A004216 -id:A004216 - cp's OEIS frontend

A076786 Squarefree numbers such that in decimal representation all their prefixes are also squarefree.

1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 30, 31, 33, 34, 35, 37, 38, 39, 51, 53, 55, 57, 58, 59, 61, 62, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 101, 102, 103, 105, 106, 107, 109, 110, 111, 113, 114, 115, 118, 119, 130, 131, 133, 134, 137

Offset: 1

Reinhard Zumkeller, Nov 16 2002

m is a term iff A008966(floor(m/10^k))=1 for 0<=k<=A004216(n);

a(n)=A005117(n) for n<=26, but A005117(27)=41 is not a term, as 4=2^2 is not squarefree.

			143=11*13 is a term, as also 14=2*7 and 1 are squarefree.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Select[Range[200],AllTrue[FromDigits/@Table[Take[IntegerDigits[#],{1,n}],{n,IntegerLength[#]}],SquareFreeQ]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Aug 23 2020 *)

A077429 a(n) = floor(log_10(n^2)).

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4

Offset: 1

Reinhard Zumkeller, Nov 05 2002

nonn

a(n) = A004216(n^2).

Cf. A004216, A077430.

Table[Floor[Log10[n^2]],{n,120}] (* Harvey P. Dale, Sep 17 2020 *)

A123119 Number of digits in sum of first n primes (A007504).

Original entry on oeis.org

1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5

Offset: 1

Jonathan Vos Post, Sep 28 2006

Since A007504(n) has the asymptotic expression ~ n^2 * log(n) / 2, a(n) has the asymptotic expression n^2 * log(n) / 2 = floor(log_10(10* n^2 * log(n) / 2)) = floor(log_10(5* n^2 * log(n))) = floor(log_10(5) + log_10(n^2) + log_10(log(n))) = floor(0.698970004 + 2*log_10(n) + log_10(log(n))). What is the smallest n such that a(n) = 5, 6, 7, ...?

			a(3) = 2 because 2 + 3 + 5 = 10 has 2 digits in its decimal expansion.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A000040, A000041, A004216, A004218, A034386, A055642, A111287.

f[n_] := Floor[ Log[10, Sum[Prime@i, {i, n}]] + 1]; Array[f, 105] (* Robert G. Wilson v *)
f[n_] := IntegerLength[Total[Prime[Range[n]]]]; Array[f, 105] (* Jan Mangaldan, Jan 04 2017 *)
IntegerLength/@Accumulate[Prime[Range[110]]] (* Harvey P. Dale, Jan 26 2019 *)

a(n) = A055642(A007504(n)) = floor(log_10(10*A007504(n))) = A004216(A007504(n)) + 1 = A004218(A007504(n) + 1).

More terms from Robert G. Wilson v, Oct 05 2006

A118115 Partial sums of n concatenated n times.

Original entry on oeis.org

1, 23, 356, 4800, 60355, 727021, 8504798, 97393686, 1097393685, 10101010102107494695, 1121212121213218605806, 122333333333334430727018, 13253646464646465743858331, 1427395060606060607157999745, 152942546575757575758673151260, 16314558708191919191920289312876

Offset: 1

Jonathan Vos Post, May 11 2006

			a(2) = 1 + 22 = 23 is prime.
a(6) = 1 + 22 + 333 + 4444 + 55555 + 666666 = 727021 is prime.
For what value of n is the next prime a(n)?
a(158), which has 474 digits, is prime. - _Harvey P. Dale_, Oct 17 2011

F. Smarandache, "Properties of the numbers", Univ. of Craiova Archives, 1975; Arizona State University Special Collections, Tempe, AZ.

Cf. A000461 (concatenate n n times), A004216, A048376, A053422.

Accumulate[FromDigits/@Table[Flatten[IntegerDigits/@PadLeft[{},n,n]], {n,15}]] (* Harvey P. Dale, Oct 17 2011 *)

a(n) = Sum_{i=1..n} A000461(i). a(n) = Sum_{i=1..n} i*(10^(i*L(i))-1)/(10^L(i)-1) where L(i) = A004216(i) + 1 = floor(log_10(10i)).

A118117 Concatenate n F(n) times.

Original entry on oeis.org

1, 2, 33, 444, 55555, 66666666, 7777777777777, 888888888888888888888, 9999999999999999999999999999999999, 10101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010

Offset: 1

Jonathan Vos Post, May 11 2006

A000461 Concatenate n n times.

			a(6) = 6 concatenated F(6) times = 6 concatenated 8 times = 66666666, where F(n) = the n-th Fibonacci number.

F. Smarandache, "Properties of the numbers", Univ. of Craiova Archives, 1975; Arizona State University Special Collections, Tempe, AZ.

Cf. A000045, A000461, A004216, A048376, A053422.

Table[FromDigits[Flatten[IntegerDigits/@PadRight[{},Fibonacci[n],n]]],{n,10}] (* Harvey P. Dale, Aug 09 2020 *)

a(n) = n concatenated A000045(n) times. a(n) = A000027(n) concatenated A000045(n) times.

