cp's OEIS Frontend

A249151 Largest m such that m! divides the product of elements on row n of Pascal's triangle: a(n) = A055881(A001142(n)).

%I A249151 #38 Aug 24 2025 11:29:43
%S A249151 1,1,2,1,4,2,6,1,2,4,10,7,12,6,4,1,16,2,18,4,6,10,22,11,4,12,2,6,28,
%T A249151 25,30,1,10,16,6,36,36,18,12,40,40,6,42,10,23,22,46,19,6,4,16,12,52,2,
%U A249151 10,35,18,28,58,47,60,30,63,1,12,10,66,16,22,49,70,41,72,36,4,18,10,12,78,80,2
%N A249151 Largest m such that m! divides the product of elements on row n of Pascal's triangle: a(n) = A055881(A001142(n)).
%C A249151 A000225 gives the positions of ones.
%C A249151 A006093 seems to give all such k, that a(k) = k.
%H A249151 Chai Wah Wu, <a href="/A249151/b249151.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..4096 from Antti Karttunen)
%F A249151 a(n) = A055881(A001142(n)).
%e A249151               Binomial coeff.   Their product  Largest k!
%e A249151                  A007318          A001142(n)   which divides
%e A249151 Row 0                1                    1        1!
%e A249151 Row 1              1   1                  1        1!
%e A249151 Row 2            1   2   1                2        2!
%e A249151 Row 3          1   3   3   1              9        1!
%e A249151 Row 4        1   4   6   4   1           96        4! (96 = 4*24)
%e A249151 Row 5      1   5  10  10   5   1       2500        2! (2500 = 1250*2)
%e A249151 Row 6    1   6  15  20  15   6   1   162000        6! (162000 = 225*720)
%o A249151 (PARI)
%o A249151 A249151(n) = { my(uplim,padicvals,b); uplim = (n+3); padicvals = vector(uplim); for(k=0, n, b = binomial(n, k); for(i=1, uplim, padicvals[i] += valuation(b, prime(i)))); k = 1; while(k>0, for(i=1, uplim, if((padicvals[i] -= valuation(k, prime(i))) < 0, return(k-1))); k++); };
%o A249151 \\ Alternative implementation:
%o A249151 A001142(n) = prod(k=1, n, k^((k+k)-1-n));
%o A249151 A055881(n) = { my(i); i=2; while((0 == (n%i)), n = n/i; i++); return(i-1); }
%o A249151 A249151(n) = A055881(A001142(n));
%o A249151 for(n=0, 4096, write("b249151.txt", n, " ", A249151(n)));
%o A249151 (Scheme) (define (A249151 n) (A055881 (A001142 n)))
%o A249151 (Python)
%o A249151 from itertools import count
%o A249151 from collections import Counter
%o A249151 from math import comb
%o A249151 from sympy import factorint
%o A249151 def A249151(n):
%o A249151     p = sum((Counter(factorint(comb(n,i))) for i in range(n+1)),start=Counter())
%o A249151     for m in count(1):
%o A249151         f = Counter(factorint(m))
%o A249151         if not f<=p:
%o A249151             return m-1
%o A249151         p -= f # _Chai Wah Wu_, Aug 19 2025
%Y A249151 One more than A249150.
%Y A249151 Cf. A249423 (numbers k such that a(k) = k+1).
%Y A249151 Cf. A249429 (numbers k such that a(k) > k).
%Y A249151 Cf. A249433 (numbers k such that a(k) < k).
%Y A249151 Cf. A249434 (numbers k such that a(k) >= k).
%Y A249151 Cf. A249424 (numbers k such that a(k) = (k-1)/2).
%Y A249151 Cf. A249428 (and the corresponding values, i.e. numbers n such that A249151(2n+1) = n).
%Y A249151 Cf. A249425 (record positions).
%Y A249151 Cf. A249427 (record values).
%Y A249151 Cf. A001142, A006093, A000225, A007917, A055881, A187059, A249346, A249421, A249430, A249431, A249432.
%K A249151 nonn,changed
%O A249151 0,3
%A A249151 _Antti Karttunen_, Oct 25 2014