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A353068 Irregular triangle read by rows: T(n,k) = n - multiplicity of prime(k) as a divisor of n!.

%I A353068 #15 May 03 2022 14:35:40
%S A353068 1,2,2,1,3,2,4,4,2,4,5,3,5,6,6,1,6,7,7,2,5,8,8,2,6,8,9,3,7,9,10,10,2,
%T A353068 7,10,11,11,3,8,11,12,12,12,3,9,12,12,13,13,4,9,12,13,14,14,1,10,13,
%U A353068 14,15,15,2,11,14,15,16,16,16,2,10,15,16,17,17,17,3,11,16,17,18,18,18,18
%N A353068 Irregular triangle read by rows: T(n,k) = n - multiplicity of prime(k) as a divisor of n!.
%C A353068 This is the "s" in the American Mathematical Monthly problem.
%H A353068 Michel Marcus, <a href="/A353068/b353068.txt">Table of n, a(n) for n = 2..10388</a> (Rows n=2..300).
%H A353068 R. D. Carmichael, <a href="https://doi.org/10.2307/2968497">Diophantine Analysis: 141</a>, The American Mathematical Monthly, Vol. 14, No. 5 (May, 1907), p. 107.
%F A353068 T(n,k) = n - A115627(n, k).
%e A353068 First few rows are:
%e A353068   1;
%e A353068   2, 2;
%e A353068   1, 3;
%e A353068   2, 4, 4;
%e A353068   2, 4, 5;
%e A353068   3, 5, 6, 6;
%e A353068   1, 6, 7, 7;
%e A353068   2, 5, 8, 8;
%e A353068   2, 6, 8, 9;
%e A353068   3, 7, 9, 10, 10;
%e A353068   2, 7, 10, 11, 11;
%e A353068   ...
%t A353068 T[n_, k_] := n - IntegerExponent[n!, Prime[k]]; Table[T[n, k], {n, 2, 19}, {k, 1, PrimePi[n]}] // Flatten (* _Amiram Eldar_, Apr 21 2022 *)
%o A353068 (PARI) row(n) = vector(primepi(n), k, n-valuation(n!, prime(k)));
%Y A353068 Cf. A115627, A000120 (column 1), A089792 (column 2).
%K A353068 nonn,tabf
%O A353068 2,2
%A A353068 _Michel Marcus_, Apr 21 2022