A359794 -id:A359794

Search: a359794 -id:a359794

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A108269

Numbers of the form (2*m - 1)*4^k where m >= 1, k >= 1.

+10
11

4, 12, 16, 20, 28, 36, 44, 48, 52, 60, 64, 68, 76, 80, 84, 92, 100, 108, 112, 116, 124, 132, 140, 144, 148, 156, 164, 172, 176, 180, 188, 192, 196, 204, 208, 212, 220, 228, 236, 240, 244, 252, 256, 260, 268, 272, 276, 284, 292, 300, 304, 308, 316, 320, 324, 332

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

Numbers of terms in nonnegative integer sequences the sum of which is never a square.

The sum of a sequence of consecutive nonnegative integers starting with k is never a square for any k, if and only if the number of the terms in the sequence can be expressed as (2*m - 1) * 2^(2*n), m and n being any positive integers. (Proved by Alfred Vella, Jun 14 2005.)

Odious and evil terms alternate. - Vladimir Shevelev, Jun 22 2009

Even numbers whose binary representation ends in an even number of zeros. - Amiram Eldar, Jan 12 2021

From Antti Karttunen, Jan 28 2023: (Start)

Numbers k for which the parity of k is equal to that of A048675(k).

A multiplicative semigroup; if m and n are in the sequence then so is m*n. (End)

LINKS

Amiram Eldar, Table of n, a(n) for n = 1..10000

FORMULA

a(n) = 6*n + O(log n). - Charles R Greathouse IV, Nov 03 2016 [Corrected by Amiram Eldar, Jan 12 2021]

a(n) = 2 * A036554(n) = 4 * A003159(n). - Amiram Eldar, Jan 12 2021

EXAMPLE

a( 1, 1 ) = 4, a( 2, 1) = 12, etc.

For a( 1, 1 ): the sum of 4 consecutive nonnegative integers (4k+6, if the first term is k) is never a square.

MATHEMATICA

Select[2 * Range[200], EvenQ @ IntegerExponent[#, 2] &] (* Amiram Eldar, Jan 12 2021 *)

PROG

(PARI) is(n)=my(e=valuation(n, 2)); e>1 && e%2==0 \\ Charles R Greathouse IV, Nov 03 2016

CROSSREFS

Intersection of A005843 and A003159.

Cf. A000069, A001969, A017113 (primitive terms), A036554, A328981 (characteristic function), A359794 (complement).

KEYWORD

nonn,easy

AUTHOR

Andras Erszegi (erszegi.andras(AT)chello.hu), May 30 2005

EXTENSIONS

Entry revised by N. J. A. Sloane, Jun 26 2005

More terms from Amiram Eldar, Jan 12 2021

STATUS

approved

A359832

a(n) = 1 if the 2-adic valuation of n is either 0 or odd, otherwise 0.

+10
7

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..100000

Index entries for characteristic functions

FORMULA

Multiplicative with a(2^e) = 1 if e is odd, and 0 if e is even (and > 0), with a(p^e) = 1 for all odd primes p.

a(n) = 1 - A328981(n).

a(n) = A000035(n+A048675(n)).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 5/6. - Amiram Eldar, Jan 24 2023

MATHEMATICA

a[n_] := If[(e = IntegerExponent[n, 2]) == 0 || OddQ[e], 1, 0]; Array[a, 100] (* Amiram Eldar, Jan 24 2023 *)

PROG

(PARI) A359832(n) = (!(n=valuation(n, 2))||(n%2));

(PARI) A359832(n) = { my(f=factor(n)); prod(k=1, #f~, ((2!=f[k, 1]) || (f[k, 2]%2))); };

(Python)

def A359832(n): return (n&1)|((~n & n-1).bit_length()&1) # Chai Wah Wu, Jan 24 2023

CROSSREFS

Characteristic function of A359794.

Cf. A000035, A007814, A048675, A328981 (one's complement), A359833 (Dirichlet inverse).

