A367066 -id:A367066 - cp's OEIS frontend

A367068 a(n) = ((Sum_{i=1..n} A340510(i))-1)/(n+1).

0, 1, 2, 2, 3, 4, 4, 5, 6, 6, 7, 7, 8, 9, 9, 10, 11, 11, 12, 12, 13, 14, 14, 15, 16, 16, 17, 17, 18, 19, 19, 20, 20, 21, 22, 22, 23, 24, 24, 25, 25, 26, 27, 27, 28, 29, 29, 30, 30, 31, 32, 32, 33, 33, 34, 35, 35, 36, 37, 37, 38, 38, 39, 40, 40, 41, 42, 42

Offset: 1

Zenan Sabanac, Dec 17 2023

nonn

For a positive integer k define the Avdispahić-Zejnulahi sequence AZ(k) by b(1)=k, and thereafter b(n) is the least positive integer not yet in the sequence such that Sum_{i=1..n} b(i) == k (mod n+k).
Define the Avdispahić-Zejnulahi means sequence AZM(k) by a(n) = ((Sum_{i=1..n} b(i))-k)/(n+k). This is the AZM(1) sequence.
Is this a duplicate of A005379? For n<=1300 at least a(n)=A005379(n). - R. J. Mathar, Jan 30 2024

Muharem Avdispahić and Faruk Zejnulahi, An integer sequence with a divisibility property, Fibonacci Quarterly, Vol. 58:4 (2020), 321-333.

Cf. A340510.

Cf. A073869 (AZM(0)), A367066 (AZM(2)).

A367068 := proc(n)
    add(A340510(i),i=1..n)-1 ;
    %/(n+1) ;
end proc:
seq(A367068(n),n=1..50) ; # R. J. Mathar, Jan 30 2024

zlist = {-1, 1, 3};
mlist = {-1, 0, 1};
For[n = 3, n <= 101, n++,
  If[MemberQ[zlist, mlist[[n]]], AppendTo[mlist, mlist[[n]] + 1];
    AppendTo[zlist, mlist[[n + 1]] + n];, AppendTo[mlist, mlist[[n]]];
     AppendTo[zlist, mlist[[n + 1]]];];];
mlist = Drop[mlist, 1]; mlist

z_list=[-1, 1, 3]
m_list=[-1, 0, 1]
n=2
for n in range(2, 100):
    if m_list[n] in z_list:
        m_list.append(m_list[n] + 1)
        z_list.append(m_list[n+1] + n+1)
    else:
        m_list.append(m_list[n])
        z_list.append(m_list[n+1])
print(m_list[1:])

For n>2, a(n) = a(n-1) if a(n-1) <> A340510(k) (for k=1..n-1) and a(n) = a(n-1)+1=A340510(n)-n otherwise. (See Proposition 3.1. of Avdispahić and Zejnulahi in the link above).

A367069 a(n) = ((Sum_{i=1..n} A367067(i))-3)/(n+3).

Original entry on oeis.org

0, 1, 1, 2, 2, 3, 4, 4, 5, 6, 6, 7, 7, 8, 9, 9, 10, 10, 11, 12, 12, 13, 14, 14, 15, 15, 16, 17, 17, 18, 19, 19, 20, 20, 21, 22, 22, 23, 23, 24, 25, 25, 26, 27, 27, 28, 28, 29, 30, 30, 31, 31, 32, 33, 33, 34, 35, 35, 36, 36, 37, 38, 38, 39, 40, 40, 41, 41, 42

Offset: 1

Zenan Sabanac, Dec 17 2023

nonn

For a positive integer k define the Avdispahić-Zejnulahi sequence AZ(k) by b(1)=k, and thereafter b(n) is the least positive integer not yet in the sequence such that Sum_{i=1..n} b(i) == k (mod n+k). Define the Avdispahić-Zejnulahi means sequence AZM(k) by a(n) = ((Sum_{i=1..n} b(i))-k)/(n+k). This is the AZM(3) sequence.

Muharem Avdispahić and Faruk Zejnulahi, An integer sequence with a divisibility property, Fibonacci Quarterly, Vol. 58:4 (2020), 321-333.

Cf. A367067.

Cf. A073869 (AZM(0)), A367068 (AZM(1)), A367066 (AZM(2)).

zlist = {-1, 3, 5};
mlist = {-1, 0, 1};
For[n = 3, n <= 101, n++,
  If[MemberQ[zlist, mlist[[n]]], AppendTo[mlist, mlist[[n]] + 1];
    AppendTo[zlist, mlist[[n + 1]] + n + 2];,
    AppendTo[mlist, mlist[[n]]]; AppendTo[zlist, mlist[[n + 1]]];];];
mlist = Drop[mlist, 1]; mlist

z_list=[-1, 3, 5]
m_list=[-1, 0, 1]
n=2
for n in range(2, 100):
    if m_list[n] in z_list:
        m_list.append(m_list[n] + 1)
        z_list.append(m_list[n+1] + n+3)
    else:
        m_list.append(m_list[n])
        z_list.append(m_list[n+1])
print(m_list[1:])

cp's OEIS Frontend

A367068 a(n) = ((Sum_{i=1..n} A340510(i))-1)/(n+1).

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