A096583 -id:A096583 - cp's OEIS frontend

A096584 Antidiagonal sums of the square array A096583, in which the n-th diagonal equals the convolution of the n-th row with the antidiagonal sums (this sequence).

1, 2, 5, 12, 24, 52, 90, 186, 306, 574, 942, 1690, 2618, 4600, 7092, 11772, 18022, 29312, 43894, 69938, 103854, 161382, 238148, 363818, 530428, 799366, 1157104, 1717906, 2470426, 3623286, 5170082, 7510056, 10647032, 15305918, 21573014

Offset: 0

Paul D. Hanna, Jun 28 2004

nonn

Partial sums form A096585, which is the main diagonal of array A096583.

Cf. A096583, A096585.

PARI

A096585 Main diagonal of the square array A096583, in which the n-th diagonal equals the convolution of the n-th row with the antidiagonal sums (A096584).

Original entry on oeis.org

1, 3, 8, 20, 44, 96, 186, 372, 678, 1252, 2194, 3884, 6502, 11102, 18194, 29966, 47988, 77300, 121194, 191132, 294986, 456368, 694516, 1058334, 1588762, 2388128, 3545232, 5263138, 7733564, 11356850, 16526932, 24036988, 34684020, 49989938

Offset: 0

Paul D. Hanna, Jun 28 2004

nonn

Forms the partial sums of A096584 (the antidiagonal sums of array A096583).

Cf. A096583, A096584.

PARI

A373887 a(n) is the length of the longest arithmetic progression of semiprimes ending in the n-th semiprime.

Original entry on oeis.org

1, 2, 2, 2, 3, 2, 3, 3, 2, 3, 3, 3, 3, 3, 2, 4, 3, 3, 4, 3, 5, 3, 3, 4, 3, 3, 4, 3, 4, 5, 4, 4, 3, 3, 5, 3, 4, 4, 3, 3, 3, 4, 4, 3, 3, 3, 3, 3, 5, 3, 3, 4, 5, 4, 4, 3, 4, 3, 4, 4, 4, 4, 3, 4, 5, 4, 4, 3, 4, 4, 4, 5, 3, 5, 6, 4, 4, 4, 4, 4, 4, 5, 4, 5, 5, 3, 3, 4, 4, 5, 5, 4, 4, 4, 4, 4, 5, 4, 5

Offset: 1

Robert Israel, Aug 10 2024

nonn

a(n) is the greatest k such that there exists d > 0 such that A001358(n) - j*d is in A001358 for j = 0 .. k-1.
The first appearance of m in this sequence is at n where A001358(n) = A096003(m).
Conjectures: a(n) >= 3 for n >= 16.
Limit_{n -> oo} a(n) = oo.
If A001358(n) is divisible by A000040(m), then a(n) >= A373888(m). In particular, the conjectures above are implied by the corresponding conjectures for A373888. - Robert Israel, Aug 19 2024

			a(5) = 3 because the 5th semiprime is A001358(5) = 14 and there is an arithmetic progression of 3 semiprimes ending in 14, namely 4, 9, 14, and no such arithmetic progression of 4 semiprimes.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A001358, A096583, A373888.

S:= select(t -> numtheory:-bigomega(t)=2, [$1..10^5]):
f:= proc(n) local s,i,m,d,j;
  m:= 1;
  s:= S[n];
  for i from n-1 to 1 by -1 do
    d:= s - S[i];
    if s - m*d < 4 then return m fi;
    for j from 2 while ListTools:-BinarySearch(S,s-j*d) <> 0 do od;
    m:= max(m, j);
  od;
m;
end proc:
map(f, [$1..100]);

cp's OEIS Frontend

A096584 Antidiagonal sums of the square array A096583, in which the n-th diagonal equals the convolution of the n-th row with the antidiagonal sums (this sequence).

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PARI

A096585 Main diagonal of the square array A096583, in which the n-th diagonal equals the convolution of the n-th row with the antidiagonal sums (A096584).

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Comments

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PARI

A373887 a(n) is the length of the longest arithmetic progression of semiprimes ending in the n-th semiprime.

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