A391958 First column of A391957.

0, 0, 3, 1, 13, 7, 12, 4, 41, 27, 32, 16, 46, 26, 33, 11, 113, 83, 88, 56, 102, 66, 73, 35, 138, 94, 101, 55, 119, 69, 78, 26, 289, 227, 232, 168, 246, 178, 185, 115, 282, 206, 213, 135, 231, 149, 158, 74, 370, 278, 285, 191, 303, 205, 214, 114, 347, 241, 250

Offset

0,3

Comments

Powers of 2 in the factorization of A391956.

The first 82 terms were obtained from a brute force calculation of A391956. From these results a regular pattern is observed to determine more terms.

See also A391957 for a Python program. A.H.M. Smeets, Jan 13 2026

Links

A.H.M. Smeets, Table of n, a(n) for n = 0..10000

Formula

Conjectured: this sequence can be obtained by the following two rules:

(Eq.1) a(2^n) = 2*a(2^(n-1)) + 2^(n+1) - 1, a(1) = 0 (see also A392083) and

(Eq.2) a(n*2^m + 2^(m-1)) = a(n*2^m) + a(2^(m-1)) - 2^m*Sum_{i = 1..n} A001511(i).

The second rule (Eq.2) is a generalization of formula like:

a(2*n+1) = a(2*n) - 2*Sum_{i = 1..n} A001511(i),

a(4*n+2) = a(4*n) + 3 - 4*Sum_{i = 1..n} A001511(i) and

a(8*n+4) = a(8*n) + 13 - 8*Sum_{i = 1..n} A001511(i).

Prog

(Python)
def SumA001511(n):
    s, i, Sum = [1], 2, 0
    while len(s) < n: s, i = s+[i]+s, i+1
    while n > 0: Sum, n = Sum+s[n-1], n-1
    return Sum
def A391958(n):
    d, k = 2, 0
    while n%d == 0: d, k = 2*d, k+1
    if n == d//2: return A2power(k)
    else:  # Eq.2 in formula section
        return A391958(n-d//2) + A392083(k)- 2*2**k*SumA001511((n-d//2)//(2*2**k))

Crossrefs

Cf. A001511, A391956, A391957, A392083.

Sequence in context: A341725 A258239 A376863 * A133176 A089435 A152474

Adjacent sequences: A391955 A391956 A391957 * A391959 A391960 A391961

Keyword

nonn

Author

A.H.M. Smeets, Dec 23 2025

Extensions

Terms a(21), a(22) and a(23) inserted by A.H.M. Smeets, Jan 08 2026

Status

approved