A002977 -id:A002977 - cp's OEIS frontend

A132138 Characteristic function of the set of numbers defined in A002977.

1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Reinhard Zumkeller, Aug 20 2007

nonn

a(A002977(n)) = 1 and a(A132142(n)) = 0.

R. Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for characteristic functions

a(n) = if chi(n) then 1 else 0, where chi(n) = if n<3 then (n=1) else ((n mod 2 = 1) AND chi((n-1)/2)) OR ((n mod 3 = 1) AND chi((n-1)/3)).

A085249 Terms k of A002977 such that both (k-1)/2 and (k-1)/3 are also terms of A002977.

Original entry on oeis.org

31, 175, 1039, 1471, 2191, 4495, 6223, 8815, 13135, 20479, 22639, 26815, 30703, 36031, 37327, 45967, 52879, 53743, 54031, 66703, 78799, 89023, 108175, 122863, 125887, 132799, 135679, 136687, 160879, 177151, 178159, 181183, 184207, 188095

Offset: 1

Reinhard Zumkeller, Aug 11 2003

nonn

			A002977(51) = 175: (175-1)/2 = 82 = A002977(28) and (175-1)/3 = 58 = A002977(22), therefore 175 is a term.

Amiram Eldar, Table of n, a(n) for n = 1..1000

Subsequence of A016921 and A002977.

seq[max_] := Module[{s = Flatten[NestWhileList[Flatten[{2*# + 1, 3*# + 1}] &, 1, Min[#1] < max &]], t}, t = Union[Select[s, # <= max &]]; Select[t, MemberQ[t, (# - 1)/2] && MemberQ[t, (# - 1)/3] &]]; seq[200000] (* Amiram Eldar, May 07 2022 *)

More terms from Ray Chandler, Sep 06 2003

A132142 Complement of A002977.

Original entry on oeis.org

2, 5, 6, 8, 11, 12, 14, 16, 17, 18, 20, 23, 24, 25, 26, 29, 30, 32, 33, 34, 35, 36, 37, 38, 41, 42, 44, 47, 48, 49, 50, 51, 52, 53, 54, 56, 59, 60, 61, 62, 65, 66, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 80, 83, 84, 86, 88, 89, 90, 92, 95, 96, 97, 98, 99, 100, 101, 102, 103

Offset: 1

Reinhard Zumkeller, Aug 20 2007

nonn

A132138(a(n)) = 0.

R. Zumkeller, Table of n, a(n) for n = 1..10000

A190858 Integers in (-1+A002977)/2; contains A002977 as a proper subsequence.

Original entry on oeis.org

1, 3, 4, 6, 7, 9, 10, 13, 15, 19, 21, 22, 27, 28, 31, 33, 39, 40, 42, 43, 45, 46, 55, 57, 58, 60, 63, 64, 67, 69, 79, 81, 82, 85, 87, 91, 93, 94, 96, 111, 115, 117, 118, 121, 123, 127, 129, 130, 135, 136, 139, 141, 159, 163, 165, 166, 171, 172, 175, 177, 183, 187, 189, 190, 193, 195, 202, 204, 223, 231, 235, 237, 238, 243, 244, 247, 249

Offset: 1

Clark Kimberling, May 25 2011

nonn

See A190803.

Cf. A190803, A002977, A190859.

h = 2; i = 1; j = 3; k = 1; f = 1; g = 9 ;
a = Union[Flatten[NestList[{h # + i, j # + k} &, f, g]]]  (* A002977 *)
b = (a - 1)/2; c = (a - 1)/3; r = Range[1, 300];
d = Intersection[b, r] (* A190858 *)
e = Intersection[c, r] (* A190859 *)

A058361 a(n) is the least k in A002977 with a gap of n. Also n + a(n) is the least k in A007448 which is repeated n times.

Original entry on oeis.org

3, 1, 4, 15, 22, 121, 735, 31, 46, 22143, 4468, 67, 31455, 391, 2308, 447, 94, 33151, 16383, 139, 202, 7551, 5224, 787, 1595391, 3685, 580, 30591, 418, 42495, 1791, 607, 1342, 3217407, 1095166, 283, 398847, 32767, 365311, 88575, 1174, 6925, 12304383

Offset: 1

Robert G. Wilson v, Dec 16 2000

nonn

Amiram Eldar, Table of n, a(n) for n = 1..84

Cf. A002977, A007448.

k = {1}; Do[ k = Union[ Join[ k, 2k + 1, 3k + 1 ] ]; l = Length[ k ]; i = 1; While[ i < l && k[ [ i ] ] < 10^9, i++ ]; k = Take[ k, {1, i} ], {n, 1, 30} ]; f[ n_Integer ] := (i = 1; While[ k[ [ i + 1 ] ] - k[ [ i ] ] != n, i++ ]; k[ [ i ] ]); Table[ f[ n ], {n, 1, 84} ]

A076291 a(n) = A002977(n+1) - A002977(n).

