[Sum[(n - i - j - k - l - m), {i, j, Floor[(n - j - k - l - m)/2]}], {j, k, Floor[(n - k - l - m)/3]}], {k, l, Floor[(n - l - m)/4]}], {l, m, Floor[(n - m)/5]}], {m, Floor[n/6]}], {n, 0, 50}]

Formula

a(n) = Sum_{m=1..floor(n/6)} Sum_{l=m..floor((n-m)/5)} Sum_{k=l..floor((n-l-m)/4)} Sum_{j=k..floor((n-k-l-m)/3)} Sum_{i=j..floor((n-j-k-l-m)/2)} (n-i-j-k-l-m).
a(n) = A308867(n) - A308868(n) - A308869(n) - A306670(n) - A306671(n) - A308872(n).

A334662 a(n) = Sum_{d|n} gcd(tau(d), pod(d)), where pod(k) is the product of the divisors of k (A007955).

Original entry on oeis.org

1, 3, 2, 4, 2, 8, 2, 8, 5, 8, 2, 15, 2, 8, 4, 9, 2, 17, 2, 11, 4, 8, 2, 27, 3, 8, 6, 11, 2, 22, 2, 11, 4, 8, 4, 33, 2, 8, 4, 23, 2, 22, 2, 11, 10, 8, 2, 30, 3, 11, 4, 11, 2, 26, 4, 23, 4, 8, 2, 43, 2, 8, 10, 12, 4, 22, 2, 11, 4, 22, 2, 57, 2, 8, 8, 11, 4, 22
Offset: 1

Views

Author

Jaroslav Krizek, May 07 2020

Keywords

Comments

Inverse Möbius transform of A306671. - Antti Karttunen, May 19 2020

Examples

			a(6) = gcd(tau(1), pod(1)) + gcd(tau(2), pod(2)) + gcd(tau(3), pod(3)) + gcd(tau(6), pod(6)) = gcd(1, 1) + gcd(2, 2) + gcd(2, 3) + gcd(4, 36) = 1 + 2 + 1 + 4 = 8.

Links

Crossrefs

Cf. A334579 (Sum_{d|n} gcd(tau(d), sigma(d))), A334663 (Sum_{d|n} gcd(sigma(d), pod(d))).
Cf. A000005 (tau(n)), A007955 (pod(n)), A306671 (gcd(tau(n), pod(n))).

Programs

  • Magma
    [&+[GCD(#Divisors(d), &*Divisors(d)): d in Divisors(n)]: n in [1..100]]
  • PARI
    a(n) = sumdiv(n, d, gcd(numdiv(d), vecprod(divisors(d)))); \\ Michel Marcus, May 08 2020

Formula

a(p) = 2 for p = odd primes (A065091).

A324555 a(n) = the smallest number m such that gcd(tau(m), pod(m)) = n 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, 9, 6, 400, 12, 3136, 24, 36, 80, 123904, 60, 692224, 448, 2025, 120, 18939904, 180, 94633984, 240, 35721, 11264, 2218786816, 360, 10000, 53248, 900, 1344, 225754218496, 720, 1031865892864, 840, 7144929, 1114112, 1960000, 1260, 94076963651584, 4980736
Offset: 1

Views

Author

Jaroslav Krizek, Mar 05 2019

Keywords

Comments

a(n) = the smallest number m such that A306671(m) = n.
If a(17) exists, it must be bigger than 10^7.

Examples

			For n=3; a(3) = 9 because gcd(tau(9), pod(9)) = gcd (3, 27) = 3 and 9 is the smallest.

Crossrefs

Cf. A000005, A007955, A306671.

Programs

  • Magma
    [Min([n: n in[1..10^6] | GCD(NumberOfDivisors(n), &*[d: d in Divisors(n)]) eq k]): k in [1..16]]
  • Mathematica
    Array[Block[{m = 1}, While[GCD[DivisorSigma[0, m], Times @@ Divisors@ m] != #, m++]; m] &, 16] (* Michael De Vlieger, Mar 24 2019 *)
  • PARI
    a(n) = {my(k=1, vk = divisors(k)); while(gcd(#vk, vecprod(vk)) != n, k++; vk = divisors(k)); k;} \\ Michel Marcus, Mar 06 2019

Extensions

a(17)-a(38) from Jon E. Schoenfield, Mar 07 2019

A334730 a(n) = Product_{d|n} gcd(tau(d), pod(d)) where tau(k) is the number of divisors of k (A000005) and pod(k) is the product of divisors of k (A007955).

