login
Fortunate semiprimes: least m > 1 such that m + sp(n)# is semiprime, where sp# denotes the product of the semiprimes <= sp.
0

%I #32 Jul 07 2023 11:14:30

%S 2,2,2,5,2,3,2,7,3,19,11,3,23,5,61,29,31,3,29,31,13,19,5,7,23,47,3,53,

%T 47,19,13,7,41,53,2,43,7,103,2,61,59,71,17,59,79,43,167,71,97,7,151,

%U 37,103,83,127,103,11,53,29,7,67,83,151,107,37

%N Fortunate semiprimes: least m > 1 such that m + sp(n)# is semiprime, where sp# denotes the product of the semiprimes <= sp.

%C This is the semiprime analogous to A005235.

%C Just like the Fortunate primes (A005235) we conjecture that all terms are prime!

%C If instead of semiprimorials (A112141), we use primorials the sequence would be {2, 3, 3, 3, 5, 1, 5, 1, 1, 3, 4, 4, 11, 1, 4, 7, 4, 1, 1, 23, 1, 29, 1, 9, 32, 1, 71, 31, 4, 32, 23, 5, 125, 1, 97, 1, 11, 7, 27, 1, 29, 61, 11}; not very interesting.

%C If instead of m > 1, we start with m > 0, we identify the semiprimorials +1 which are semiprimes by index: 2, 3, 8, 9, 15, 16, 19, 21, 23, 27, 29, ....

%C A112141(66) + 197 has been completed through the 2030 curve.

%H Dario Alejandro Alpern, <a href="https://www.alpertron.com.ar/ECM.HTM"> Factorization using the Elliptic Curve Method</a>

%H Cyril Banderier, <a href="https://lipn.univ-paris13.fr/~banderier/Computations/prime_factorial.html">Fortune's conjecture (Fortunate and unfortunate primes: Nearest primes from a prime factorial)</a>

%F The difference between the n-th semiprimorial and the next semiprime greater than that semiprimorial plus 1.

%e a(3) = 3 since sp(3) = 4*6*9*10 = 2160 and the least number greater than the fourth semiprimorial which is the semiprime is 2165 = 5*433. Therefore the difference is a(3) which equals 3.

%e A112141(47) + a(47) = 24011725937636436154291480954413133199 * 68433092684820794078956407785220072996675433.

%e A112141(55) + a(55) = 795251036594717254131632161591406578993 * 650971642564884068706166933685477027845256102005635827825839.

%e A112141(63) + a(63) = 168586841653003537 * 40363128914158968243564625304355041082304983807201816858670871447070744 83558441664261096307889392423.

%t NextSemiPrime[n_, k_: 1] := Block[{c = 0, sgn = Sign[k]}, sp = n + sgn; While[c < Abs[k], While[ PrimeOmega[sp] != 2, If[sgn < 0, sp--, sp++]]; If[sgn < 0, sp--, sp++]; c++]; sp + If[sgn < 0, 1, -1]]; f[n_] := Block[{k = 1, m = Times @@ NestList[ NextSemiPrime, 2^2, n-1]}, While[ PrimeOmega[m + k] != 2, k++]; k]; Do[ Print@ f[n], {n, 50}]

%Y Cf. A005235.

%K nonn,hard

%O 1,1

%A _Sean A. Irvine_, _Jonathan Vos Post_ and _Robert G. Wilson v_, Jun 09 2013