A277569 -id:A277569 - cp's OEIS frontend

A277573 a(n) = (1/3)*A277569(n).

1, 3, 5, 6, 7, 9, 11, 13, 15, 17, 18, 19, 21, 23, 25, 27, 29, 30, 31, 33, 35, 36, 37, 39, 41, 42, 43, 45, 47, 49, 51, 53, 54, 55, 57, 59, 61, 63, 65, 66, 67, 69, 71, 73, 75, 77, 78, 79, 81, 83, 85, 87, 89, 90, 91, 93, 95, 97, 99, 101, 102, 103, 105, 107, 108

Offset: 1

Clark Kimberling, Nov 01 2016

From Amiram Eldar, Jan 16 2022: (Start)
Numbers whose 3-adic valuation is not smaller than their 2-adic valuation.
The asymptotic density of this sequence is 3/5. (End)

Clark Kimberling, Table of n, a(n) for n = 1..10000

Cf. A007814, A007949.

Cf. A277568, A277569, A277572, A277574.

z = 260; a[b_] := Table[Mod[n/b^IntegerExponent[n, b], b], {n, 1, z}]
p[b_, d_] := Flatten[Position[a[b], d]]
p[6, 3]/3
(* second program *)
Select[Range[200], IntegerExponent[#, 3] >= IntegerExponent[#, 2] &] (* Amiram Eldar, Jan 16 2022 *)

A337377 Primorial deflation (denominator) of Doudna-tree.

Original entry on oeis.org

1, 1, 2, 1, 3, 1, 4, 1, 5, 3, 2, 1, 9, 2, 8, 1, 7, 5, 10, 3, 3, 1, 4, 1, 25, 9, 6, 1, 27, 4, 16, 1, 11, 7, 14, 5, 21, 5, 20, 3, 5, 3, 2, 1, 9, 2, 8, 1, 49, 25, 50, 9, 15, 3, 4, 1, 125, 27, 18, 2, 81, 8, 32, 1, 13, 11, 22, 7, 33, 7, 28, 5, 55, 21, 14, 5, 63, 10, 40, 3, 7, 5, 10, 3, 3, 1, 4, 1, 25, 9, 6, 1, 27, 4, 16, 1, 121

Offset: 0

Antti Karttunen, Michael De Vlieger and Peter Munn, Aug 25 2020

Like A005940, also this irregular table can be represented as a binary tree:
                                      1
                                      |
                   ...................1...................
                  2                                       1
        3......../ \........1                   4......../ \........1
       / \                 / \                 / \                 / \
      /   \               /   \               /   \               /   \
     /     \             /     \             /     \             /     \
    5       3           2       1           9       2           8       1
   7 5    10 3         3 1     4 1        25 9     6 1        27 4    16 1
etc.
A194602 gives the positions of nodes that have value 1. They correspond to terms of A005940 that are products of primorials (A025487). The first 2^k nodes contain A000041(k+1) 1's.
a(n) is even if and only if A005940(1+n) occurs in A277569.

Antti Karttunen, Table of n, a(n) for n = 0..8191
Antti Karttunen, Data supplement: n, a(n) computed for n = 0..65537
Index entries for fraction trees

Cf. A337376 (numerators).
A003961, A005940, A006519, A026741, A064989, A319627 are used in a formula defining this sequence.
Cf. A000041, A025487, A277569.
Positions of 1's: A194602.
Cf. also A329886, A346097.

Array[#2/GCD[#1, #2] & @@ {#, Apply[Times, Map[If[#1 <= 2, 1, NextPrime[#1, -1]]^#2 & @@ # &, FactorInteger[#]]]} &@ Function[p, Times @@ Flatten@ Table[Prime[Count[Flatten[#], 0] + 1]^#[[1, 1]] &@ Take[p, -i], {i, Length[p]}]]@ Partition[Split[Join[IntegerDigits[# - 1, 2], {2}]], 2] &[# + 1] &, 96] (* Michael De Vlieger, Aug 27 2020 *)

A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); (t); };
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A319627(n) = (A064989(n) / gcd(n, A064989(n)));
A337377(n) = A319627(A005940(1+n));

a(n) = A319627(A005940(1+n)).
For n >= 1, a(2*n) = A003961(a(n)) * A006519(n+1).
a(2*n+1) = A026741(a(n)).

A277568 Numbers k such that k/6^m == 2 (mod 6), where 6^m is the greatest power of 6 that divides k.

Original entry on oeis.org

2, 8, 12, 14, 20, 26, 32, 38, 44, 48, 50, 56, 62, 68, 72, 74, 80, 84, 86, 92, 98, 104, 110, 116, 120, 122, 128, 134, 140, 146, 152, 156, 158, 164, 170, 176, 182, 188, 192, 194, 200, 206, 212, 218, 224, 228, 230, 236, 242, 248, 254, 260, 264, 266, 272, 278

Offset: 1

Clark Kimberling, Nov 01 2016

Positions of 2 in A277544.

Numbers having 2 as rightmost nonzero digit in base 6. This is one sequence in a 5-way splitting of the positive integers; the other four are indicated in the Mathematica program. Every term is even; see A277572.

Clark Kimberling, Table of n, a(n) for n = 1..10000

Cf. A277544, A277567, A277569, A277572.

z = 260; a[b_] := Table[Mod[n/b^IntegerExponent[n, b], b], {n, 1, z}]
p[b_, d_] := Flatten[Position[a[b], d]]
p[6, 1] (* A277567 *)
p[6, 2] (* A277568 *)
p[6, 3] (* A277569 *)
p[6, 4] (* A277570 *)
p[6, 5] (* A277571 *)

is(n)=(n/6^valuation(n,6))%6==2 \\ Charles R Greathouse IV, Nov 03 2016

a(n) = 5n + O(log n). - Charles R Greathouse IV, Nov 03 2016

A277570 Numbers k such that k/6^m == 4 (mod 6), where 6^m is the greatest power of 6 that divides k.

