A099407 -id:A099407 - cp's OEIS frontend

A083553 Product of prime(n+1)-1 and prime(n)-1.

2, 8, 24, 60, 120, 192, 288, 396, 616, 840, 1080, 1440, 1680, 1932, 2392, 3016, 3480, 3960, 4620, 5040, 5616, 6396, 7216, 8448, 9600, 10200, 10812, 11448, 12096, 14112, 16380, 17680, 18768, 20424, 22200, 23400, 25272, 26892, 28552, 30616, 32040

Offset: 1

Labos Elemer, May 22 2003

nonn

The conductor of x*prime(n) + y*prime(n+1); that is, for all k >= a(n), there exist nonnegative integers x and y such that k = x*prime(n) + y*prime(n+1). - T. D. Noe, Sep 22 2004

			n=25: a(25) = (97-1)*(101-1) = 9600.

David Bressoud and Stan Wagon, A Course in Computational Number Theory, Key College Pub., 2000, p. 46.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000040, A006093, A058263, A083538-A083555, A099407 (terms halved), A172042 [= A000010(a(n))], A256617.
One more than A037165.
Column 3 of A379010.

f[x_] := Prime[x]-1; Table[f[w+1]*f[w], {w, 1, 128}]

A083553(n) = ((prime(1+n)-1)*(prime(n)-1)); \\ Antti Karttunen, Dec 14 2024

a(n) = A006093(n+1)*A006093(n) = (prime(n+1)-1)*(prime(n)-1).
a(n) = A037165(n) + 1.
a(n) = 2*A099407(n). - Antti Karttunen, Dec 14 2024

A201498 a(n) = (prime(n) - 1)*(prime(n+1) - 1)/2 + 3.

Original entry on oeis.org

4, 7, 15, 33, 63, 99, 147, 201, 311, 423, 543, 723, 843, 969, 1199, 1511, 1743, 1983, 2313, 2523, 2811, 3201, 3611, 4227, 4803, 5103, 5409, 5727, 6051, 7059, 8193, 8843, 9387, 10215, 11103, 11703, 12639, 13449, 14279, 15311, 16023, 17103, 18243, 18819, 19407

Offset: 1

Zak Seidov, Dec 02 2011

nonn

Consider strictly increasing sequence with the rule:
a(n) is the smallest pairwise sum s of all previous terms such that s > a(n-1).
We start with some pair of coprime positive integers b < c, a(1)=b, a(2)=c; from now on, to find a(n) we use the above-mentioned rule. We observe that, for any seeds b,c, after some term, a(n) = a(n-1) + 1.
E.g., for b=7, c=12, we get 7, 12,1 9, 26, 31, 33, 38, 40, 43, 45, 47, 50, 52, 54, 55, 57, 59, 61, 62, 64, 66, 67, 68, 69, 71, 73, 74, 75, 76, 78, 79, 80, 81, 82, 83, 85, 86, 87, 88, 89, 90, 91, 92, ...
We stop at the term a(L=36) = 85 after which a(n) = a(n-1) + 1.
In the general case of arbitrary coprime b < c, the length of the sequence is L = 3 + (b-1)(c-1)/2, and a(L) = b*c + 1.
In A201498, we present the dependence of L(n) for the particular case b=prime(n) and c=prime(n+1).

#/2+3&/@(Times@@@Partition[Prime[Range[50]]-1,2,1])  (* Harvey P. Dale, Jun 01 2015 *)

p=2;forprime(q=3,1e3,print1((p-1)*(q-1)/2+3", ");p=q) \\ Charles R Greathouse IV, Dec 05 2011

a(n) = A099407(n) + 3.

cp's OEIS Frontend

A083553 Product of prime(n+1)-1 and prime(n)-1.

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