A071621 -id:A071621 - cp's OEIS frontend

A071318 Lesser of 2 consecutive numbers which are cubefree and not squarefree, i.e., numbers k such that both k and k+1 are in A067259.

44, 49, 75, 98, 99, 116, 147, 171, 244, 260, 275, 315, 332, 363, 387, 475, 476, 507, 524, 531, 548, 549, 603, 604, 636, 692, 724, 725, 747, 764, 774, 819, 844, 845, 846, 867, 908, 924, 931, 963, 980, 1035, 1075, 1083, 1179, 1196, 1251, 1274, 1275, 1324

Offset: 1

Labos Elemer, May 29 2002

nonn

The numbers of terms not exceeding 10^k, for k = 1, 2, ..., are 0, 5, 41, 407, 4125, 41215, 412331, 4123625, 41236308, ... . Apparently, the asymptotic density of this sequence exists and equals 0.041236... . - Amiram Eldar, Jan 18 2023

The asymptotic density of this sequence is Product_{p prime} (1 - 2/p^3) - 2 * Product_{p prime} (1 - 1/p^2 - 1/p^3) + Product_{p prime} (1 - 2/p^2) = 0.041236147082334172926... . - Amiram Eldar, Jan 05 2024

			75 is a term since 75 = 3*5^2 and 76 = 2^2*19.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)

Subsequence of A067259, A340152.
Cf. A007674, A063528, A071619, A071620, A071621, A071622, A071623.
Cf. A008966, A051903, A212793.

a071318 n = a071318_list !! (n-1)
a071318_list = [x | x <- [1..],  a212793 x == 1, a008966 x == 0,
                    let y = x+1, a212793 y == 1, a008966 y == 0]
-- Reinhard Zumkeller, May 27 2012

With[{s = Select[Range[1350], And[MemberQ[#, 2], FreeQ[#, k_ /; k > 2]] &@ FactorInteger[#][[All, -1]] &]}, Function[t, Part[s, #] &@ Position[t, 1][[All, 1]]]@ Differences@ s] (* Michael De Vlieger, Jul 30 2017 *)

isok(n) = (n>1) && (vecmax(factor(n)[, 2])==2) && (vecmax(factor(n+1)[, 2])==2); \\ Michel Marcus, Aug 02 2017

A051903(k) = A051903(k+1) = 2 when k is a term.

A224534 Prime numbers that are the sum of three distinct prime numbers.

Original entry on oeis.org

19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307

Offset: 1

T. D. Noe, Apr 15 2013

nonn

Similar to Goldbach's weak conjecture.
Primes in A124867, and by the comment in A124867 also the set of all primes >=19. - R. J. Mathar, Apr 19 2013
"Goldbach's original conjecture (sometimes called the 'ternary' Goldbach conjecture), written in a June 7, 1742 letter to Euler, states 'at least it seems that every number that is greater than 2 is the sum of three primes' (Goldbach 1742; Dickson 2005, p. 421). Note that here Goldbach considered the number 1 to be a prime, a convention that is no longer followed." [Weisstein] - Jonathan Vos Post, May 15 2013

			19 = 3 + 5 + 11.

H. A. Helfgott and David J. Platt, Numerical Verification of the Ternary Goldbach Conjecture up to 8.875e30, arXiv:1305.3062 [math.NT], 2013-2014.
H. A. Helfgott and David J. Platt, Numerical verification of the Ternary Goldbach Conjecture up to 8.875*10^30, Exp. Math. 22 (4) (2013) 406-409.
Eric W. Weisstein, Goldbach conjecture
Wikipedia, Goldbach's conjecture
Wikipedia, Goldbach's weak conjecture

Cf. A002372, A002375, A024684 (number of sums), A224535, A166063, A166061, A071621.

Union[Select[Total /@ Subsets[Prime[Range[2, 30]], {3}], PrimeQ]]

cp's OEIS Frontend

A071318 Lesser of 2 consecutive numbers which are cubefree and not squarefree, i.e., numbers k such that both k and k+1 are in A067259.

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