A366370 -id:A366370 - cp's OEIS frontend

A365805 a(n) = largest exponent m for which a representation of the form A163511(n) = k^m exists (for some k). a(0) = 0 by convention.

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

Offset: 0

Antti Karttunen, Oct 01 2023

nonn

Equivalently, the largest exponent m for which a representation of the form A332214(n) = k^m exists (for some k), or similarly, for any other such variant of A163511, like A332817.

Antti Karttunen, Table of n, a(n) for n = 0..65537
Index entries for sequences related to binary expansion of n

Cf. A052409, A163511, A332214, A332817, A365807, A366370.

Cf. A365808 (positions of even terms), A365801 (multiples of 3), A365802 (multiples of 5), A366287 (multiples of 7), A366391 (multiples of 11).

A052409(n) = { my(k=ispower(n)); if(k, k, n>1); };
A163511(n) = if(!n, 1, my(p=2, t=1); while(n>1, if(!(n%2), (t*=p), p=nextprime(1+p)); n >>= 1); (t*p));
A365805(n) = A052409(A163511(n));

a(n) = A052409(A163511(n)).

If a(n) > 1 (or A052409(n) > 1), then a(n) <> A052409(n). [Consider A366370]

A285097 a(n) = difference between the positions of two least significant 1-bits in base-2 representation of n, or 0 if there are less than two 1-bits in n (when n is either zero or a power of 2).

Original entry on oeis.org

0, 0, 0, 1, 0, 2, 1, 1, 0, 3, 2, 1, 1, 2, 1, 1, 0, 4, 3, 1, 2, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 0, 5, 4, 1, 3, 2, 1, 1, 2, 3, 2, 1, 1, 2, 1, 1, 1, 4, 3, 1, 2, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 0, 6, 5, 1, 4, 2, 1, 1, 3, 3, 2, 1, 1, 2, 1, 1, 2, 4, 3, 1, 2, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 5, 4, 1, 3, 2, 1, 1, 2, 3, 2, 1, 1, 2, 1, 1, 1, 4, 3, 1, 2, 2, 1, 1, 1

Offset: 0

Antti Karttunen, Apr 20 2017

a(1+n) is the length of the least significant run of 0-bits in n, or 0 if n is one of terms of A000225. - Antti Karttunen, Oct 14 2023

			For n = 3, "11" in binary, the second least significant 1-bit (the second 1-bit from the right) is at position 1 and the rightmost 1-bit is at position 0, thus a(3) = 1-0 = 1.
For n = 4, "100" in binary, there is just one 1-bit present, thus a(4) = 0.
For n = 5, "101" in binary, the second 1-bit from the right is at position 2, and the least significant 1 is at position 0, thus a(5) = 2-0 = 2.
For n = 26, "11010" in binary, the second 1-bit from the right is at position 3, and the least significant 1 is at position 1, thus a(26) = 3-1 = 2.

Antti Karttunen, Table of n, a(n) for n = 0..8192
Index entries for sequences related to binary expansion of n

Cf. A000120, A000225, A000265, A007814, A119387, A129760, A285099, A366370.

a007814[n_]:=IntegerExponent[n, 2]; a285099[n_]:=If[DigitCount[n, 2, 1]<2, 0, a007814[BitAnd[n, n - 1]]]; a[n_]:=If[DigitCount[n, 2, 1]<2, 0,a285099[n] - a007814[n]]; Table[a[n], {n, 0, 150}] (* Indranil Ghosh, Apr 20 2017 *)

A285097(n) = if(!n || !bitand(n,n-1), 0, valuation((n>>valuation(n,2))-1, 2)); \\ Antti Karttunen, Oct 14 2023

import math
def a007814(n): return int(math.log(n - (n & n - 1), 2))
def a285099(n): return 0 if bin(n)[2:].count("1") < 2 else a007814(n & (n - 1))
def a(n): return 0 if bin(n)[2:].count("1")<2 else a285099(n) - a007814(n) # Indranil Ghosh, Apr 20 2017

(define (A285097 n) (if (<= (A000120 n) 1) 0 (- (A285099 n) (A007814 n))))

If A000120(n) < 2, a(n) = 0, otherwise a(n) = A285099(n) - A007814(n) = A007814(A129760(n)) - A007814(n).

a(n) = 0 if n is 0 or of the form 2^k, (k>=0), otherwise a(n) = v_2(A000265(n)-1), where v_2(i) = A007814(i). - Ridouane Oudra, Oct 20 2019

cp's OEIS Frontend

A365805 a(n) = largest exponent m for which a representation of the form A163511(n) = k^m exists (for some k). a(0) = 0 by convention.

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