A123384 -id:A123384 - cp's OEIS frontend

A226721 Position of 2^n in the joint ranking of all the numbers 2^j for j>=0 and 5^k for k>=1; complement of A123384.

2, 3, 5, 6, 8, 9, 11, 12, 13, 15, 16, 18, 19, 21, 22, 23, 25, 26, 28, 29, 31, 32, 33, 35, 36, 38, 39, 41, 42, 43, 45, 46, 48, 49, 51, 52, 53, 55, 56, 58, 59, 61, 62, 63, 65, 66, 68, 69, 71, 72, 73, 75, 76, 78, 79, 81, 82, 83, 85, 86, 88, 89, 91, 92, 93, 95

Offset: 1

Clark Kimberling, Jun 16 2013

			The joint ranking of the powers of 2 and of 5 begins like this: 1, 2, 4, 5, 8, 16, 25, 32, 64, 125, 128, 256, 512.  The numbers 2^n for n >= 1 are in positions 2, 3, 5, 6, 8, 9, 11, 12, 13.

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

Cf. A123384, A226720.

b = 2; c=5; Floor[1 + Range[0, 100]*(1 + Log[b, c])]  (* A123384 *)
Floor[1 + Range[1, 100]*(1 + Log[c, b])]  (* A226721 *)

a(n) = 1 + A066344(n).

a(n) = 1 + floor(n*(1 + log_5(2))).

A352378 Irregular triangle read by rows: T(n,k) is the (n-th)-to-last digit of 2^p such that p == k + A123384(n-1) (mod A005054(n)); k >= 0.

Original entry on oeis.org

2, 4, 8, 6, 1, 3, 6, 2, 5, 1, 2, 4, 9, 9, 8, 6, 3, 7, 4, 8, 7, 5, 0, 0, 1, 2, 5, 0, 0, 0, 1, 3, 7, 5, 0, 1, 2, 5, 1, 3, 6, 2, 4, 8, 7, 4, 9, 8, 6, 2, 5, 1, 3, 7, 4, 9, 8, 7, 5, 1, 2, 4, 8, 6, 3, 6, 3, 6, 2, 4, 9, 9, 9, 9, 8, 7, 4, 9, 9, 9, 8, 6, 2, 4, 9, 8, 7, 4, 8, 6, 3, 7, 5, 1, 2, 5, 0, 1, 3, 7, 4, 8, 6, 2, 5, 0, 1

Offset: 1

Davis Smith, Mar 14 2022

The n-th row of this triangle is the cycle of the (n-th)-to-last digit of powers of 2.

The period of the last n digits of powers of 2 where the exponent is greater than or equal to n is A005054(n). As a result, this triangle can be used to get the (n-th)-to-last digit of a large power of 2; if p == k + A123384(n-1) (mod A005054(n)), then the (n-th)-to-last digit (base 10) of 2^p is T(n,k). For example, for n = 1, if p == 1 (mod 4), then 2^p == 2 (mod 10) and if p == 3 (mod 4), then 2^p == 8 (mod 10). For n = 2, if p == 4 (mod 20), then the second-to-last digit of 2^p (base 10) is 1 and if p == 7 (mod 20), then the second-to-last digit of 2^p (base 10) is 2.

			Irregular triangle begins:
n/k| 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ... | Number of terms:
---+---------------------------------------+-----------------
1  | 2, 4, 8, 6;                           |                4
2  | 1, 3, 6, 2, 5, 1, 2, 4, 9, 9,  8, ... |               20
3  | 1, 2, 5, 0, 0, 0, 1, 3, 7, 5,  0, ... |              100
4  | 1, 2, 4, 8, 6, 2, 5, 1, 2, 4,  8, ... |              500
5  | 1, 3, 6, 3, 6, 2, 4, 9, 9, 8,  7, ... |             2500
6  | 1, 2, 5, 0, 0, 1, 3, 7, 5, 1,  2, ... |            12500
...

Cf. A005054, A123384, A216099.

The (n-th)-to-last digit of a power of 2: A000689 (n=1), A160590 (n=2).

A352378_rows(n)=my(N=logint(10^(n-1),2),k=4*5^(n-1)); vector(k,v,floor(lift(Mod(2,10^n)^(v+N))/(10^(n-1))))

For n > 1, T(n,0) = 1.

A253635 Rectangular array read by upwards antidiagonals: a(n,k) = index of largest term <= 10^k in row n of A253572, n >= 1, k >= 0.