Original entry on oeis.org

2, 1, 3, 2, 1, 3, 2, 4, 2, 1, 5, 1, 3, 8, 1, 3, 2, 1, 9, 2, 1, 5, 1, 3, 12, 2, 1, 3, 2, 4, 2, 1, 17, 4, 2, 1, 3, 6, 2, 1, 5, 1, 3, 20, 4, 2, 1, 5, 1, 3, 8, 4, 2, 1, 3, 9, 21, 8, 4, 2, 1, 5, 1, 3, 8, 1, 3, 2, 1, 9, 2, 1, 5, 1, 3, 36, 8, 4, 2, 1, 9, 2, 1, 5, 1, 3, 9, 3, 8, 4, 2, 1, 5, 1, 3, 14, 1, 3, 9, 29, 16

Offset: 1

Benoit Cloitre, Nov 06 2002

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000
Benoit Cloitre, Illustration of initial terms

Cf. A002977.

seq[max_] := Module[{s = Flatten[NestWhileList[Flatten[{2*# + 1, 3*# + 1}] &, 1, Min[#1] < max &]]}, Differences[Union[Select[s, # <= max &]]]]; seq[400000] (* Amiram Eldar, May 07 2022 *)

A190859 Integers in (-1+A002977)/3; contains A002977 as a proper subsequence.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 9, 10, 13, 14, 15, 18, 19, 21, 22, 26, 27, 28, 30, 31, 38, 39, 40, 42, 43, 45, 46, 54, 55, 57, 58, 62, 63, 64, 67, 74, 78, 79, 81, 82, 85, 86, 87, 90, 91, 93, 94, 106, 110, 111, 114, 115, 117, 118, 121, 122, 126, 127, 129, 130, 135, 136, 139, 154, 158, 159, 162, 163, 165, 166, 170, 171, 172, 174, 175, 182, 183, 186

Offset: 1

Clark Kimberling, May 25 2011

nonn

See A190803.

Cf. A190803, A002977, A190858.

Mathematica
```
(See A190858.)
```

A073942 Numbers k such that A002977(k) - A002977(k-1) = 1.

Original entry on oeis.org

3, 6, 11, 13, 16, 19, 22, 24, 28, 33, 37, 41, 43, 48, 50, 55, 62, 64, 67, 70, 73, 75, 81, 84, 86, 93, 95, 98, 106, 109, 111, 115, 120, 124, 127, 129, 138, 142, 145, 147, 150, 158, 162, 164, 168, 170, 173, 182, 186, 189, 191, 196, 198, 203, 211, 215, 218, 220, 223

Offset: 1

Benoit Cloitre, Nov 20 2002

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000
Amiram Eldar, Plot of a(n)/n for n = 20000..650000

Cf. A002977.

max = 1500; s = Union @ Flatten[NestWhileList[Flatten[{2*# + 1, 3*# + 1}] &, 1, Min[#1] < max &]]; 1 + Position[Differences[Select[s, # <= max &]], 1] // Flatten (* Amiram Eldar, May 07 2022 *)

Conjecture : a(n) is asymptotic to c*n where c is around 3.8...

A076827 Number of odd terms minus number of even terms in the set S(n) = ( A002977(k) : k<=n).

Original entry on oeis.org

1, 2, 1, 2, 3, 2, 3, 4, 5, 6, 5, 6, 5, 6, 7, 6, 7, 8, 7, 8, 9, 8, 9, 8, 9, 10, 11, 10, 11, 12, 13, 14, 13, 14, 15, 16, 15, 16, 17, 18, 17, 18, 17, 18, 19, 20, 21, 20, 21, 20, 21, 22, 23, 24, 23, 24, 23, 24, 25, 26, 27, 26, 27, 26, 27, 28, 27, 28, 29, 28, 29, 30, 29, 30, 29, 30, 31

Offset: 1

Benoit Cloitre, Nov 20 2002

nonn

			S(12) = (1,3,4,7,9,10,13,15,19,21,22,27) and there are 9 odd terms and 3 even terms in S(12), hence a(12) = 9-3 = 6.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Amiram Eldar, Plot of a(n)/n for n = 1000..10000

Cf. A002977.

seq[max_] := Module[{s = Flatten[NestWhileList[Flatten[{2*# + 1, 3*# + 1}] &, 1, Min[#1] < max &]], t}, t = Union[Select[s, # <= max &]]; FoldList[Plus, -(-1)^t]]; seq[300] (* Amiram Eldar, May 07 2022 *)

a(n) = (-1)*Sum_{k=1..n} (-1)^A002977(k).

a(n) > 0 and a(n) seems to be asymptotic to c*n where c = 0.4...

