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URL: https://oeis.org/A392806

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A392806
Irregular triangle read by rows: exponents of the primes in the prime factorization of A392805(n) (primes in increasing order).
7
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).
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
KEYWORD
nonn,tabf
AUTHOR
A.H.M. Smeets, Jan 23 2026
STATUS
approved