A306044 -id:A306044 - cp's OEIS frontend

A226722 Positions of the numbers 2^n, for n >=0, in the joint ranking of all the numbers 2^h, 3^k, 5^k, for h >= 0, k >= 1.

1, 2, 4, 6, 8, 11, 12, 15, 17, 18, 21, 22, 25, 27, 29, 31, 33, 35, 37, 39, 41, 44, 45, 47, 50, 51, 54, 56, 58, 60, 61, 64, 66, 68, 70, 73, 74, 76, 78, 80, 83, 84, 87, 89, 90, 93, 95, 97, 99, 101, 103, 105, 107, 109, 112, 113, 116, 117, 119, 122, 123, 126

Offset: 1

Clark Kimberling, Jun 16 2013

			The joint ranking of the powers of 2, 3, 5 begins like this: 1, 2, 3, 4, 5, 8, 9, 16, 25, 27, 32, 64, 81, 125, 128, 243, 256, 512.  The numbers 2^n for n >= 0 are in positions 1, 2, 4, 6, 8, 11, 12, 15, 17, 18.

Clark Kimberling, Table of n, a(n) for n = 1..2000

Cf. A123384, A226720, A226723, A226724, A306044.

z = 120; b = 2; c = 3; d = 5; f[x_]:=Floor[x];
Table[n + f[(n-1)*Log[c, b]] + f[(n-1)*Log[d, b]], {n, 1, z}]  (* this sequence *)
Table[1 + n + f[n*Log[b, c]] + f[n*Log[d, c]], {n, 1, z}]  (* A226723 *)
Table[1 + n + f[n*Log[b, d]] + f[n*Log[c, d]], {n, 1, z}]  (* A226724 *)

a(n) = n + floor((n-1)*log_3(2)) + floor((n-1)*log_5(2)). [corrected by Jason Yuen, Nov 02 2024]

A226723 Positions of the numbers 3^n, for n >= 1, in the joint ranking of all the numbers 2^h, 3^k, 5^k, for h >= 0, k >= 1.

Original entry on oeis.org

3, 7, 10, 13, 16, 20, 23, 26, 30, 32, 36, 40, 42, 46, 49, 52, 55, 59, 62, 65, 69, 72, 75, 79, 82, 85, 88, 92, 94, 98, 102, 104, 108, 111, 114, 118, 121, 124, 127, 131, 133, 137, 141, 144, 147, 150, 154, 157, 160, 164, 166, 170, 174, 176, 180, 183, 186, 189

Offset: 1

Clark Kimberling, Jun 16 2013

			The joint ranking of the powers of 2, 3, 5 begins like this: 1, 2, 3, 4, 5, 8, 9, 16, 25, 27, 32, 64, 81, 125, 128, 243, 256, 512.  The numbers 3^n for n >= 1 are in positions 3, 7, 10, 13, 16.

Clark Kimberling, Table of n, a(n) for n = 1..2000

Cf. A123384, A226720, A226722, A226724, A306044.

z = 120; b = 2; c = 3; d = 5; f[x_]:=Floor[x];
Table[1 + n + f[n*Log[c, b]] + f[n*Log[d, b]], {n, 0, z}]  (* A226722 *)
Table[1 + n + f[n*Log[b, c]] + f[n*Log[d, c]], {n, 1, z}]  (* A226723 *)
Table[1 + n + f[n*Log[b, d]] + f[n*Log[c, d]], {n, 1, z}]  (* A226724 *)

a(n) = 1 + n + floor(n*log_2(3)) + floor(n*log_5(3)).

A226724 Positions of the numbers 5^n, for n >= 1, in the joint ranking of all the numbers 2^h, 3^k, 5^k, for h >= 0, k >= 1.

Original entry on oeis.org

5, 9, 14, 19, 24, 28, 34, 38, 43, 48, 53, 57, 63, 67, 71, 77, 81, 86, 91, 96, 100, 106, 110, 115, 120, 125, 129, 135, 139, 143, 148, 153, 158, 162, 168, 172, 177, 182, 187, 191, 197, 201, 205, 211, 215, 220, 225, 230, 234, 240, 244, 249, 254, 259, 263, 269

Offset: 1

Clark Kimberling, Jun 16 2013

			The joint ranking of the powers of 2, 3, 5 begins like this: 1, 2, 3, 4, 5, 8, 9, 16, 25, 27, 32, 64, 81, 125, 128, 243, 256, 512.  The numbers 5^n for n >= 0 are in positions 5, 9, 14.

Clark Kimberling, Table of n, a(n) for n = 1..2000

Cf. A123384, A226720, A226722, A226723, A306044.

z = 120; b = 2; c = 3; d = 5; f[x_]:=Floor[x];
Table[1 + n + f[n*Log[c, b]] + f[n*Log[d, b]], {n, 0, z}]  (* A226722 *)
Table[1 + n + f[n*Log[b, c]] + f[n*Log[d, c]], {n, 1, z}]  (* A226723 *)
Table[1 + n + f[n*Log[b, d]] + f[n*Log[c, d]], {n, 1, z}]  (* A226724 *)

a(n) = 1 + n + floor(n*log_2(5)) + floor(n*log_3(5)).

A374277 Numbers k divisible by exactly one of the prime factors of 30.

Original entry on oeis.org

2, 3, 4, 5, 8, 9, 14, 16, 21, 22, 25, 26, 27, 28, 32, 33, 34, 35, 38, 39, 44, 46, 51, 52, 55, 56, 57, 58, 62, 63, 64, 65, 68, 69, 74, 76, 81, 82, 85, 86, 87, 88, 92, 93, 94, 95, 98, 99, 104, 106, 111, 112, 115, 116, 117, 118, 122, 123, 124, 125, 128, 129, 134

Offset: 1

Michael De Vlieger, Jul 26 2024

Numbers k congruent to r (mod 30), where r is in {2, 3, 4, 5, 8, 9, 14, 16, 21, 22, 25, 26, 27, 28}, residues r = p^m mod 30 and r = (30 - p^m) mod 30.

The asymptotic density of this sequence is 7/15. - Amiram Eldar, Jul 26 2024

			8 is in this sequence since it is even and a multiple of neither 3 nor 5.
10 is not in this sequence since 10 = 2*5; both 2 and 5 divide 30.
14 is in this sequence since it is even and a multiple of neither 3 nor 5, etc.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,0,0,0,0,0,1,-1).

Cf. A047228, A051037, A306044.

s = Prime@ Range[3]; k = Times @@ s; r = Union[#, k - #] &@ Flatten@ Map[PowerRange[#, k, #] &, s]; m = Length[r]; Array[k*#1 + r[[1 + #2]] & @@ QuotientRemainder[# - 1, m] &, 60]

is(k) = isprime(gcd(k, 30)); \\ Amiram Eldar, Jul 26 2024

Intersection of this sequence and 5-smooth numbers (A051037) is A306044 \ {1}.

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A226722 Positions of the numbers 2^n, for n >=0, in the joint ranking of all the numbers 2^h, 3^k, 5^k, for h >= 0, k >= 1.

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A226723 Positions of the numbers 3^n, for n >= 1, in the joint ranking of all the numbers 2^h, 3^k, 5^k, for h >= 0, k >= 1.

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A226724 Positions of the numbers 5^n, for n >= 1, in the joint ranking of all the numbers 2^h, 3^k, 5^k, for h >= 0, k >= 1.

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A374277 Numbers k divisible by exactly one of the prime factors of 30.

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