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A370129
Triangle read by rows: T(n,k) = A003415(A002110(n)+A002110(k)), 0 <= k <= n; arithmetic derivatives of the sums of two primorial numbers.
6
1, 1, 4, 1, 12, 16, 1, 80, 60, 92, 1, 216, 540, 608, 704, 1, 3740, 3100, 4548, 6324, 8164, 568, 60080, 40060, 56292, 116208, 61768, 110752, 33975, 1021040, 1041768, 794468, 2415104, 1091004, 1357128, 1942844, 28300, 9789116, 29099520, 19722884, 18576860, 35347200, 35779644, 26575580, 37935056, 704080, 335024060
OFFSET
0,3
COMMENTS
Apart from those positions (A014545) at the left edge where a(n) = 1, a(n) <= A087112(1+n) only at n=2, 4 and 5, i.e., never after the third row.
FORMULA
a(n) = A003415(A370121(n)).
For n, k >= 1, T(n,k) = A002110(k)*A370136(n,k) + A024451(k)*A370135(n,k).
EXAMPLE
Triangle begins as:
1;
1, 4;
1, 12, 16;
1, 80, 60, 92;
1, 216, 540, 608, 704;
1, 3740, 3100, 4548, 6324, 8164;
568, 60080, 40060, 56292, 116208, 61768, 110752;
33975, 1021040, 1041768, 794468, 2415104, 1091004, 1357128, 1942844;
28300, 9789116, 29099520, 19722884, 18576860, 35347200, 35779644, 26575580, 37935056;
PROG
(PARI)
A002110(n) = prod(i=1, n, prime(i));
A370121(n) = { my(c = (sqrtint(8*n + 1) - 1) \ 2); (A002110(c) + A002110(n - binomial(c + 1, 2))); };
A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
CROSSREFS
Cf. A014545 (positions of 1's at the left edge), A087112.
Cf. also A024451 (arithmetic derivatives of primorials).
Sequence in context: A338864 A078219 A373547 * A187541 A117413 A322970
KEYWORD
nonn,tabl
AUTHOR
Antti Karttunen, Feb 29 2024
STATUS
approved