A190803 Increasing sequence generated by these rules: a(1)=1, and if x is in a then 2x-1 and 3x-1 are in a.

Original entry on oeis.org

1, 2, 3, 5, 8, 9, 14, 15, 17, 23, 26, 27, 29, 33, 41, 44, 45, 50, 51, 53, 57, 65, 68, 77, 80, 81, 86, 87, 89, 98, 99, 101, 105, 113, 122, 129, 131, 134, 135, 149, 152, 153, 158, 159, 161, 170, 171, 173, 177, 194, 195, 197, 201, 203, 209, 225, 230, 239, 242

Offset: 1

Clark Kimberling, May 25 2011

nonn

This sequence represents a class of sequences generated by rules of the form "a(1)=1, and if x is in a then hx+i and jx+k are in a, where h,i,j,k are integers." If m>1, at least one of the numbers b(m)=(a(m)-i)/h and c(m)=(a(m)-k)/j is in the set N of natural numbers. Let d(n) be the n-th b(m) in N, and let e(n) be the n-th c(m) in N. Note that a is a subsequence of both d and e.
Examples, where [A......] indicates a conjecture:
A190803: (h,i,j,k)=(2,-1,3,-1); d=A190841, e=A190842
A190804: (h,i,j,k)=(2,-1,3,0); d=[A190803], e=A190844
A190805: (h,i,j,k)=(2,-1,3,1); d=A190845, e=[A190808]
A190806: (h,i,j,k)=(2,-1,3,2); d=[A190804], e=A190848
...
A190807: (h,i,j,k)=(2,0,3,-1); d=A190849, e=A190850
A003586: (h,i,j,k)=(2,0,3,0); d=e=A003586
A190808: (h,i,j,k)=(2,0,3,1); d=A190851, e=A190852
A190809: (h,i,j,k)=(2,0,3,2); d=A190853, e=A190854
...
A190810: (h,i,j,k)=(2,1,3,-1); d=A190855, e=A190856
A190811: (h,i,j,k)=(2,1,3,0); d=A002977, e=A190857
A002977: (h,i,j,k)=(2,1,3,1); d=A190858, e=A190859
A190812: (h,i,j,k)=(2,1,3,2); d=A069353, e=[A190812]
...
For h=j=3, see A191106; for h=3 and j=4, see A191113.

			1 -> 2 -> 3,5 -> 8,9,14 -> 15,17,23,26,27,41 -> ...

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
David Garth and Adam Gouge, Affinely Self-Generating Sets and Morphisms, Journal of Integer Sequences, 10 (2007) 1-13.

Cf. A002977, A003586, A190804-A190812, A190841-A190860.

import Data.Set (singleton, deleteFindMin, insert)
a190803 n = a190803_list !! (n-1)
a190803_list = 1 : f (singleton 2)
   where f s = m : (f $ insert (2*m-1) $ insert (3*m-1) s')
             where (m, s') = deleteFindMin s
-- Reinhard Zumkeller, Jun 01 2011

h = 2; i = -1; j = 3; k = -1; f = 1; g = 10;
a = Union[Flatten[NestList[{h # + i, j # + k} &, f, g]]]  (* A190803 *)
b = (a + 1)/2; c = (a + 1)/3; r = Range[1, 300];
d = Intersection[b, r] (* A190841 *)
e = Intersection[c, r] (* A190842 *)
(* Regarding this program - useful for many choices of h,i,j,k,f,g - the depth g must be chosen with care - that is, large enough.  Assuming that h<=j, the least new terms in successive nests a are typically iterates of hx+i, starting from x=1.  If, for example, h=2 and i=0, the least terms are 2,4,8,...,2^g, so that g>=9 ensures inclusion of all the desired terms <=256. *)

a(34)=225 inserted by Reinhard Zumkeller, Jun 01 2011

cp's OEIS Frontend

A132138 Characteristic function of the set of numbers defined in A002977.

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A085249 Terms k of A002977 such that both (k-1)/2 and (k-1)/3 are also terms of A002977.

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Mathematica

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A132142 Complement of A002977.

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A190858 Integers in (-1+A002977)/2; contains A002977 as a proper subsequence.

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Mathematica

A058361 a(n) is the least k in A002977 with a gap of n. Also n + a(n) is the least k in A007448 which is repeated n times.

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Mathematica

A076291 a(n) = A002977(n+1) - A002977(n).

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Mathematica

A190859 Integers in (-1+A002977)/3; contains A002977 as a proper subsequence.

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Mathematica

A073942 Numbers k such that A002977(k) - A002977(k-1) = 1.

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Mathematica

Formula

A076827 Number of odd terms minus number of even terms in the set S(n) = ( A002977(k) : k<=n).

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Mathematica

Formula

A190803 Increasing sequence generated by these rules: a(1)=1, and if x is in a then 2x-1 and 3x-1 are in a.

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