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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A326671 Number of factorizations of 2^n into factors > 1 with even integer average.

Original entry on oeis.org

1, 2, 2, 3, 4, 5, 7, 8, 11, 14, 14, 20, 27, 31, 41, 47, 57, 75, 95, 102, 155, 170, 195, 239, 327, 331, 483, 517, 617, 740, 952, 942, 1406, 1484, 1742, 2023, 2652, 2688, 3680, 3892, 4729, 5375, 6689, 6911, 9437, 9938, 11754, 13529, 16710, 17419, 22346, 24230
Offset: 1

Views

Author

Gus Wiseman, Jul 17 2019

Keywords

Comments

Also the number of integer partitions y of n such that the average of the multiset {2^(s - 1): s in y} is an integer.

Examples

			The a(1) = 1 through a(8) = 8 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (111)  (22)    (32)     (33)      (43)       (44)
                    (1111)  (311)    (42)      (52)       (53)
                            (11111)  (222)     (331)      (62)
                                     (111111)  (511)      (422)
                                               (3211)     (2222)
                                               (1111111)  (4211)
                                                          (11111111)

Crossrefs

The strict case is A326670.
Factorizations with integer average are A326622.
Partitions with integer average are A067538.
Cf. A001055, A051293, A102627, A326028, A326622, A326647, A326666, A326667, A326668.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],IntegerQ[Mean[2^(#-1)]]&]],{n,30}]

A326670 Number of strict integer partitions y of n such that the average of the set {2^(s - 1): s in y} is an integer.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 3, 3, 5, 4, 6, 6, 8, 7, 10, 9, 13, 12, 15, 16, 23, 22, 27, 31, 41, 41, 50, 57, 74, 75, 90, 99, 133, 127, 158, 167, 226, 203, 278, 262, 371, 325, 457, 387, 622, 484, 715, 606, 969, 672, 1178, 866, 1428, 1050, 1776, 1142, 2276, 1459, 2514, 1792
Offset: 1

Views

Author

Gus Wiseman, Jul 17 2019

Keywords

Examples

			The a(1) = 1 through a(12) = 6 partitions (A = 10, B = 11, C = 12):
  (1)  (2)  (3)  (4)  (5)   (6)   (7)   (8)   (9)    (A)   (B)    (C)
                      (32)  (42)  (43)  (53)  (54)   (64)  (65)   (75)
                                  (52)  (62)  (63)   (73)  (74)   (84)
                                              (72)   (82)  (83)   (93)
                                              (531)        (92)   (A2)
                                                           (731)  (642)

Crossrefs

The non-strict case is A326671.
Strict factorizations with integer average are A326668.
Strict partitions with integer average are A102627.
Cf. A001055, A051293, A067538, A102627, A326028, A326622, A326647, A326666, A326667.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&IntegerQ[Mean[2^(#-1)]]&]],{n,30}]

A326642 Number of non-constant integer partitions of n whose mean and geometric mean are both integers.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 1, 0, 0, 1, 1, 0, 0, 0, 4, 2, 0, 0, 3, 1, 2, 1, 2, 0, 7, 0, 4, 2, 2, 4, 7, 0, 0, 4, 12, 0, 9, 0, 2, 11, 0, 0, 17, 6, 14, 4, 8, 0, 13, 6, 27, 6, 2, 0, 36, 0, 0, 35, 32, 8, 20, 0, 11, 6, 56, 0, 91, 0, 2, 17
Offset: 0

Views

Author

Gus Wiseman, Jul 16 2019

Keywords

Comments

The Heinz numbers of these partitions are given by A326646.

Examples

			The a(30) = 7 partitions:
  (27,3)
  (24,6)
  (24,3,3)
  (16,8,2,2,2)
  (9,9,9,1,1,1)
  (8,8,8,2,2,2)
  (8,8,4,4,1,1,1,1,1,1)

Links

Crossrefs

Partitions with integer mean and geometric mean are A326641.
Heinz numbers of non-constant partitions with integer mean and geometric mean are A326646.
Non-constant partitions with integer geometric mean are A326624.
Cf. A051293, A067538, A067539, A102627, A316413, A326029, A326643, A326644, A326645, A326647.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],!SameQ@@#&&IntegerQ[Mean[#]]&&IntegerQ[GeometricMean[#]]&]],{n,0,30}]

Formula

a(n) = A326641(n) - A000005(n).

A327777 Prime numbers whose binary indices have integer mean and integer geometric mean.

Original entry on oeis.org

2, 257, 8519971, 36574494881, 140739702949921, 140773995710729, 140774004099109
Offset: 1

Views

Author

Gus Wiseman, Sep 27 2019

Keywords

Comments

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.
Conjecture: This sequence is infinite.

Examples

			The initial terms together with their binary indices:
                2: {2}
              257: {1,9}
          8519971: {1,2,6,9,18,24}
      36574494881: {1,6,8,16,18,27,32,36}
  140739702949921: {1,6,12,27,32,48}
  140773995710729: {1,4,9,12,18,32,36,48}
  140774004099109: {1,3,6,12,18,24,32,36,48}

Links

Crossrefs

A subset of A327368.
The binary weight of prime(n) is A014499(n), with binary length A035100(n).
Heinz numbers of partitions with integer mean: A316413.
Heinz numbers of partitions with integer geometric mean: A326623.
Heinz numbers with both: A326645.
Subsets with integer mean: A051293
Subsets with integer geometric mean: A326027
Subsets with both: A326643
Partitions with integer mean: A067538
Partitions with integer geometric mean: A067539
Partitions with both: A326641
Strict partitions with integer mean: A102627
Strict partitions with integer geometric mean: A326625
Strict partitions with both: A326029
Factorizations with integer mean: A326622
Factorizations with integer geometric mean: A326028
Factorizations with both: A326647
Numbers whose binary indices have integer mean: A326669
Numbers whose binary indices have integer geometric mean: A326673
Numbers whose binary indices have both: A327368
Cf. A000120, A029931, A034797, A048793, A070939, A096111, A291166, A326031, A326644, A326699/A326700.

Programs

  • Mathematica
    bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Select[Prime[Range[1000]],IntegerQ[Mean[bpe[#]]]&&IntegerQ[GeometricMean[bpe[#]]]&]

Extensions

a(4)-a(7) from Giovanni Resta, Dec 01 2019
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