Original entry on oeis.org

4, 10, 16, 22, 24, 28, 34, 40, 46, 52, 58, 60, 64, 70, 76, 82, 88, 94, 96, 100, 106, 112, 118, 124, 130, 132, 136, 142, 144, 148, 154, 160, 166, 168, 172, 178, 184, 190, 196, 202, 204, 208, 214, 220, 226, 232, 238, 240, 244, 250, 256, 262, 268, 274, 276, 280

Offset: 1

Clark Kimberling, Nov 01 2016

Positions of 4 in A277544.

Numbers having 4 as rightmost nonzero digit in base 6. This is one sequence in a 5-way splitting of the positive integers; the other four are indicated in the Mathematica program. Every term is even; see A277574.

Clark Kimberling, Table of n, a(n) for n = 1..10000

Cf. A277544, A277567, A277571, A277574.

z = 260; a[b_] := Table[Mod[n/b^IntegerExponent[n, b], b], {n, 1, z}]
p[b_, d_] := Flatten[Position[a[b], d]]
p[6, 1] (* A277567 *)
p[6, 2] (* A277568 *)
p[6, 3] (* A277569 *)
p[6, 4] (* A277570 *)
p[6, 5] (* A277571 *)

is(n)=(n/6^valuation(n,6))%6==4 \\ Charles R Greathouse IV, Nov 03 2016

a(n) = 5n + O(log n). - Charles R Greathouse IV, Nov 03 2016

A277567 Numbers k such that k/6^m == 1 (mod 6), where 6^m is the greatest power of 6 that divides k.

Original entry on oeis.org

1, 6, 7, 13, 19, 25, 31, 36, 37, 42, 43, 49, 55, 61, 67, 73, 78, 79, 85, 91, 97, 103, 109, 114, 115, 121, 127, 133, 139, 145, 150, 151, 157, 163, 169, 175, 181, 186, 187, 193, 199, 205, 211, 216, 217, 222, 223, 229, 235, 241, 247, 252, 253, 258, 259, 265

Offset: 1

Clark Kimberling, Nov 01 2016

Positions of 1 in A277544. Numbers having 1 as rightmost nonzero digit in base 6. This is one sequence in a 5-way splitting of the positive integers; the other four are indicated in the Mathematica program.

Clark Kimberling, Table of n, a(n) for n = 1..10000

Cf. A277544, A277568, A277569.

z = 260; a[b_] := Table[Mod[n/b^IntegerExponent[n, b], b], {n, 1, z}]
p[b_, d_] := Flatten[Position[a[b], d]]
p[6, 1] (* A277567 *)
p[6, 2] (* A277568 *)
p[6, 3] (* A277569 *)
p[6, 4] (* A277570 *)
p[6, 5] (* A277571 *)
Select[Range[300],Mod[#/6^IntegerExponent[#,6],6]==1&] (* Harvey P. Dale, Sep 27 2023 *)

is(n)=(n/6^valuation(n,6))%6==1 \\ Charles R Greathouse IV, Nov 03 2016

a(n) = 5n + O(log n). - Charles R Greathouse IV, Nov 03 2016

A277571 Numbers k such that k/6^m == 5 (mod 6), where 6^m is the greatest power of 6 that divides k.

Original entry on oeis.org

5, 11, 17, 23, 29, 30, 35, 41, 47, 53, 59, 65, 66, 71, 77, 83, 89, 95, 101, 102, 107, 113, 119, 125, 131, 137, 138, 143, 149, 155, 161, 167, 173, 174, 179, 180, 185, 191, 197, 203, 209, 210, 215, 221, 227, 233, 239, 245, 246, 251, 257, 263, 269, 275, 281

Offset: 1

Clark Kimberling, Nov 01 2016

Positions of 5 in A277544.

Numbers having 5 as rightmost nonzero digit in base 6. This is one sequence in a 5-way splitting of the positive integers; the other four are indicated in the Mathematica program.

Clark Kimberling, Table of n, a(n) for n = 1..10000

Cf. A277544, A277567, A277570.

z = 260; a[b_] := Table[Mod[n/b^IntegerExponent[n, b], b], {n, 1, z}]
p[b_, d_] := Flatten[Position[a[b], d]]
p[6, 1] (* A277567 *)
p[6, 2] (* A277568 *)
p[6, 3] (* A277569 *)
p[6, 4] (* A277570 *)
p[6, 5] (* A277571 *)
Select[Range[300],Mod[#/6^IntegerExponent[#,6],6]==5&] (* Harvey P. Dale, Feb 14 2025 *)

is(n) = Mod(n/6^valuation(n, 6), 6)==5 \\ Felix Fröhlich, Nov 02 2016

a(n) = 5n + O(log n). - Charles R Greathouse IV, Nov 03 2016

cp's OEIS Frontend

A277573 a(n) = (1/3)*A277569(n).

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A337377 Primorial deflation (denominator) of Doudna-tree.

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A277568 Numbers k such that k/6^m == 2 (mod 6), where 6^m is the greatest power of 6 that divides k.

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A277570 Numbers k such that k/6^m == 4 (mod 6), where 6^m is the greatest power of 6 that divides k.

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A277567 Numbers k such that k/6^m == 1 (mod 6), where 6^m is the greatest power of 6 that divides k.

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A277571 Numbers k such that k/6^m == 5 (mod 6), where 6^m is the greatest power of 6 that divides k.

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