Original entry on oeis.org

1, 1, 4, 1, 7, 7, 1, 9, 20, 10, 1, 10, 34, 40, 14, 1, 10, 46, 86, 67, 17, 1, 10, 55, 141, 175, 101, 20, 1, 10, 62, 192, 338, 313, 142, 24, 1, 10, 67, 242, 522, 694, 507, 190, 27, 1, 10, 72, 287, 733, 1197, 1273, 768, 244, 30

Offset: 1

L. Edson Jeffery, Jan 07 2015

Or a(n,k) = the number of positive integers less than or equal to 10^k that are divisible by no prime exceeding prime(n).

			Array begins:
{1,  4,  7,  10,   14,   17,    20,    24,    27,     30, ...}
{1,  7, 20,  40,   67,  101,   142,   190,   244,    306, ...}
{1,  9, 34,  86,  175,  313,   507,   768,  1105,   1530, ...}
{1, 10, 46, 141,  338,  694,  1273,  2155,  3427,   5194, ...}
{1, 10, 55, 192,  522, 1197,  2432,  4520,  7838,  12867, ...}
{1, 10, 62, 242,  733, 1848,  4106,  8289, 15519,  27365, ...}
{1, 10, 67, 287,  945, 2579,  6179, 13389, 26809,  50351, ...}
{1, 10, 72, 331, 1169, 3419,  8751, 20198, 42950,  85411, ...}
{1, 10, 76, 369, 1385, 4298, 11654, 28434, 63768, 133440, ...}
{1, 10, 79, 402, 1581, 5158, 14697, 37627, 88415, 193571, ...}

Cf. A123384, A100752, A106598, A106600, A107352, A106629, A108275, A108276, A108277 (rows 1-9).

Cf. A011557, A066343, A253572, A253573.

r = 10; y[1] = t = Table[2^j, {j, 0, 39}]; max = 10^13; len = 10^10; prev = 0; For[n = 2, n <= r, n++, next = 0; For[k = 1, k <= 43, k++, If[Prime[n]^k < max, t = Union[t, Prime[n]*t]; s = FirstPosition[t, v_ /; v > len, 0]; t = Take[t, s[[1]] - 1]; If[t[[-1]] > len, t = Delete[t, -1]]; next = Length[t]; If[next == prev, Break, prev = next], Break]]; y[n] = t]; b[i_, j_] := FirstPosition[y[i], v_ /; v > 10^j][[1]]; a253635[n_, j_] := If[IntegerQ[b[n, j]], b[n, j] - 1, 0]; Flatten[Table[a253635[n - j, j], {n, r}, {j, 0, n - 1}]] (* array antidiagonals flattened *)

A226720 Complement of A122437.

Original entry on oeis.org

2, 4, 5, 7, 9, 10, 12, 14, 15, 17, 18, 20, 22, 23, 25, 27, 28, 30, 31, 33, 35, 36, 38, 40, 41, 43, 45, 46, 48, 49, 51, 53, 54, 56, 58, 59, 61, 62, 64, 66, 67, 69, 71, 72, 74, 76, 77, 79, 80, 82, 84, 85, 87, 89, 90, 92, 93, 95, 97, 98, 100, 102, 103, 105, 107

Offset: 1

Clark Kimberling, Jun 16 2013

Suppose that b and c are integers satisfying 1 < b < c. Let x = 1 + log_b(c) and y = 1 + log_c(b). Jointly rank all the numbers b^k for k>=0 and c^k for k>=1; then for n >= 0, the position of b^n is 1 + floor(n*y), and for n >=1, the position of c^n is 1+ floor(n*x).

These position sequences are closely related to the Beatty sequences given by floor(n*x) and floor(n*y).

			The joint ranking of the powers of 2 and of 3 begins like this: 1, 2, 3, 4, 8, 9, 16, 27, 32, 64. The numbers 2^n for n >= 1 are in positions 2, 4, 5, 7, 9, 10.

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

Cf. A123384, A226721.

b = 2; c=3; Floor[1 + Range[0, 100]*(1 + Log[b, c])]  (* A123384 *)
Floor[1 + Range[1, 100]*(1 + Log[c, b])]  (* A226721 *)

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.

Original entry on oeis.org

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)).

A108276 Number of positive integers <= 10^n that are divisible by no prime exceeding 19.

Original entry on oeis.org

1, 10, 72, 331, 1169, 3419, 8751, 20198, 42950, 85411, 160626, 288126, 496303, 825326, 1330766, 2088013, 3197529, 4791093, 7039193, 10159603, 14427309, 20186026, 27861175, 37974797, 51162295, 68191379, 89983125, 117635672

Offset: 0

Robert G. Wilson v, May 31 2005

nonn

Row 8 of A253635.

