A001511 -id:A001511 - cp's OEIS frontend

A349346 Dirichlet inverse of A181988, where A181988(n) = A001511(n)*A003602(n).

1, -2, -2, 1, -3, 4, -4, 0, -1, 6, -6, -2, -7, 8, 4, 0, -9, 2, -10, -3, 5, 12, -12, 0, -4, 14, -2, -4, -15, -8, -16, 0, 7, 18, 6, -1, -19, 20, 8, 0, -21, -10, -22, -6, 3, 24, -24, 0, -9, 8, 10, -7, -27, 4, 8, 0, 11, 30, -30, 4, -31, 32, 4, 0, 9, -14, -34, -9, 13, -12, -36, 0, -37, 38, 8, -10, 9, -16, -40, 0, -4, 42

Offset: 1

Antti Karttunen, Nov 15 2021

sign

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A001511, A003602, A181988, A349347.

Cf. also A347957, A349134, A349344.

up_to = 20000;
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(dA001511(n) = 1+valuation(n,2);
A003602(n) = (1+(n>>valuation(n,2)))/2;
A181988(n) = (A001511(n)*A003602(n));
v349346 = DirInverseCorrect(vector(up_to,n,A181988(n)));
A349346(n) = v349346[n];

a(1) = 1; a(n) = -Sum_{d|n, d < n} A181988(n/d) * a(d).

a(n) = A349347(n) - A181988(n).

A349445 Dirichlet convolution of A001511 (the 2-adic valuation of 2n) with A349134 (the Dirichlet inverse of Kimberling's paraphrases).

Original entry on oeis.org

1, 1, -1, 1, -2, -1, -3, 1, -2, -2, -5, -1, -6, -3, 0, 1, -8, -2, -9, -2, 0, -5, -11, -1, -6, -6, -4, -3, -14, 0, -15, 1, 0, -8, 0, -2, -18, -9, 0, -2, -20, 0, -21, -5, 2, -11, -23, -1, -12, -6, 0, -6, -26, -4, 0, -3, 0, -14, -29, 0, -30, -15, 3, 1, 0, 0, -33, -8, 0, 0, -35, -2, -36, -18, 4, -9, 0, 0, -39, -2, -8

Offset: 1

Antti Karttunen, Nov 18 2021

sign

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A001511, A003602, A349134, A349444 (Dirichlet inverse), A349446 (sum with it).

Cf. also A349432, A349448.

k[n_] := (n / 2^IntegerExponent[n, 2] + 1)/2; kinv[1] = 1; kinv[n_] := kinv[n] = -DivisorSum[n, kinv[#]*k[n/#] &, # < n &]; a[n_] := DivisorSum[n, IntegerExponent[2*#, 2]*kinv[n/#] &]; Array[a, 100] (* Amiram Eldar, Nov 19 2021 *)

A001511(n) = (1+valuation(n,2));
A003602(n) = (1+(n>>valuation(n,2)))/2;
memoA349134 = Map();
A349134(n) = if(1==n,1,my(v); if(mapisdefined(memoA349134,n,&v), v, v = -sumdiv(n,d,if(dA003602(n/d)*A349134(d),0)); mapput(memoA349134,n,v); (v)));
A349445(n) = sumdiv(n,d,A001511(n/d)*A349134(d));

a(n) = Sum_{d|n} A001511(n/d) * A349134(d).

If p odd prime, a(p) = (1-p)/2. - Bernard Schott, Nov 19 2021

A373347 Positive integers k such that A000120(k) > A001511(k).

Original entry on oeis.org

3, 5, 7, 9, 11, 13, 14, 15, 17, 19, 21, 22, 23, 25, 26, 27, 29, 30, 31, 33, 35, 37, 38, 39, 41, 42, 43, 45, 46, 47, 49, 50, 51, 53, 54, 55, 57, 58, 59, 60, 61, 62, 63, 65, 67, 69, 70, 71, 73, 74, 75, 77, 78, 79, 81, 82, 83, 85, 86, 87, 89, 90, 91, 92, 93, 94, 95, 97, 98, 99

Offset: 1

Paolo Xausa, Jun 01 2024

Numbers whose binary expansion does not encode for any Schreier set (cf. A371176 and A373345).

All odd numbers > 1 are terms.

Paolo Xausa, Table of n, a(n) for n = 1..10000

Complement of A371176.

Cf. A000120, A001511, A008466, A373345, A373360 (first differences).

Select[Range[100], DigitSum[#, 2] > IntegerExponent[#, 2] + 1 &]

isok(k) = hammingweight(k) > valuation(2*k, 2); \\ Michel Marcus, Jun 07 2024

def isa(n): return (n - 1).bit_count() < ((n.bit_count() - 1) << 1)
print([n for n in range(100) if isa(n)])  # Peter Luschny, Jun 07 2024

a(k) = 2^(n+1) - 1; a(k+1) = 2^(n+1) + 1, where k = A008466(n+1).

