A226525 Fortunate semiprimes: least m > 1 such that m + sp(n)# is semiprime, where sp# denotes the product of the semiprimes <= sp.

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, 47, 19, 13, 7, 41, 53, 2, 43, 7, 103, 2, 61, 59, 71, 17, 59, 79, 43, 167, 71, 97, 7, 151, 37, 103, 83, 127, 103, 11, 53, 29, 7, 67, 83, 151, 107, 37

Offset

1,1

Comments

This is the semiprime analogous to A005235.

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

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.

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, ....

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

Links

Table of n, a(n) for n=1..65.

Dario Alejandro Alpern, Factorization using the Elliptic Curve Method

Cyril Banderier, Fortune's conjecture (Fortunate and unfortunate primes: Nearest primes from a prime factorial)

Formula

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

Example

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.
A112141(47) + a(47) = 24011725937636436154291480954413133199 * 68433092684820794078956407785220072996675433.
A112141(55) + a(55) = 795251036594717254131632161591406578993 * 650971642564884068706166933685477027845256102005635827825839.
A112141(63) + a(63) = 168586841653003537 * 40363128914158968243564625304355041082304983807201816858670871447070744 83558441664261096307889392423.

Mathematica

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}]

Crossrefs

Cf. A005235.

Sequence in context: A350403 A113516 A376336 * A120642 A216624 A183413

Adjacent sequences: A226522 A226523 A226524 * A226526 A226527 A226528

Keyword

nonn,hard

Author

Sean A. Irvine, Jonathan Vos Post and Robert G. Wilson v, Jun 09 2013

Status

approved