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A090623 Triangle of T(n,k) = [n/k] + [n/k^2] + [n/k^3] + [n/k^4] + ... for n, k > 1.

%I A090623 #30 Sep 02 2025 09:15:03
%S A090623 1,1,1,3,1,1,3,1,1,1,4,2,1,1,1,4,2,1,1,1,1,7,2,2,1,1,1,1,7,4,2,1,1,1,
%T A090623 1,1,8,4,2,2,1,1,1,1,1,8,4,2,2,1,1,1,1,1,1,10,5,3,2,2,1,1,1,1,1,1,10,
%U A090623 5,3,2,2,1,1,1,1,1,1,1,11,5,3,2,2,2,1,1,1,1,1,1,1,11,6,3,3,2,2,1,1,1
%N A090623 Triangle of T(n,k) = [n/k] + [n/k^2] + [n/k^3] + [n/k^4] + ... for n, k > 1.
%H A090623 Zhuorui He, <a href="/A090623/b090623.txt">Table of n, a(n) for n = 2..11326</a>
%H A090623 Wenguang Zhai, <a href="http://dx.doi.org/10.1016/j.jnt.2009.02.016">On the prime power factorization of n!</a>, Journal of Number Theory, Volume 129, Issue 8, August 2009, Pages 1820-1836.
%F A090623 For p prime, T(n, p) = A090622(n, p) is the number of times that p is a factor of n!.
%F A090623 T(n,k) = (n - A240236(n, k))/(k - 1). - _Zhuorui He_, Aug 25 2025
%e A090623 Rows start:
%e A090623   1;
%e A090623   1,1;
%e A090623   3,1,1;
%e A090623   3,1,1,1;
%e A090623   4,2,1,1,1;
%e A090623   4,2,1,1,1,1;
%e A090623   7,2,2,1,1,1,1;
%e A090623   7,4,2,1,1,1,1,1;
%e A090623   8,4,2,2,1,1,1,1,1;
%e A090623   ...
%t A090623 A090623[n_, k_] := Quotient[n - DigitSum[n, k], k - 1];
%t A090623 Table[A090623[n, k], {n, 2, 15}, {k, 2, n}] (* _Paolo Xausa_, Sep 02 2025 *)
%o A090623 (PARI) T(n,k) = {my(s = 0, j = 1); while(p=n\k^j, s += p; j++); s;} \\ _Michel Marcus_, Feb 02 2016
%o A090623 (PARI) T(n,k) = (n - sumdigits(n,k))/(k-1) \\ _Zhuorui He_, Aug 25 2025
%Y A090623 Columns include A011371, A054861, A054893, A027868, A054895, A054896, A054897, A054898, A054899, A064458, A064459, A090620, A054900.
%Y A090623 Row sums: A078632.
%Y A090623 Cf. A240236.
%K A090623 nonn,tabl,changed
%O A090623 2,4
%A A090623 _Henry Bottomley_, Dec 06 2003
%E A090623 a(41) onward corrected by _Zhuorui He_, Aug 25 2025