A286253 Compound filter: a(n) = P(A055396(n), A001511(1+n)), where P(n,k) is sequence A000027 used as a pairing function.

Original entry on oeis.org

0, 1, 8, 1, 9, 1, 25, 1, 5, 1, 26, 1, 27, 1, 17, 1, 35, 1, 53, 1, 5, 1, 75, 1, 9, 1, 8, 1, 65, 1, 131, 1, 5, 1, 13, 1, 90, 1, 12, 1, 104, 1, 134, 1, 5, 1, 186, 1, 14, 1, 8, 1, 152, 1, 18, 1, 5, 1, 188, 1, 189, 1, 30, 1, 9, 1, 229, 1, 5, 1, 273, 1, 252, 1, 8, 1, 14, 1, 347, 1, 5, 1, 323, 1, 9, 1, 12, 1, 324, 1, 19, 1, 5, 1, 31, 1, 350, 1, 8, 1, 377, 1, 462, 1, 5

Offset: 1

Antti Karttunen, May 07 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000
MathWorld, Pairing Function

Cf. A000027, A001511, A055396, A286164, A286251, A286252, A286254.

A001511(n) = (1+valuation(n,2));
A055396(n) = if(n==1, 0, primepi(factor(n)[1, 1])); \\ This function from Charles R Greathouse IV, Apr 23 2015
A286253(n) = (2 + ((A055396(n)+A001511(1+n))^2) - A055396(n) - 3*A001511(1+n))/2;
for(n=1, 10000, write("b286253.txt", n, " ", A286253(n)));

from sympy import primepi, isprime, primefactors
def T(n, m): return ((n + m)**2 - n - 3*m + 2)/2
def a049084(n): return primepi(n)*(1*isprime(n))
def a055396(n): return 0 if n==1 else a049084(min(primefactors(n)))
def a001511(n): return 2 + bin(n - 1)[2:].count("1") - bin(n)[2:].count("1")
def a(n): return T(a055396(n), a001511(n + 1)) # Indranil Ghosh, May 07 2017

(define (A286253 n) (* (/ 1 2) (+ (expt (+ (A055396 n) (A001511 (+ 1 n))) 2) (- (A055396 n)) (- (* 3 (A001511 (+ 1 n)))) 2)))

a(n) = (1/2)*(2 + ((A055396(n)+A001511(1+n))^2) - A055396(n) - 3*A001511(1+n)).

A286254 Compound filter: a(n) = P(A001511(n), A055396(1+n)), where P(n,k) is sequence A000027 used as a pairing function.

Original entry on oeis.org

1, 5, 1, 13, 1, 12, 1, 14, 1, 17, 1, 31, 1, 5, 1, 60, 1, 38, 1, 9, 1, 47, 1, 19, 1, 5, 1, 69, 1, 68, 1, 27, 1, 8, 1, 94, 1, 5, 1, 124, 1, 107, 1, 9, 1, 122, 1, 33, 1, 5, 1, 156, 1, 8, 1, 14, 1, 155, 1, 193, 1, 5, 1, 43, 1, 192, 1, 9, 1, 212, 1, 280, 1, 5, 1, 18, 1, 255, 1, 20, 1, 278, 1, 13, 1, 5, 1, 355, 1, 12, 1, 9, 1, 8, 1, 441, 1, 5, 1, 381, 1, 380, 1, 14

Offset: 1

Antti Karttunen, May 07 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000
MathWorld, Pairing Function

Cf. A000027, A001511, A055396, A253790, A286161, A286251, A286252, A286253.

A001511(n) = (1+valuation(n,2));
A055396(n) = if(n==1, 0, primepi(factor(n)[1, 1])); \\ This function from Charles R Greathouse IV, Apr 23 2015
A286254(n) = (2 + ((A001511(n)+A055396(1+n))^2) - A001511(n) - 3*A055396(1+n))/2;
for(n=1, 10000, write("b286254.txt", n, " ", A286254(n)));

from sympy import primepi, isprime, primefactors
def T(n, m): return ((n + m)**2 - n - 3*m + 2)/2
def a049084(n): return primepi(n)*(1*isprime(n))
def a055396(n): return 0 if n==1 else a049084(min(primefactors(n)))
def a001511(n): return 2 + bin(n - 1)[2:].count("1") - bin(n)[2:].count("1")
def a(n): return T(a001511(n), a055396(n + 1)) # Indranil Ghosh, May 07 2017

(define (A286254 n) (* (/ 1 2) (+ (expt (+ (A001511 n) (A055396 (+ 1 n))) 2) (- (A001511 n)) (- (* 3 (A055396 (+ 1 n)))) 2)))

a(n) = (1/2)*(2 + ((A001511(n)+A055396(1+n))^2) - A001511(n) - 3*A055396(1+n)).

