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 1-8 of 8 results.

A063752 Numbers k such that cototient(k) is a square.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 8, 11, 13, 17, 19, 21, 23, 24, 27, 28, 29, 31, 32, 37, 41, 43, 47, 53, 54, 59, 61, 67, 68, 69, 71, 73, 79, 83, 89, 96, 97, 101, 103, 107, 109, 112, 113, 124, 125, 127, 128, 131, 133, 137, 139, 141, 149, 151, 157, 163, 167, 173, 179, 181, 189, 191
Offset: 1

Views

Author

Jason Earls, Aug 11 2001

Keywords

Comments

Some different families and subsequences of integers belong to this sequence, see the file "Subfamilies and subsequences" for more details, with data, comments, proofs, formulas and examples. - Bernard Schott, Mar 05 2019

Links

Crossrefs

Cf. A000010, A051953.
Subsequences: A000396, A246551, A323916, A323917, A323918, A306670.

Programs

  • Magma
    [n: n in [1..200] | IsSquare(n - EulerPhi(n))]; // Vincenzo Librandi, Jan 11 2019
  • Mathematica
    Select[Range[200], IntegerQ[Sqrt[# - EulerPhi[#]]]&] (* Jean-François Alcover, Nov 06 2016 *)
  • PARI
    j=[]; for(n=1,400,x=n-eulerphi(n); if(issquare(x),j=concat(j,n))); j
  • PARI
    { n=0; for (m=1, 10^9, if (issquare(m - eulerphi(m)), write("b063752.txt", n++, " ", m); if (n==1000, break)) ) } \\ Harry J. Smith, Aug 29 2009

Formula

a(n) seems to be asymptotic to c * n * log(n) with c = 1.7... (all primes are in the sequence since cototient(p) = 1). - Benoit Cloitre, Sep 08 2002

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).

A323918 Numbers k with exactly two distinct prime divisors and such that cototient(k) is a square, where: k = p^(2s) * q^(2t+1) with s >= 1, t >= 0, p <> q primes and such that p * (p+q-1) = M^2.

Original entry on oeis.org

28, 68, 112, 124, 272, 284, 388, 448, 496, 508, 657, 796, 964, 1025, 1088, 1136, 1348, 1372, 1552, 1792, 1796, 1984, 2032, 2169, 2308, 2588, 3184, 3524, 3856, 3868, 4352, 4544, 4604, 4996, 5392, 5488, 5913, 6025, 6057, 6208, 6268, 7168, 7184, 7936, 8128, 9232, 9244
Offset: 1

Views

Author

Bernard Schott, Feb 09 2019

Keywords

Comments

This is the second subsequence of A323916, the first one is A323917.
Some values of (k,p,q,M): (28,2,7,2), (68,2,17,3), (124,2,31,4), (284,2,71,6), (388,97,7), (657,3,73,5).
The primitive terms of this sequence are the products p^2 * q, with p,q which satisfy p*(p+q-1) = M^2; the first ones are 28, 68, 124, 284, 388, 508, 657, 796. Then, the integers (p^2 * q) * p^2 and (p^2 * q) * q^2 are new terms of the general sequence.
Except 6, all the even perfect numbers of A000396 belong to this sequence.
See the file "Subfamilies of terms" in A063752 for more details, proofs with data, comments, formulas and examples.

Examples

			272 = 2^4 * 17, M = 2*(2+17-1) = 6^2 and cototient(272) = (2^1 * 17^0 * 6)^2 = 12^2.
1025 = 5^2 * 41 and cototient(1025) = 5 * (5+41-1) = 15^2.
Perfect number: 8128 = 2^6 * 127 and cototient(8128) = 64^2.

Crossrefs

Cf. A000396, A051953, A063752, A246551, A323916, A323917, A306670.

Programs

  • PARI
    isok(n) = (omega(n)==2) && issquare(n - eulerphi(n)) && ((factor(n)[1,2] % 2) != (factor(n)[2,2] % 2)); \\ Michel Marcus, Feb 10 2019

Formula

cototient(p^2 * q) = p * (p + q - 1) = M^2;
cototient(k) = (p^(s-1) * q^t * M)^2 with k as in the name of this sequence.

A308871 Sum of the third largest parts in the partitions of n into 6 parts.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 1, 2, 4, 7, 11, 19, 26, 40, 57, 81, 109, 153, 198, 264, 342, 442, 556, 710, 875, 1093, 1338, 1638, 1975, 2398, 2855, 3416, 4040, 4779, 5595, 6573, 7627, 8875, 10244, 11822, 13549, 15553, 17707, 20187, 22883, 25935, 29239, 32991, 37010
Offset: 0

Views

Author

Wesley Ivan Hurt, Jun 29 2019

Keywords

Links

Crossrefs

Cf. A238340, A308867, A308868, A308869, A306670, A308872, A308873.

Programs

  • Mathematica
    Table[Sum[Sum[Sum[Sum[Sum[j, {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)} j.
a(n) = A308867(n) - A308868(n) - A308869(n) - A306670(n) - A308872(n) - A308873(n).
Showing 1-8 of 8 results.

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