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A353068
Irregular triangle read by rows: T(n,k) = n - multiplicity of prime(k) as a divisor of n!.
1
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, 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, 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
OFFSET
2,2
COMMENTS
This is the "s" in the American Mathematical Monthly problem.
LINKS
Michel Marcus, Table of n, a(n) for n = 2..10388 (Rows n=2..300).
R. D. Carmichael, Diophantine Analysis: 141, The American Mathematical Monthly, Vol. 14, No. 5 (May, 1907), p. 107.
FORMULA
T(n,k) = n - A115627(n, k).
EXAMPLE
First few rows are:
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, 7, 10, 11, 11;
...
MATHEMATICA
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 *)
PROG
(PARI) row(n) = vector(primepi(n), k, n-valuation(n!, prime(k)));
CROSSREFS
Cf. A115627, A000120 (column 1), A089792 (column 2).
Sequence in context: A375300 A291712 A074945 * A367292 A323479 A276427
KEYWORD
nonn,tabf
AUTHOR
Michel Marcus, Apr 21 2022
STATUS
approved