A318450 Denominators of the sequence whose Dirichlet convolution with itself yields A001511, the 2-adic valuation of 2n.

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 2, 1, 8, 2, 2, 2, 2, 2, 4, 1, 2, 8, 2, 2, 4, 2, 2, 2, 8, 2, 16, 2, 2, 4, 2, 1, 4, 2, 4, 8, 2, 2, 4, 2, 2, 4, 2, 2, 16, 2, 2, 2, 8, 8, 4, 2, 2, 16, 4, 2, 4, 2, 2, 4, 2, 2, 16, 1, 4, 4, 2, 2, 4, 4, 2, 8, 2, 2, 16, 2, 4, 4, 2, 2, 128, 2, 2, 4, 4, 2, 4, 2, 2, 16, 4, 2, 4, 2, 4, 2, 2, 8, 16, 8, 2, 4, 2, 2, 8

Offset: 1

Antti Karttunen and Andrew Howroyd, Aug 29 2018

The sequence seems to give the denominators of several other similarly constructed "Dirichlet Square Roots".

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A001511, A318449 (numerators), A318451.

a1511[n_] := IntegerExponent[2n, 2];
f[1] = 1; f[n_] := f[n] = 1/2 (a1511[n] - Sum[f[d] f[n/d], {d, Divisors[ n][[2 ;; -2]]}]);
Table[f[n] // Denominator, {n, 1, 105}] (* Jean-François Alcover, Sep 13 2018 *)

up_to = 65537;
A001511(n) = 1+valuation(n,2);
DirSqrt(v) = {my(n=#v, u=vector(n)); u[1]=1; for(n=2, n, u[n]=(v[n]/v[1] - sumdiv(n, d, if(d>1&&dA317937.
v318449_51 = DirSqrt(vector(up_to, n, A001511(n)));
A318450(n) = denominator(v318449_51[n]);

a(n) = denominator of f(n), where f(1) = 1, f(n) = (1/2) * (A001511(n) - Sum_{d|n, d>1, d 1.

a(n) = 2^A318451(n).

A324725 a(n) = sign(A324543(n)) * A001511(A324543(n)), with a(n) = 0 if A324543(n) = 0.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Mar 17 2019

sign

Antti Karttunen, Table of n, a(n) for n = 1..10000 (based on Hans Havermann's factorization of A156552)
Index entries for sequences related to binary expansion of n
Index entries for sequences computed from indices in prime factorization
Index entries for sequences related to sigma(n)

Cf. A001511, A324343, A324543, A324724, A324817, A324828.

A324543(n) = sumdiv(n,d,moebius(n/d)*A323243(d));
A001511ext(n) = if(!n,n,sign(n)*(1+valuation(n,2))); \\ Like A001511 but gives 0 for 0 and -A001511(-n) for negative numbers.
A324725(n) = A001511ext(A324543(n));

If A324543(n) = 0, then a(n) = 0, otherwise a(n) = sign(A324543(n)) * A001511(A324543(n)).
a(p) = 1 for all primes p.
A324828(n) = [1 == abs(a(n))], where [ ] is the Iverson bracket.

A324882 a(1) = 0; for n > 1, a(n) = A001511(A324866(n)), where A324866(n) = A156552(n) OR (A323243(n) - A156552(n)).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Mar 28 2019

nonn

Terms 0 .. k occur for the first time at n = 1, 2, 9, 25, 133, 253, 559, 2159, 2489, 3151, 5597, 7967, ..., which after 2 seem all to be semiprimes, that is, A156552(n) has binary weight 2.

Antti Karttunen, Table of n, a(n) for n = 1..10000 (based on Hans Havermann's factorization of A156552)
Index entries for sequences computed from indices in prime factorization
Index entries for sequences related to sigma(n)

Cf. A001511, A156552, A323243, A324866, A324883, A324884.

A156552(n) = {my(f = factor(n), p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res}; \\ From A156552
A324866(n) = { my(k=A156552(n)); bitor(k,(A323243(n)-k)); }; \\ Needs also code from A323243.
A001511ext(n) = if(!n,n,sign(n)*(1+valuation(n,2))); \\ Like A001511 but gives 0 for 0 and -A001511(-n) for negative numbers.
A324882(n) = A001511ext(A324866(n));

a(1) = 0; for n > 1, a(n) = A001511(A324866(n)).
a(n) = A324884(n) - A324883(n).
a(p) = 1 for all primes p.

