A004555 -id:A004555 - cp's OEIS frontend

A068432 Expansion of golden ratio (1 + sqrt(5))/2 in base 2.

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

Offset: 1

Benoit Cloitre, Mar 09 2002

Differs from A004555 in the 2nd digit. - R. J. Mathar, Dec 15 2008

			1.1001111000110111011110011011100101...

Cf. A001622, A004555, A004714, A169868.

RealDigits[ (1+Sqrt[5])/2, 2, 1000][[1]]

concat(binary((1+sqrt(5))/2)) \\ Michel Marcus, Dec 14 2017

a(n) = floor(quadgen(5)*2^(n-1))%2 \\ Chittaranjan Pardeshi, Feb 06 2023

A096432 Let n = 2^e_2 * 3^e_3 * 5^e_5 * ... be the prime factorization of n; sequence gives n such that 1 + max{e_2, e_3, ...} is a prime.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 25, 26, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 55, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83

Offset: 1

N. J. A. Sloane, Sep 18 2008

nonn

The old entry with this sequence number was a duplicate of A004555.
Sequence is of positive density. - Charles R Greathouse IV, Dec 07 2012
The asymptotic density of this sequence is Sum_{p prime} (1/zeta(p) - 1/zeta(p-1)) = 0.8817562193... - Amiram Eldar, Oct 18 2020

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..881 from R. J. Mathar)

Cf. A060476, A074661.

(Maple code for this entry and A074661)
M:=2000; ans1:=[]; ans2:=[];
for n from 1 to M do
t1:=op(2..-1, ifactors(n)); t2:=nops(t1);
m1:=0; for i from 1 to t2 do m1:=max(m1,t1[i][2]); od:
if isprime(1+m1) then ans1:=[op(ans1),n]; fi;
if isprime(m1) then ans2:=[op(ans2),n]; fi;
od:

Select[Range[2, 100], PrimeQ[1 + Max[FactorInteger[#][[;; , 2]]]] &] (* Amiram Eldar, Oct 18 2020 *)

isA096432(n) = if(n<2,0,isprime(vecmax(factor(n)[,2])+1))

A004559 Expansion of sqrt(5) in base 6.

Original entry on oeis.org

2, 1, 2, 2, 5, 5, 3, 5, 5, 3, 1, 5, 1, 3, 0, 3, 3, 4, 3, 1, 2, 4, 5, 1, 4, 3, 2, 0, 3, 4, 0, 2, 4, 0, 1, 3, 4, 5, 4, 0, 2, 5, 2, 1, 3, 2, 2, 3, 2, 0, 3, 3, 2, 5, 0, 2, 1, 5, 4, 4, 1, 1, 0, 1, 3, 2, 1, 5, 5, 0, 1, 0, 0, 0, 4, 5, 3, 1, 4, 1, 1, 2, 5, 1, 4, 2, 5, 0, 0, 0, 0, 1, 1, 3, 4, 5, 1, 3, 5

Offset: 1

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 1..5000
Jason Kimberley, Index of expansions of sqrt(d) in base b

Cf. A002163, A004555, A004556, A004557, A004558.

Cf. A004540, A004549, A004575.

Prune(Reverse(IntegerToSequence(Isqrt(5*6^200), 6))); // Vincenzo Librandi, Jan 08 2018

RealDigits[Sqrt[5],6,120][[1]] (* Harvey P. Dale, Mar 24 2012 *)

Updated by Alois P. Heinz at the suggestion of Kevin Ryde, Feb 19 2012

A316997 Number of 1's in the first n digits of the binary expansion of sqrt(n).

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 4, 3, 5, 2, 5, 5, 9, 7, 11, 13, 1, 7, 9, 9, 12, 9, 11, 14, 10, 2, 13, 13, 16, 12, 16, 12, 16, 19, 18, 15, 2, 21, 18, 20, 19, 25, 19, 20, 25, 26, 19, 24, 26, 3, 20, 25, 25, 31, 28, 36, 30, 33, 33, 37, 38

Offset: 0

Rainer Rosenthal, Dec 14 2018

			For n = 7 we have sqrt(7) = 2.64575131... with binary expansion 10.1010010.... Of the first 7 digits there are a(7) = 3 digits equal to 1.

Rainer Rosenthal, Table of n, a(n) for n = 0..1000

Cf. A004539, A004547, A004555, A004609, A004569, A004585.

Cf. A000120, A000290.

zaehle := proc(n) local e, p, c, i, z, m; Digits := n+5; e := evalf(sqrt(n)); p := [op(convert(e, binary))]; c := convert(p[1], base, 10); z := 0; m := min(n, nops(c)); for i to m do if c[-i] = 1 then z := z+1; fi; od; return z; end: seq(zaehle(n), n=0..60); # Rainer Rosenthal, Dec 14 2018
a := n -> StringTools:-CountCharacterOccurrences(convert(convert(evalf(sqrt(n), n+5), binary, n), string), "1"): seq(a(n),n=0..60); # Peter Luschny, Dec 15 2018

a[n_] := Count[RealDigits[Sqrt[n], 2, n][[1]], 1]; Array[a, 60, 0] (* Amiram Eldar, Dec 14 2018 *)

a(n)=my(v=concat(binary(sqrt(n))));hammingweight(v[1..n]) \\ Hugo Pfoertner, Dec 16 2018

a(n^2) = A000120(n). - Michel Marcus, Dec 15 2018

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A068432 Expansion of golden ratio (1 + sqrt(5))/2 in base 2.

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A096432 Let n = 2^e_2 * 3^e_3 * 5^e_5 * ... be the prime factorization of n; sequence gives n such that 1 + max{e_2, e_3, ...} is a prime.

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A004559 Expansion of sqrt(5) in base 6.

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A316997 Number of 1's in the first n digits of the binary expansion of sqrt(n).

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