Original entry on oeis.org

1, 2, 1, 2, 1, 8, 1, 8, 3, 8, 1, 48, 1, 8, 1, 8, 1, 144, 1, 16, 1, 8, 1, 1536, 1, 8, 3, 16, 1, 256, 1, 16, 1, 8, 1, 7776, 1, 8, 1, 512, 1, 256, 1, 16, 9, 8, 1, 3072, 1, 16, 1, 16, 1, 1152, 1, 512, 1, 8, 1, 36864, 1, 8, 9, 16, 1, 256, 1, 16, 1, 256, 1, 2985984, 1, 8, 3, 16, 1, 256, 1
Offset: 1

Views

Author

Jaroslav Krizek, May 09 2020

Keywords

Examples

			a(6) = gcd(tau(1), pod(1)) * gcd(tau(2), pod(2)) * gcd(tau(3), pod(3)) * gcd(tau(6), pod(6)) = gcd(1, 1) * gcd(2, 2) * gcd(2, 3) * gcd(4, 36) = 1 * 2 * 1 * 4 = 8.

Links

Crossrefs

Cf. A334729 (Product_{d|n} gcd(tau(d), sigma(d))), A334662 (Sum_{d|n} gcd(tau(d), pod(d))).
Cf. A000005 (tau(n)), A007955 (pod(n)), A306671 (gcd(tau(n), pod(n))).

Programs

  • Magma
    [&*[GCD(#Divisors(d), &*Divisors(d)): d in Divisors(n)]: n in [1..100]]
  • Mathematica
    a[n_] := Product[GCD[DivisorSigma[0, d], d^(DivisorSigma[0, d]/2)], {d, Divisors[n]}]; Array[a, 100] (* Amiram Eldar, May 09 2020 *)
  • PARI
    pod(n) = vecprod(divisors(n));
a(n) = my(d=divisors(n)); prod(k=1, #d, gcd(numdiv(d[k]), pod(d[k]))); \\ Michel Marcus, May 09-11 2020

Formula

a(p) = 1 for p = odd primes (A065091).

A308870 Sum of the fourth largest parts in the partitions of n into 6 parts.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 6, 9, 15, 20, 31, 42, 61, 80, 112, 143, 191, 243, 316, 393, 501, 613, 767, 930, 1141, 1367, 1659, 1967, 2354, 2769, 3279, 3824, 4491, 5196, 6047, 6956, 8031, 9181, 10536, 11971, 13647, 15434, 17497, 19690, 22211, 24880, 27929
Offset: 0

Views

Author

Wesley Ivan Hurt, Jun 29 2019

Keywords

Links

Crossrefs

Cf. A238340, A308867, A308868, A308869, A306671, A308872, A308873.

Programs

  • Mathematica
    Table[Sum[Sum[Sum[Sum[Sum[k, {i, j, Floor[(n - j - k - l - m)/2]}], {j, k, Floor[(n - k - l - m)/3]}], {k, l, Floor[(n - l - m)/4]}], {l, m, Floor[(n - m)/5]}], {m, Floor[n/6]}], {n, 0, 50}]
Table[Total[IntegerPartitions[n,{6}][[;;,4]]],{n,0,50}] (* Harvey P. Dale, Jul 30 2024 *)

Formula

a(n) = Sum_{m=1..floor(n/6)} Sum_{l=m..floor((n-m)/5)} Sum_{k=l..floor((n-l-m)/4)} Sum_{j=k..floor((n-k-l-m)/3)} Sum_{i=j..floor((n-j-k-l-m)/2)} k.
a(n) = A308867(n) - A308868(n) - A308869(n) - A306671(n) - A308872(n) - A308873(n).
Showing 1-10 of 10 results.

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