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 1-6 of 6 results.

A326667 Number of factorizations of 2^n into factors > 1 with integer average.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 11, 15, 19, 21, 29, 37, 44, 58, 67, 86, 105, 136, 146, 219, 236, 295, 327, 473, 469, 694, 707, 932, 1020, 1398, 1340, 2023, 2059, 2636, 2816, 3887, 3855, 5377, 5467, 7095, 7611, 9924, 9992, 13795, 14205, 17728, 19315, 24803, 25452, 33026
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: s in y} is an integer.

Examples

			The a(1) = 1 through a(8) = 11 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (21)   (22)    (32)     (33)      (43)       (44)
             (111)  (31)    (41)     (42)      (52)       (53)
                    (1111)  (311)    (51)      (61)       (62)
                            (11111)  (222)     (331)      (71)
                                     (2211)    (511)      (422)
                                     (111111)  (3211)     (2222)
                                               (1111111)  (3311)
                                                          (4211)
                                                          (311111)
                                                          (11111111)

Crossrefs

The strict case is A326668.
Factorizations with integer average are A326622.
Partitions with integer average are A067538.
Subsets with integer average are A051293.
Cf. A001055, A102627, A326028, A326647, A326666, A326670, A326671.

Programs

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

A326668 Number of strict factorizations of 2^n into factors > 1 with integer average.

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 4, 4, 6, 5, 7, 7, 9, 9, 12, 12, 17, 17, 21, 24, 33, 33, 42, 46, 63, 61, 81, 82, 118, 106, 149, 137, 213, 172, 263, 221, 363, 266, 453, 335, 594, 409, 735, 484, 968, 594, 1139, 731, 1486, 813, 1801, 1026, 2177, 1230, 2667, 1348, 3334, 1693
Offset: 1

Views

Author

Gus Wiseman, Jul 17 2019

Keywords

Comments

Also the number of strict integer partitions y of n such that the average of the set {2^s: s in y} is an integer.

Examples

			The a(1) = 1 through a(11) = 7 partitions (A = 10, B = 11):
  (1)  (2)  (3)   (4)   (5)   (6)   (7)   (8)   (9)    (A)   (B)
            (21)  (31)  (32)  (42)  (43)  (53)  (54)   (64)  (65)
                        (41)  (51)  (52)  (62)  (63)   (73)  (74)
                                    (61)  (71)  (72)   (82)  (83)
                                                (81)   (91)  (92)
                                                (531)        (A1)
                                                             (731)

Crossrefs

The non-strict case is A326667.
Factorizations with integer average are A326622.
Strict partitions with integer average are A102627.
Subsets with integer average are A051293.
Cf. A001055, A067538, A102627, A326028, A326647, A326666, A326670, A326671.

Programs

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

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}]

A327906 Numbers with only one factorization into factors > 1 with integer mean (namely, as a singleton).

Original entry on oeis.org

2, 3, 5, 6, 7, 10, 11, 13, 14, 17, 18, 19, 22, 23, 26, 29, 30, 31, 34, 37, 38, 41, 43, 46, 47, 53, 58, 59, 61, 62, 66, 67, 70, 71, 73, 74, 79, 82, 83, 86, 89, 90, 94, 97, 98, 101, 102, 103, 106, 107, 109, 113, 118, 122, 127, 130, 131, 134, 137, 138, 139, 142
Offset: 1

Views

Author

Gus Wiseman, Sep 30 2019

Keywords

Examples

			There are 4 factorizations of 24 with integer mean, namely:
  (24)
  (4*6)
  (2*12)
  (2*3*4)
so 24 is not in the sequence.

Links

Crossrefs

Complement of A327907.
Positions of 1's in A326622.
Cf. A001055, A051293, A067538, A078175, A123528/A123529, A316413, A326515, A326516, A326666, A326667, A326671.

Programs

  • Mathematica
    facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Select[Range[100],Length[Select[facs[#],IntegerQ[Mean[#]]&]]==1&]
  • PARI
    A326622(n, m=n, facsum=0, facnum=0) = if(1==n,facnum > 0 && 1==denominator(facsum/facnum), my(s=0); fordiv(n, d, if((d>1)&&(d<=m), s += A326622(n/d, d, facsum+d, facnum+1))); (s)); \\ Antti Karttunen, Nov 10 2024
isA327906(n) = (1==A326622(n)); \\ Antti Karttunen, Nov 10 2024

A327907 Numbers with more than one factorization into at factors > 1 with integer mean.

Original entry on oeis.org

4, 8, 9, 12, 15, 16, 20, 21, 24, 25, 27, 28, 32, 33, 35, 36, 39, 40, 42, 44, 45, 48, 49, 50, 51, 52, 54, 55, 56, 57, 60, 63, 64, 65, 68, 69, 72, 75, 76, 77, 78, 80, 81, 84, 85, 87, 88, 91, 92, 93, 95, 96, 99, 100, 104, 105, 108, 110, 111, 112, 114, 115, 116
Offset: 1

Views

Author

Gus Wiseman, Sep 30 2019

Keywords

Examples

			There are 6 factorizations of 60 with integer mean, namely:
  (60)
  (2*30)
  (6*10)
  (3*4*5)
  (2*3*10)
  (2*2*3*5)
so 60 is in the sequence.

Crossrefs

Complement of A327906.
Positions of terms > 1 in A326622.
Cf. A001055, A051293, A067538, A078175, A123528/A123529, A316413, A326515, A326516, A326666, A326667, A326671.

Programs

  • Mathematica
    facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Select[Range[100],Length[Select[facs[#],IntegerQ[Mean[#]]&]]>1&]
Showing 1-6 of 6 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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