A364543 -id:A364543 - cp's OEIS frontend

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

Antti Karttunen, Jul 28 2023

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)

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

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences related to binary expansion of n

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.

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 *)

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);

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

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

a(n) = -A364559(A005940(n)).
For all n >= 1, a(2*n) = 2*a(n).
For all n >= 1, a(A029747(n)) = 0.

A364563 Difference k - A005940(k) computed for those odd numbers k for which the difference is nonnegative.

Original entry on oeis.org

0, 0, 0, 2, 6, 20, 2, 48, 28, 4, 110, 80, 48, 18, 18, 234, 202, 166, 110, 132, 12, 64, 42, 24, 484, 446, 402, 348, 360, 238, 50, 68, 276, 132, 246, 186, 204, 240, 994, 940, 884, 824, 830, 690, 462, 526, 722, 560, 240, 192, 680, 300, 596, 194, 602, 614, 696, 2012, 1958, 1898, 1794, 1840, 1624, 1336, 1442, 1724, 1458

Offset: 1

Antti Karttunen, Aug 06 2023

Antti Karttunen, Table of n, a(n) for n = 1..11023

Cf. A005940, A364499, A364543.

Cf. also A364294.

a(n) = -A364499(A364543(n)).

cp's OEIS Frontend

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

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