A344587 Deficiency of prime-shifted n: a(n) = 2*A003961(n) - sigma(A003961(n)).

1, 2, 4, 5, 6, 6, 10, 14, 19, 10, 12, 12, 16, 18, 22, 41, 18, 26, 22, 22, 38, 22, 28, 30, 41, 30, 94, 42, 30, 18, 36, 122, 46, 34, 58, 47, 40, 42, 62, 58, 42, 42, 46, 52, 102, 54, 52, 84, 109, 66, 70, 72, 58, 126, 70, 114, 86, 58, 60, 6, 66, 70, 178, 365, 94, 54, 70, 82, 110, 78, 72, 110, 78, 78, 148, 102, 118, 78

Offset: 1

Author

Antti Karttunen, May 28 2021

Keywords

• sign
• look

Comments

First negative value occurs as a(120) = -30.

Questions: Which subsets of natural numbers generate the "cut sigmoid" graph(s) that cross the X-axis in the (lowermost) scatter plot?

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537
• Index entries for sequences computed from indices in prime factorization
• Index entries for sequences related to sigma(n)

Crossrefs

Cf. A000203, A003961, A003973, A033879, A153881, A336851, A337386 (positions of terms <= 0), A346246 (Dirichlet inverse), A349387, A378216, A378231 [= a(n^2)].

Inverse Möbius transform of A337544.

Programs

PARI

A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A344587(n) = { my(u=A003961(n)); (u+u - sigma(u)); };

Formula

a(n) = A033879(A003961(n)) = 2*A003961(n) - A003973(n).

a(n) = Sum_{d|n} A337544(d).

From Antti Karttunen, Nov 23 2024: (Start)

a(n) = Sum_{d|n} A003961(d)*A153881(n/d) = A003961(n) - A336851(n).

a(n) = Sum_{d|n} A033879(d)*A349387(n/d).

a(n) = Sum_{d|n} A003972(d)*A378216(n/d).

(End)