KEYWORD

nonn,mult

AUTHOR

Antti Karttunen, Jan 24 2023

STATUS

approved

A373368

a(n) = gcd(n, A059975(n)), where A059975 is fully additive with a(p) = p-1.

+10
6

1, 1, 1, 2, 1, 3, 1, 1, 1, 5, 1, 4, 1, 7, 3, 4, 1, 1, 1, 2, 1, 11, 1, 1, 1, 13, 3, 4, 1, 1, 1, 1, 3, 17, 5, 6, 1, 19, 1, 1, 1, 3, 1, 4, 1, 23, 1, 6, 1, 1, 3, 2, 1, 1, 1, 1, 1, 29, 1, 4, 1, 31, 1, 2, 1, 1, 1, 2, 3, 1, 1, 1, 1, 37, 5, 4, 1, 3, 1, 8, 1, 41, 1, 2, 5, 43, 3, 1, 1, 9, 1, 4, 1, 47, 1, 1, 1, 1, 1, 10

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,4

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..65537

PROG

(PARI)

A059975(n) = {my(f = factor(n)); sum(i = 1, #f~, f[i, 2]*(f[i, 1] - 1)); };

A373368(n) = gcd(n, A059975(n));

CROSSREFS

Cf. A059975, A108269 (positions of even terms), A359794 (of odd terms), A359832 (parity of terms).

Cf. also A082299, A373361, A373369.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Jun 05 2024

STATUS

approved

A373361

a(n) = gcd(n, A276085(n)), where A276085 is the primorial base log-function.

+10
4

1, 1, 1, 2, 1, 3, 1, 1, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 3, 4, 1, 3, 1, 1, 1, 1, 1, 6, 1, 19, 1, 1, 1, 3, 1, 4, 5, 1, 1, 6, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 10, 1, 1, 1, 2, 1, 3, 1, 4, 1, 1, 1, 1, 1, 1, 1, 4, 1, 3, 1, 10, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 13, 4, 1, 1, 1, 1, 1, 1, 1, 2, 1, 3, 1, 1, 1

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,4

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..65537

Index entries for sequences related to primorial numbers

PROG

(PARI)

A002110(n) = prod(i=1, n, prime(i));

A276085(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*A002110(primepi(f[k, 1])-1)); };

A373361(n) = gcd(n, A276085(n));

CROSSREFS

Cf. A002110, A276085.

Cf. A108269 (positions of even terms), A328981 (their characteristic function), A359794 (positions of odd terms), A359832 (their characteristic function, parity of terms).

Cf. also A324198, A373145, A373362.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Jun 02 2024

STATUS

approved

A370596

Numbers k such that A007814(k) is a prime number.

+10
2

4, 8, 12, 20, 24, 28, 32, 36, 40, 44, 52, 56, 60, 68, 72, 76, 84, 88, 92, 96, 100, 104, 108, 116, 120, 124, 128, 132, 136, 140, 148, 152, 156, 160, 164, 168, 172, 180, 184, 188, 196, 200, 204, 212, 216, 220, 224, 228, 232, 236, 244, 248, 252, 260, 264, 268, 276

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

Numbers whose binary representation has a prime number of trailing 0's.

a(n)-1 is the sequence of numbers whose binary representation has a prime number of trailing 1's.

Numbers of the form (2^(p+1))*k + 2^p = 2^p * (2*k + 1), where p is prime and k >= 0.

All the terms are divisible by 4.

The asymptotic density of this sequence is Sum_{p prime} 1/2^(p+1) = 0.20734125492555583012... = A051006 / 2.

LINKS

Amiram Eldar, Table of n, a(n) for n = 1..10000

MATHEMATICA

Select[Range[300], PrimeQ[IntegerExponent[#, 2]] &]

PROG

(PARI) is(n) = isprime(valuation(n, 2));

CROSSREFS

Cf. A007814, A051006, A359794.

Subsequences: A017113, A051062.

KEYWORD

nonn,easy

AUTHOR

Amiram Eldar, Feb 23 2024

STATUS

approved

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