A326668 -id:A326668 - cp's OEIS frontend

A326674 GCD of the set of positions of 1's in the reversed binary expansion of n.

1, 2, 1, 3, 1, 1, 1, 4, 1, 2, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 6, 1, 2, 1, 3, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Gus Wiseman, Jul 17 2019

a(n) is even if and only if n is in A062880. - Robert Israel, Oct 13 2020

			The reversed binary expansion of 40 is (0,0,0,1,0,1), with positions of 1's being {4,6}, so a(40) = GCD(4,6) = 2.

Robert Israel, Table of n, a(n) for n = 1..10000

Positions of 1's are A291166, and non-1's are A291165.
GCDs of prime indices are A289508.
GCDs of strict partitions encoded by FDH numbers are A319826.
Numbers whose binary positions are pairwise coprime are A326675.
Cf. A000120, A051293, A062880, A070939, A326667, A326668, A326669, A326670, A326672, A326673.

f:= proc(n) local B;
  B:= convert(n,base,2);
  igcd(op(select(t -> B[t]=1, [$1..ilog2(n)+1])))
end proc:
map(f, [$1..100]); # Robert Israel, Oct 13 2020

Table[GCD@@Join@@Position[Reverse[IntegerDigits[n,2]],1],{n,100}]

Trivially, a(n) <= log_2(n). - Charles R Greathouse IV, Nov 15 2022

A326669 Numbers k such that the average position of the ones in the binary expansion of k is an integer.

Original entry on oeis.org

1, 2, 4, 5, 7, 8, 10, 14, 16, 17, 20, 21, 27, 28, 31, 32, 34, 35, 39, 40, 42, 49, 54, 56, 57, 62, 64, 65, 68, 70, 73, 78, 80, 84, 85, 93, 98, 99, 107, 108, 112, 114, 119, 124, 127, 128, 130, 133, 136, 140, 141, 146, 147, 155, 156, 160, 161, 167, 168, 170, 175

Offset: 1

Gus Wiseman, Jul 17 2019

These are numbers whose exponents in their representation as a sum of distinct powers of 2 have integer average.

			42 is in the sequence because 42 = 2^1 + 2^3 + 2^5 and the average of {1,3,5} is 3, an integer.

Cf. A000120, A051293, A070939, A102627, A291165, A291166, A316413, A326622, A326668, A326672, A326673.

Select[Range[100],IntegerQ[Mean[Join@@Position[IntegerDigits[#,2],1]]]&]

isok(m) = my(b=binary(m)); denominator(vecsum(Vec(select(x->(x==1), b, 1)))/hammingweight(m)) == 1; \\ Michel Marcus, Jul 02 2021

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

Gus Wiseman, Jul 17 2019

nonn

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

			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)

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.

Table[Length[Select[IntegerPartitions[n],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

Gus Wiseman, Jul 17 2019

nonn

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.

			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)

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.

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

Gus Wiseman, Jul 17 2019

nonn

			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)

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.

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

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A326674 GCD of the set of positions of 1's in the reversed binary expansion of n.

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A326669 Numbers k such that the average position of the ones in the binary expansion of k is an integer.

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

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

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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.

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