A237040 -id:A237040 - cp's OEIS frontend

A242262 Semiprimes of the form k^3 - 1.

26, 215, 511, 1727, 2743, 7999, 13823, 54871, 157463, 238327, 511999, 728999, 1330999, 2628071, 3374999, 4410943, 4741631, 7077887, 7301383, 20123647, 21484951, 30959143, 36594367, 42144191, 63044791, 64964807, 81746503, 124999999, 187149247, 264609287, 267089983

Offset: 1

K. D. Bajpai, May 09 2014

nonn

From Jianing Song, Aug 01 2018: (Start)
k^3 - 1 is a term iff k - 1 and k^2 + k + 1 are both prime.
Is this sequence infinite? That is, are there infinitely many primes p such that p^2 + 3*p + 3 is also prime?
(End)

			a(1) = 26 = 3^3 - 1 = 26 = 2 * 13, is a semiprime.
a(2) = 215 = 6^3 - 1 = 215 = 5 * 43, is a semiprime.

K. D. Bajpai, Table of n, a(n) for n = 1..2474

Cf. A001358, A096175.

Cf. A237040 (semiprimes of the form k^3 + 1).

with(numtheory): A242262:= proc() local k; k:= x^3-1; if bigomega(k) = 2  then RETURN (k); fi; end: seq(A242262 (),x=1..1000);

Select[Table[n^3 - 1, {n, 100}], PrimeOmega[#] == 2 &]
Select[Range[700]^3-1,PrimeOmega[#]==2&] (* Harvey P. Dale, Jan 25 2019 *)

a(n) = A096175(n-1)^3 - 1 for n > 1. - Jianing Song, Aug 01 2018

First Mathematica program corrected by Harvey P. Dale, Jan 25 2019

A237117 Remainder mod p of the smallest semiprime of the form k^p+1, where p = prime(n); or -1 if no such semiprime exists.

Original entry on oeis.org

0, 0, 3, 3, 3, 3, 3, 3, 3, 24, 3, 17, 26, 3, 7, 11, 7, 3, 11, 47, 19, 3, 5, 17, 71, 3, 97, 7, 13, 32, 3, 97, 67, 31, 17, 48, 23, 53, 3, 17, 157, 108, 3, 13, 53, 3, 67, 47, 23, 97, 88, 127, 106, 17, 37, 97, 145, 89, 73, 53, 173, 11, 17, 106, 3, 17, 47, 323, 3, 112, 23, 314, 37, 29, 331, 174, 266, 194, 226, 397, 29, 16, 176, 45, 44, 152, 373, 349, 101, 143, 53, 386, 133, 29, 345, 1

Offset: 1

Jonathan Sondow, Feb 06 2014

nonn

It appears that a(n) > 0 for all n > 2. See the comments in A237114.

			Prime(2)=3 and the smallest semiprime of the form k^3+1 is 2^3+1 = 9 = 3*3, so a(2) = 9 mod 3 = 0.
Prime(3)=5 and the smallest semiprime of the form k^5+1 is 2^5+1 = 33 = 3*11, so a(3) = 33 mod 5 = 3.

Eric Weisstein's World of Mathematics, Semiprime
Wikipedia, Semiprime

Cf. A001358, A006881, A103795, A123627, A123628, A237040, A237114, A237115, A237116.

L = {0}; Do[p = Prime[k]; n = 1; q = Prime[n] - 1; cp = (q^p + 1)/(q + 1); While[! PrimeQ[cp], n = n + 1; q = Prime[n] - 1; cp = (q^p + 1)/(q + 1)]; L = Append[L, Mod[q^p + 1, p]], {k, 2, 87}]; L

a(n) = A237114(n) mod prime(n) = A237115(n) mod prime(n), if A237114(n)>0.

A259189 Semiprimes of the form n^3 + 2.

Original entry on oeis.org

10, 218, 514, 731, 1333, 2199, 2746, 3377, 4915, 5834, 6861, 8002, 9263, 12169, 15627, 29793, 35939, 42877, 54874, 59321, 68923, 117651, 125002, 132653, 148879, 185195, 205381, 314434, 405226, 421877, 474554, 531443, 592706, 658505, 704971

Offset: 1

Morris Neene, Jun 20 2015

nonn

Intersection of A001358 and A084380. - Michel Marcus, Jun 20 2015
Since there are no squares of the form n^3 + 2, all semiprimes in this sequence are products of distinct primes.
No term in A040034 divides any term in this sequence.

Jon E. Schoenfield, Table of n, a(n) for n = 1..10000

Cf. A001358 (semiprimes), A084380 (n^3+2), A144953 (primes of same form).

Cf. A237040 (similar sequence with n^3+1).

IsSP:=func;[r:n in [1..1000]|IsSP(r) where r is 2+n^3];

Select[Range[100]^3 + 2, PrimeOmega[#] == 2 &] (* Alonso del Arte, Jun 20 2015 *)

is(n)=bigomega(n^3 + 2)==2 \\ Anders Hellström, Sep 07 2015

use ntheory ":all"; my @sp = grep { scalar(factor($))==2 } map { $**3+2 } 1..100; say "@sp"; # Dana Jacobsen, Sep 07 2015

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A242262 Semiprimes of the form k^3 - 1.

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A237117 Remainder mod p of the smallest semiprime of the form k^p+1, where p = prime(n); or -1 if no such semiprime exists.

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A259189 Semiprimes of the form n^3 + 2.

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