A364499 a(n) = A005940(n) - n.

0, 0, 0, 0, 0, 0, 2, 0, -2, 0, 4, 0, 12, 4, 12, 0, -6, -4, 2, 0, 14, 8, 22, 0, 24, 24, 48, 8, 96, 24, 50, 0, -20, -12, -2, -8, 18, 4, 24, 0, 36, 28, 62, 16, 130, 44, 88, 0, 72, 48, 96, 48, 192, 96, 170, 16, 286, 192, 316, 48, 564, 100, 180, 0, -48, -40, -28, -24, -4, -4, 28, -16, 18, 36, 90, 8, 198, 48, 110, 0, 62

Offset

1,7

Comments

Compare to the scatter plot of A364563.

From Antti Karttunen, Aug 11 2023: (Start)

Can be computed as a certain kind of bitmask transformation of A364568 (analogous to the inverse Möbius transform that is appropriate for A156552-encoding of n).

See A364572, A364573 (and also A364576) for n (apart from those in A029747) where a(n) comes relatively close to the X-axis.

(End)

Links

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

Index entries for sequences related to binary expansion of n

Formula

a(n) = -A364559(A005940(n)).

For all n >= 1, a(2*n) = 2*a(n).

For all n >= 1, a(A029747(n)) = 0.

Example

A005940(528577) = 528581, therefore a(528577) = 528581 - 528577 = 4. (See A364576).
A005940(2109697) = 2109629, therefore a(2109697) = 2109629 - 2109697 = -68.

Mathematica

nn = 81; Array[Set[a[#], #] &, 2]; Do[If[EvenQ[n], Set[a[n], 2 a[n/2]], Set[a[n], Times @@ Power @@@ Map[{Prime[PrimePi[#1] + 1], #2} & @@ # &, FactorInteger[a[(n + 1)/2]]]]], {n, 3, nn}]; Array[a[#] - # &, nn] (* Michael De Vlieger, Jul 28 2023 *)

Prog

(PARI)
A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t };
A364499(n) = (A005940(n)-n);
(PARI) A364499(n) = { my(m=1, p=2, x=0, z=1); n--; while(n, if(!(n%2), p=nextprime(1+p), x += m; z *= p); n>>=1; m <<=1); (z-x)-1; }; \\ Antti Karttunen, Aug 06 2023
(Python)
from math import prod
from itertools import accumulate
from collections import Counter
from sympy import prime
def A364499(n): return prod(prime(len(a)+1)**b for a, b in Counter(accumulate(bin(n-1)[2:].split('1')[:0:-1])).items())-n # Chai Wah Wu, Aug 07 2023

Crossrefs

Cf. A005940, A364500 [= gcd(n,a(n))], A364559, A364572, A364573, A364576.

Cf. A029747 (known positions of 0's), A364540 (positions of terms < 0), A364541 (of terms <= 0), A364542 (of terms >= 0), A364563 [= -a(A364543(n))].

Cf. also A364258, A364568.

Sequence in context: A052120 A179461 A364568 * A238763 A029186 A283023

Adjacent sequences: A364496 A364497 A364498 * A364500 A364501 A364502

Keyword

sign,look

Author

Antti Karttunen, Jul 28 2023

Status

approved