A324884 a(1) = 0; for n > 1, a(n) = A001511(A324819(n)), where A324819(n) = 2*A156552(n) OR A323243(n).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Mar 28 2019

nonn

Terms 0 .. k occur for the first time at n = 1, 2, 4, 9, 85, 133, 451, 1469, 2159, 2489, 4393, 7279, ..., which after 2 seem all to be semiprimes, that is, A156552(n) has binary weight 2.

Antti Karttunen, Table of n, a(n) for n = 1..10000 (based on Hans Havermann's factorization of A156552)
Index entries for sequences computed from indices in prime factorization
Index entries for sequences related to sigma(n)

Cf. A001511, A156552, A323243, A324819, A324882, A324883.

A156552(n) = {my(f = factor(n), p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res}; \\ From A156552
A324819(n) = bitor(2*A156552(n),A323243(n)); \\ Needs code also from A323243.
A001511ext(n) = if(!n,n,sign(n)*(1+valuation(n,2))); \\ Like A001511 but gives 0 for 0 and -A001511(-n) for negative numbers.
A324884(n) = A001511ext(A324819(n));

a(1) = 0; for n > 1, a(n) = A001511(A324819(n)).
a(n) = A324882(n) + A324883(n).
a(p) = 1 for all primes p.

A094290 a(n) = prime(A001511(n)), where A001511 is one more than the 2-adic valuation of n.

Original entry on oeis.org

2, 3, 2, 5, 2, 3, 2, 7, 2, 3, 2, 5, 2, 3, 2, 11, 2, 3, 2, 5, 2, 3, 2, 7, 2, 3, 2, 5, 2, 3, 2, 13, 2, 3, 2, 5, 2, 3, 2, 7, 2, 3, 2, 5, 2, 3, 2, 11, 2, 3, 2, 5, 2, 3, 2, 7, 2, 3, 2, 5, 2, 3, 2, 17, 2, 3, 2, 5, 2, 3, 2, 7, 2, 3, 2, 5, 2, 3, 2, 11, 2, 3, 2, 5, 2, 3, 2, 7, 2, 3, 2, 5, 2, 3, 2, 13, 2, 3, 2, 5, 2, 3

Offset: 1

Amarnath Murthy, Apr 28 2004

nonn

Originally defined as: a(1) = 2 = prime(1). Then the first occurrence of prime(n) followed by all previous terms. i.e. If the index of first occurrence of prime(n) is k then the next k-1 terms are defined as a(k+r) = a(r), r = 1 to k-1. and a(2k) = prime(n+1) and so on.
Index of the first occurrence of prime(n)= 2^(n-1). Subsidiary sequences: If prime(n) is replaced by f(n) a large number of sequences can be obtained choosing f(n) = composite(n), f(n) = n^2,f(n) = n^r, r =3,4,5,..., f(n) = tau(n), f(n) = sigma(n), f(n) = n!, f(n) = Fibonacci(n), f(n) = T(n), triangular number, f(n) = n-th Bell, etc. each giving a distinct fascinating music.
The lexicographically earliest sequence such that no product of consecutive terms is a perfect square. - Joshua Zucker, Apr 30 2011

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A000040, A001511.

Cf. also A115364.

Array[Prime[IntegerExponent[#, 2] + 1] &, 102] (* Michael De Vlieger, Nov 02 2018 *)

A094290(n) = prime(1+valuation(n,2)); \\ Antti Karttunen, Nov 02 2018

from sympy import prime
def A094290(n): return prime((~n & n-1).bit_length()+1) # Chai Wah Wu, Jul 02 2022

a(n) = A000040(A001511(n)). - Omar E. Pol, Sep 13 2013

Replaced the name with a formula given by Omar E. Pol, which is equivalent to the original definition. - Antti Karttunen, Nov 02 2018

cp's OEIS Frontend

A349346 Dirichlet inverse of A181988, where A181988(n) = A001511(n)*A003602(n).

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A349445 Dirichlet convolution of A001511 (the 2-adic valuation of 2n) with A349134 (the Dirichlet inverse of Kimberling's paraphrases).

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A373347 Positive integers k such that A000120(k) > A001511(k).

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A286253 Compound filter: a(n) = P(A055396(n), A001511(1+n)), where P(n,k) is sequence A000027 used as a pairing function.

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A286254 Compound filter: a(n) = P(A001511(n), A055396(1+n)), where P(n,k) is sequence A000027 used as a pairing function.

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A318450 Denominators of the sequence whose Dirichlet convolution with itself yields A001511, the 2-adic valuation of 2n.

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A324725 a(n) = sign(A324543(n)) * A001511(A324543(n)), with a(n) = 0 if A324543(n) = 0.

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A324882 a(1) = 0; for n > 1, a(n) = A001511(A324866(n)), where A324866(n) = A156552(n) OR (A323243(n) - A156552(n)).

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