A392806 Irregular triangle read by rows: exponents of the primes in the prime factorization of A392805(n) (primes in increasing order).

2, 0, 2, 6, 4, 1, 0, 2, 9, 4, 4, 0, 8, 6, 2, 14, 0, 8, 4, 9, 8, 0, 6, 25, 16, 4, 8, 0, 1, 8, 10, 2, 20, 11, 12, 12, 4, 13, 21, 16, 0, 6, 2, 35, 2, 0, 4, 8, 4, 0, 14, 6, 8, 10, 6, 30, 26, 12, 12, 12, 8, 25, 0, 18, 16, 14, 10, 2, 57, 16, 24, 20, 16, 12, 4, 11, 32

Offset

3,1

Comments

All prime factors p_i of A391956(n) satisfy p_i < n, so the row length is A000720(n-1).

Links

A.H.M. Smeets, Table of n, a(n) for n = 3..10327, row indices 2..300

A.H.M. Smeets, First 200 rows, printed row by row

Formula

In the following, p_m is the m-th prime, A000040(m).

T((p_m)*k,m) = 0.

T(k*p_m,m) = 0, 0 <= k <= p_m.

T((1+p_m)*p_m,m) = 1.

T(1+p_m,m) = 2.

T((2+p_m)*p_m,m) = 2.

Row(n) = A067255(A392805(n)).

Example

The irregular triangle T(n,m) begins:
   \   m   1   2   3   4   5   6   7   8   9
   n\p_m   2   3   5   7  11  13  17  19  23 = A000040(m)
   3       2
   4       0   2
   5       6   4
   6       1   0   2
   7       9   4   4
   8       0   8   6   2
   9      14   0   8   4
  10       9   8   0   6
  11      25  16   4   8
  12       0   1   8  10   2
  13      20  11  12  12   4
  14      13  21  16   0   6   2
  15      35   2   0   4   8   4
  16       0  14   6   8  10   6
  17      30  26  12  12  12   8

Prog

(PARI) default(realprecision, 1000);
L(n, z) = subst(pollaguerre(n), 'x, z);
wl(n) = my(r=polrootsreal(pollaguerre(n))); vector(n, i, r[i]/((n+1)^2*L(n+1, r[i])^2));
np(n) = my(vw=wl(n)); prod(i=1, n, 'x-vw[i])/prod(i=1, n, vw[i]);
v(n) = my(v=apply(bestappr, Vec(np(n)))); v*lcm(apply(denominator, v));
row(n) = my(m=(-1)^n*v(n)[n+1], f=factor(m)); vector(primepi(vecmax(f[, 1])), i, valuation(m, prime(i))); \\ Michel Marcus, Feb 05 2026
(Python)
from sympy import prime
def Sum(n, p):
    s = 0
    while n > 0: s, n = s+n, n//p
    return s
def PrimePower(p, e):
    if e == 0: return 0
    else: return p*PrimePower(p, e-1) + 2*p**e - 1
def A392806(n, m):
    p, d = prime(m), []
    while n > 0: d, n = [n%p]+d, n//p
    L = len(d)-1
    a, k, ns = 0, 0, 0
    while k < L:
        ns, k = ns*p+d[k], k+1
        if d[k] > 0: a = a + d[k]*max(0, (-PrimePower(p, L-k)+2*p**(L-k)*Sum(ns, p)))
    return a # A.H.M. Smeets, Feb 05 2026

Crossrefs

Cf. A000040, A000720, A067255, A392805.

Sequence in context: A388953 A325199 A388351 * A185197 A327116 A157491

Adjacent sequences: A392803 A392804 A392805 * A392807 A392808 A392809

Keyword

nonn,tabf

Author

A.H.M. Smeets, Jan 23 2026

Status

approved