A329644 Möbius transform of A323244, the deficiency of A156552(n).

0, 1, 1, 1, 1, 2, 1, 4, -1, 3, 1, 5, 1, 14, 0, 0, 1, 9, 1, 12, -5, 16, 1, 8, -5, 44, 4, 5, 1, 2, 1, 24, 12, 80, -4, -4, 1, 254, -14, 0, 1, 22, 1, 47, 7, 224, 1, 24, -13, 19, 6, 83, 1, 12, -21, 44, -14, 746, 1, 14, 1, 1360, 20, -8, 8, 9, 1, 131, 252, 24, 1, 12, 1, 3836, 13, 149, -12, 71, 1, 56, -16, 5456, 1, -21, -74, 12248, -350, -40, 1

Offset: 1

Author

Antti Karttunen, Nov 21 2019

Keywords

• sign

Comments

The first eleven zeros occur at n = 1, 15, 16, 40, 96, 119, 120, 160, 893, 2464, 6731. There are 3091 negative terms among the first 10000 terms.

Applying this function to the divisors of the first four terms of A324201 reveals the following pattern:

------------------------------------------------------------------------------------

A324201(n) divisors a(n) applied Sum of positive

to each: terms, A329610

9: [1, 3, 9] -> [0, 1, -1] 1

125: [1, 5, 25, 125] -> [0, 1, -5, 4] 5

161051: [1, 11, 121, 1331, 14641, 161051] -> [0, 1, -29, 4, -240, 264] 269

410338673: [1, 17, 289, 4913, 83521, 1419857, 24137569, 410338673]

-> [0, 1, -125, 4, -1008, 1032, -5048, 5144] 6181

The positive and negative terms seem to alternate, and the fourth term (from case n=125 onward) is always 4. See also array A329637.

Links

• 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)

Crossrefs

Cf. A000203, A008683, A036563, A156552, A297112, A297113, A323243, A323244, A324543, A329610, A329637, A329638, A329639, A329645, A329646.

Programs

PARI

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
A323244(n) = if(1==n, 0, my(k=A156552(n)); (2*k)-sigma(k));
A329644(n) = sumdiv(n,d,moebius(n/d)*A323244(d));

Formula

a(n) = Sum_{d|n} A008683(n/d) * A323244(d).

a(n) = Sum_{d|n} A008683(n/d) * (2*A156552(d) - A323243(d)).

a(1) = 0; for n > 1, a(n) = 2*A297112(n) - A324543(n) = 2^A297113(n) - A324543(n).

a(n) = A329642(n) - A329643(n).

For all n >= 1, a(A000040(n)^2) = A323244(A000040(n)^2)-1 = -A036563(n).

For all primes p, a(p^3) = A323244(p^3) - A323244(p^2) = 4.