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 1-3 of 3 results.

A328023 Heinz number of the multiset of differences between consecutive divisors of n.

Original entry on oeis.org

1, 2, 3, 6, 7, 20, 13, 42, 39, 110, 29, 312, 37, 374, 261, 798, 53, 2300, 61, 3828, 903, 1426, 79, 18648, 497, 2542, 2379, 21930, 107, 86856, 113, 42294, 4503, 5546, 2247, 475800, 151, 7906, 8787, 370620, 173, 843880, 181, 249798, 92547, 12118, 199, 5965848
Offset: 1

Views

Author

Gus Wiseman, Oct 02 2019

Keywords

Comments

The Heinz number of an integer partition or multiset {y_1,...,y_k} is prime(y_1)*...*prime(y_k).

Examples

			The sequence of terms together with their prime indices begins:
            1: ()
            2: (1)
            3: (2)
            6: (2,1)
            7: (4)
           20: (3,1,1)
           13: (6)
           42: (4,2,1)
           39: (6,2)
          110: (5,3,1)
           29: (10)
          312: (6,2,1,1,1)
           37: (12)
          374: (7,5,1)
          261: (10,2,2)
          798: (8,4,2,1)
           53: (16)
         2300: (9,3,3,1,1)
           61: (18)
         3828: (10,5,2,1,1)
For example, the divisors of 6 are {1,2,3,6}, with differences {1,1,3}, with Heinz number 20, so a(6) = 20.

Crossrefs

The sorted version is A328024.
a(n) is the Heinz number of row n of A193829, A328025, or A328027.
Cf. A000005, A027750, A056239, A060682, A060683, A112798, A129308, A193829.

Programs

  • Mathematica
    Table[Times@@Prime/@Differences[Divisors[n]],{n,100}]

Formula

A056239(a(n)) = n - 1. In words, the integer partition with Heinz number a(n) is an integer partition of n - 1.
A055396(a(n)) = A060680(n).
A061395(a(n)) = A060681(n).
A001221(a(n)) = A060682(n).
A001222(a(n)) = A000005(n).

A328024 Heinz numbers of multisets representing the differences between some positive integer's consecutive divisors.

Original entry on oeis.org

1, 2, 3, 6, 7, 13, 20, 29, 37, 39, 42, 53, 61, 79, 107, 110, 113, 151, 173, 181, 199, 239, 261, 271, 281, 312, 317, 349, 359, 374, 397, 421, 457, 497, 503, 541, 557, 577, 593, 613, 701, 733, 769, 787, 798, 857, 863, 903, 911, 953, 983, 1021, 1061, 1069, 1151
Offset: 1

Views

Author

Gus Wiseman, Oct 02 2019

Keywords

Comments

The Heinz number of an integer partition or multiset {y_1,...,y_k} is prime(y_1)*...*prime(y_k).
There is exactly one entry with any given sum of prime indices A056239.

Examples

			The sequence of terms together with their prime indices begins:
     1: {}
     2: {1}
     3: {2}
     6: {1,2}
     7: {4}
    13: {6}
    20: {1,1,3}
    29: {10}
    37: {12}
    39: {2,6}
    42: {1,2,4}
    53: {16}
    61: {18}
    79: {22}
   107: {28}
   110: {1,3,5}
   113: {30}
   151: {36}
   173: {40}
   181: {42}
   199: {46}
   239: {52}
   261: {2,2,10}
   271: {58}
   281: {60}
   312: {1,1,1,2,6}
For example, the divisors of 8 are {1,2,4,8}, with differences {1,2,4}, with Heinz number 42, so 42 belongs to the sequence.

Crossrefs

A permutation of A328023.
Also the set of possible Heinz numbers of rows of A193829, A328025, or A328027.
Cf. A001222, A000005, A027750, A056239, A060680, A060681, A060682, A112798.

Programs

  • Mathematica
    nn=1000;
Select[Union[Table[Times@@Prime/@Differences[Divisors[n]],{n,nn}]],#<=nn&]

A328027 Irregular triangle read by rows where row n lists, in weakly increasing order, the differences between consecutive divisors of n.

Original entry on oeis.org

1, 2, 1, 2, 4, 1, 1, 3, 6, 1, 2, 4, 2, 6, 1, 3, 5, 10, 1, 1, 1, 2, 6, 12, 1, 5, 7, 2, 2, 10, 1, 2, 4, 8, 16, 1, 1, 3, 3, 9, 18, 1, 1, 2, 5, 10, 2, 4, 14, 1, 9, 11, 22, 1, 1, 1, 2, 2, 4, 12, 4, 20, 1, 11, 13, 2, 6, 18, 1, 2, 3, 7, 14, 28, 1, 1, 1, 2, 4, 5, 15
Offset: 1

Views

Author

Gus Wiseman, Oct 02 2019

Keywords

Examples

			Triangle begins:
   {}
   1
   2
   1  2
   4
   1  1  3
   6
   1  2  4
   2  6
   1  3  5
  10
   1  1  1  2  6
  12
   1  5  7
   2  2 10
   1  2  4  8
  16
   1  1  3  3  9
  18
   1  1  2  5 10
   2  4 14
   1  9 11
  22
   1  1  1  2  2  4 12
For example, the divisors of 18 are {1,2,3,6,9,18}, with differences {1,1,3,3,9}, which is row 18.

Crossrefs

Same as A193829 with rows sorted in increasing order.
Same as A328025 with rows reversed.
Row sums are A001477.
Row lengths are A000005.
First column is A060680.
Cf. A027750, A060681, A060682, A060683, A070211, A129308, A328024.

Programs

  • Mathematica
    Table[Sort[Differences[Divisors[n]]],{n,30}]
Showing 1-3 of 3 results.

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