A211665 Minimal number of iterations of log_10 applied to n until the result is < 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2

Offset: 1

Hieronymus Fischer, Apr 30 2012

Different from A055642 and A138902, cf. Example.

Instead the real-valued log function one can consider only the integer part (i.e., A004216), since log_b(x) < k <=> x < b^k <=> floor(x) < b^k for any integer k >= 0; that's also why the first 2, 3, 4, ... appears exactly for 10, 10^10, 10^(10^10) etc. - M. F. Hasler, Dec 12 2018

			a(n) = 1, 2, 3, 4 for n = 1, 10, 10^10, 10^(10^10), i.e., n = 1, 10, 10000000000, 10^10000000000.
a(n) = 2 for all n >= 10, n < 10^10.

Cf. A001069, A010096, A211661, A211663, A211666, A211668, A211670.

a[n_] := Length[NestWhileList[Log10, n, # >= 1 &]] - 1; Array[a, 100] (* Amiram Eldar, Dec 08 2018 *)

a(n,i=1)={while(n=logint(n,10),i++);i} \\ M. F. Hasler, Dec 07 2018

With E_{i=1..n} c(i) := c(1)^(c(2)^(c(3)^(...(c(n-1)^(c(n)))...))); E_{i=1..0} := 1; example: E_{i=1..3} 10 = 10^(10^10) = 10^10000000000, we have:
a(E_{i=1..n} 10) = a(E_{i=1..n-1} 10) + 1, for n >= 1.
G.f.: g(x) = (1/(1-x))*Sum_{k>=0} x^(E_{i=1..k} 10) = (x + x^10 + x^(10^10) + ...)/(1-x).

Name reworded by M. F. Hasler, Dec 12 2018

A333128 Ending position of the first occurrence of n in the decimal expansion of Pi.

Original entry on oeis.org

33, 2, 7, 1, 3, 5, 8, 14, 12, 6, 51, 96, 150, 112, 3, 5, 42, 97, 426, 39, 55, 95, 137, 18, 294, 91, 8, 30, 35, 188, 66, 2, 17, 26, 88, 11, 287, 48, 19, 45, 72, 4, 94, 25, 61, 62, 21, 121, 89, 59, 33, 50, 174, 10, 193, 132, 212, 406, 12, 6, 129, 221, 22, 314

Offset: 0

Francesco Vissani, Apr 08 2020

Variant of A032445, considering the position of the least-significant digit of n in the decimal expansion of Pi (A000796).

Cf. A000796, A004216, A032445.

n = 1000;
intPi = Ceiling[N[Pi, n]*10^(n-1)];
piString = ToString[intPi];
Table[StringPosition[piString, ToString[n]][[1, 2]] , {n, 0, 70}]

a(n) = A032445(n) for n=0..9.
a(n) = 1 + A032445(n) for n = 10..99.
a(n) = A004216(n) + A032445(n).

A348960 a(n) = floor(log_10(Pi*n!)).

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 16, 17, 18, 20, 21, 22, 24, 25, 27, 28, 29, 31, 32, 34, 35, 37, 38, 40, 42, 43, 45, 46, 48, 50, 51, 53, 54, 56, 58, 59, 61, 63, 64, 66, 68, 70, 71, 73, 75, 77, 78, 80, 82, 84, 85, 87, 89, 91, 93, 95, 96, 98, 100

Offset: 0

Paul F. Marrero Romero, Nov 05 2021

nonn

Cf. A004216, A034886, A053511.

Array[Floor@ Log10[Pi*#!] &, 71, 0] (* Michael De Vlieger, Nov 05 2021 *)

a(n) = floor(log_10(Pi*n!)).

a(n) = floor(A053511 + log_10(n!)).

A371672 a(n) = floor(log_phi(n)) with phi = A001622.

Original entry on oeis.org

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

Offset: 1

Aleksandar V. Petojevic, Apr 02 2024

nonn

Aleksandar Petojević, Marjana Gorjanac Ranitović, Dragan Rastovac, and Milinko Mandić, The Golden Ratio, Factorials, and the Lambert W Function, Journal of Integer Sequences, Vol. 27 (2024), Article 24.5.7. See p. 4.

Cf. A000195, A000523, A001622, A004216, A362692.

Table[IntegerPart[Log[n]/Log[GoldenRatio]],{n,1,130}]

cp's OEIS Frontend

A076786 Squarefree numbers such that in decimal representation all their prefixes are also squarefree.

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A077429 a(n) = floor(log_10(n^2)).

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A123119 Number of digits in sum of first n primes (A007504).

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A118115 Partial sums of n concatenated n times.

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A118117 Concatenate n F(n) times.

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A211665 Minimal number of iterations of log_10 applied to n until the result is < 1.

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PARI

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A333128 Ending position of the first occurrence of n in the decimal expansion of Pi.

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A348960 a(n) = floor(log_10(Pi*n!)).

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