Bassam Abdul-Baki - cp's OEIS frontend

User: Bassam Abdul-Baki

Bassam Abdul-Baki has authored 2 sequences.

A383174 Permutation of the natural numbers formed by ordering by max(gpfi,bigomega), then bigomega, then numerically, where gpfi(k) = A061395(k) and bigomega(k) = A001222(k).

1, 2, 3, 4, 6, 9, 5, 10, 15, 25, 8, 12, 18, 20, 27, 30, 45, 50, 75, 125, 7, 14, 21, 35, 49, 28, 42, 63, 70, 98, 105, 147, 175, 245, 343, 16, 24, 36, 40, 54, 56, 60, 81, 84, 90, 100, 126, 135, 140, 150, 189, 196, 210, 225, 250, 294, 315, 350, 375, 441, 490, 525

Offset: 1

Bassam Abdul-Baki, Apr 18 2025

nonn

The sequence can be constructed starting with term 1 and then:
At step number m >= 1, append terms k with gpfi(k) <= m and bigomega(k) <= m, and which have not already appeared, and ordered first by bigomega and then numerically.
Terms which have not appeared are exactly those with gpfi(k) = m or bigomega(k) = m, and in particular they start with prime(m) and end with prime(m)^m.
First differs from A344844 at n=30, with its ordering by prime exponents differing from here ordering numerically. - Michael S. Branicky, Apr 20 2025

			At step m=2, the new terms added are 3, 4, 6, 9, being those with gpfi(k) = 2, or with bigomega(k) = 2.

Michael S. Branicky, Table of n, a(n) for n = 1..12870 (all terms involving only primes <= 19, so ending with 19^8)
Index entries for sequences that are permutations of the natural numbers

Cf. A061395, A001222, A263297, A344844.

from math import prod
from sympy import nextprime
from itertools import count, islice, combinations_with_replacement as cwr
def agen(): # generator of terms
    aset, plst = set(), [1, 2]
    for n in count(1):
        row = []
        for mc in cwr(plst, n):
            p = prod(mc)
            if p not in aset:
                row.append((n-mc.count(1), p))
                aset.add(p)
        plst.append(nextprime(plst[-1]))
        yield from (p for m, p in sorted(row))
print(list(islice(agen(), 80))) # Michael S. Branicky, Apr 18 2025

a(33) and on corrected by Michael S. Branicky, Apr 19 2025

A248377 Number of compositions of 1 into parts 1/2^k with 0 <= k <= n.

Original entry on oeis.org

1, 2, 6, 56, 5272, 47350056, 3820809588459176, 24878564279781563409541239097464, 1054787931172699885204409659788147413348784265452313995416385160

Offset: 0

Bassam Abdul-Baki, Oct 05 2014

nonn

Equivalently, the number of compositions of 2^n into powers of 2.

			a(0) = 1: [1].
a(1) = 2: [1/2,1/2], [1].
a(2) = 6: [1/4,1/4,1/4,1/4], [1/2,1/4,1/4], [1/4,1/2,1/4], [1/4,1/4,1/2],  [1/2,1/2], [1].

Alois P. Heinz, Table of n, a(n) for n = 0..11
B. Abdul-Baki, Rhythmic Figures

Cf. A023359.

Row sums of A323840.

b:= proc(n) option remember; `if`(n=0, 1,
      add(b(n-2^j), j=0..ilog2(n)))
    end:
a:= n-> b(2^n):
seq(a(n), n=0..10);  # Alois P. Heinz, Oct 20 2014

$RecursionLimit = 2000; Clear[b]; b[n_] := b[n] = If[n == 0, 1, Sum[b[n - 2^j], {j, 0, Log[2, n] // Floor}]]; a[n_] := b[2^n]; Table[a[n], {n, 0, 10}] (* Jean-François Alcover, Oct 30 2014, after Alois P. Heinz *)

a(n) = A023359(2^n).

lim_{n->oo} a(n+1)/a(n)^2 = 1.704176310706592045608982.... - Bassam Abdul-Baki, Sep 03 2020

cp's OEIS Frontend

User: Bassam Abdul-Baki

A383174 Permutation of the natural numbers formed by ordering by max(gpfi,bigomega), then bigomega, then numerically, where gpfi(k) = A061395(k) and bigomega(k) = A001222(k).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Python

Extensions