Cf. A123384, A100752, A106598, A106600, A107352, A106629, A108275, A108277, A011557, A080682.

n = 9; t = Select[ Flatten[ Table[19^h*Select[ Flatten[ Table[17^g*Select[ Flatten[ Table[13^f*Select[ Flatten[ Table[11^e*Select[ Flatten[ Table[7^d*Select[ Flatten[ Table[5^c*Select[ Flatten[ Table[2^a*3^b, {a, 0, n*Log[2, 10]}, {b, 0, n*Log[3, 10]}]], # <= 10^n &], {c, 0, n*Log[5, 10]}]], # <= 10^n &], {d, 0, n*Log[7, 10]}]], # <= 10^n &], {e, 0, n*Log[11, 10]}]], # <= 10^n &], {f, 0, n*Log[13, 10]}]], # <= 10^n &], {g, 0, n*Log[17, 10]}]], # <= 10^n &], {h, 0, n*Log[19, 10]}]], # <= 10^n &]; Table[ Length[ Select[t, # <= 10^n &]], {n, 0, 10}]

a(10)-a(18) from Donovan Johnson, Sep 16 2009

a(19)-a(27) from Max Alekseyev, Apr 28 2010

A108277 Number of positive integers <= 10^n that are divisible by no prime exceeding 23.

Original entry on oeis.org

1, 10, 76, 369, 1385, 4298, 11654, 28434, 63768, 133440, 263529, 495412, 892644, 1550012, 2605342, 4254753, 6771752, 10531080, 16038303, 23965659, 35195450, 50872227, 72464493, 101837746, 141340075, 193902062, 263152095, 353549942

Offset: 0

Robert G. Wilson v, May 31 2005

nonn

Row 9 of A253635.

Cf. A123384, A100752, A106598, A106600, A107352, A106629, A108275, A108276, A011557, A080683.

n = 6; t = Select[ Flatten[ Table[23^i*Select[ Flatten[ Table[19^h*Select[ Flatten[ Table[17^g*Select[ Flatten[ Table[13^f*Select[ Flatten[ Table[11^e*Select[ Flatten[ Table[7^d*Select[ Flatten[ Table[5^c*Select[ Flatten[ Table[2^a*3^b, {a, 0, n*Log[2, 10]}, {b, 0, n*Log[3, 10]}]], # <= 10^n &], {c, 0, n*Log[5, 10]}]], # <= 10^n &], {d, 0, n*Log[7, 10]}]], # <= 10^n &], {e, 0, n*Log[11, 10]}]], # <= 10^n &], {f, 0, n*Log[13, 10]}]], # <= 10^n &], {g, 0, n*Log[17, 10]}]], # <= 10^n &], {h, 0, n*Log[19, 10]}]], # <= 10^n &], {i, 0, n*Log[23, 10]}]], # <= 10^n &]; Table[ Length[ Select[t, # <= 10^n &]], {n, 0, 10}]

a(7)-a(18) from Donovan Johnson, Sep 16 2009

a(19)-a(27) from Max Alekseyev, Apr 28 2010

A158333 Position of number of digits increment in the sequence of powers of 3.

Original entry on oeis.org

1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 89, 91, 93, 95, 97, 99, 101, 103, 105, 107, 109, 112, 114, 116, 118, 120, 122, 124, 126, 128, 130, 133, 135

Offset: 0

Julio Cesar de la Yncera, Mar 16 2009

			For n=1 a(1)=3 since the sequence of powers of 3 is 1, 3, 9, 27, 81, 243, 729 and numbers of digits increase at position 1,3,5,...

A123384

A158333 := proc(n)
        1+floor(n/log10(3)) ;
end proc:
seq(A158333(n),n=0..20) ; # R. J. Mathar, Sep 01 2014

a[x_] := 1 + Floor[x/Log[10, 3]]; Table[a[i], {i, 0, 20}]

a(n)=1+Floor(n/Log_10(3)) = 1+A054965(n).

Indices in offset, example, and formula adjusted by R. J. Mathar, May 21 2009

More terms from Robert G. Wilson v, May 29 2009

cp's OEIS Frontend

A226721 Position of 2^n in the joint ranking of all the numbers 2^j for j>=0 and 5^k for k>=1; complement of A123384.

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A352378 Irregular triangle read by rows: T(n,k) is the (n-th)-to-last digit of 2^p such that p == k + A123384(n-1) (mod A005054(n)); k >= 0.

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A253635 Rectangular array read by upwards antidiagonals: a(n,k) = index of largest term <= 10^k in row n of A253572, n >= 1, k >= 0.

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A226720 Complement of A122437.

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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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A108276 Number of positive integers <= 10^n that are divisible by no prime exceeding 19.

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A108277 Number of positive integers <= 10^n that are divisible by no prime exceeding 23.

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