A096175 -id:A096175 - cp's OEIS frontend

Showing 1-8 of 8 results.

A081257 a(n) is the greatest prime factor of (n^3 - 1).

7, 13, 7, 31, 43, 19, 73, 13, 37, 19, 157, 61, 211, 241, 13, 307, 17, 127, 421, 463, 13, 79, 601, 31, 37, 757, 271, 67, 29, 331, 151, 1123, 397, 97, 43, 67, 1483, 223, 547, 1723, 139, 631, 283, 109, 103, 61, 181, 43, 2551, 379, 919, 409, 2971, 79, 103, 3307, 163

Offset: 2

Jan Fricke, Mar 14 2003

The record values here (as well as those for A081256) appear to match the terms of A002383 for n > 1. - Bill McEachen, Jun 19 2023

			a(7)=19 because 7^3 - 1 = 342 = 2*3*3*19.

Jon E. Schoenfield, Table of n, a(n) for n = 2..10000 (first 999 terms from R. J. Mathar).

Cf. A081256, A081258, A002383.

Cf. A096175 (n^3-1 is an odd semiprime), A096176 ((n^3-1)/(n-1) is prime).

A081257 := proc(n)
        A006530( n^3-1) ;
end proc: # R. J. Mathar, Jul 18 2015

FactorInteger[#][[-1,1]]&/@(Range[2,60]^3-1) (* Harvey P. Dale, Oct 09 2017 *)

a(n)=my(f=factor(n^3-1)); f[#f~,1] \\ Charles R Greathouse IV, Mar 08 2017

from sympy import primefactors
def A081257(n): return max(primefactors(n-1)+primefactors(n*(n+1)+1)) # Chai Wah Wu, Oct 15 2022

a(n) = A006530(A068601(n)). - Michel Marcus, Jun 19 2023

More terms from Hugo Pfoertner, Jun 21 2004

A268043 Numbers k such that k^3 - 1 and k^3 + 1 are both semiprimes.

Original entry on oeis.org

6, 1092, 1932, 2730, 4158, 6552, 11172, 25998, 30492, 55440, 76650, 79632, 85092, 102102, 150990, 152082, 152418, 166782, 211218, 235662, 236208, 248640, 264600, 298410, 300300, 301182, 317772, 380310, 387198, 441798, 476028, 488418

Offset: 1

Vincenzo Librandi, Jan 25 2016

Obviously, k+1 and k-1 are always prime numbers.

If k is a term then m = (k - 1) * (k^2 + k + 1) is a term of A169635, i.e., A001157(m) = A001157(m+2) (De Koninck, 2002). - Amiram Eldar, Apr 19 2024

			a(1) = 6 because 6^3-1 = 215 = 5*43 and 6^3+1 = 217 = 7*31, therefore 215, 217 are both semiprimes.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Jean-Marie De Koninck, On the solutions of sigma_2(n) = sigma_2(n + l), Ann. Univ. Sci. Budapest Sect. Comput. 21 (2002), 127-133.

Subsequence of A014574 and A136242.
Cf. A002384, A055494, A088707, A096173, A096175, A109373.
Cf. A001157, A169635.

IsSemiprime:=func< n | &+[k[2]: k in Factorization(n)] eq 2 >; [ n: n in [2..300000] | IsSemiprime(n^3+1) and IsSemiprime(n^3-1) ];

Select[Range[500000], PrimeOmega[#^3 - 1] == PrimeOmega[#^3 + 1] == 2 &]
Select[Range[10^6], And @@ PrimeQ[{# - 1, # + 1, #^2 - # + 1, #^2 + # + 1}] &] (* Amiram Eldar, Apr 19 2024 *)

isok(n) = (bigomega(n^3-1) == 2) && (bigomega(n^3+1) == 2); \\ Michel Marcus, Jan 26 2016

is(n) = isprime(n - 1) && isprime(n + 1) && isprime(n^2 - n + 1) && isprime(n^2 + n + 1); \\ Amiram Eldar, Apr 19 2024

A096176 Numbers k such that (k^3-1)/(k-1) is prime.

Original entry on oeis.org

2, 3, 5, 6, 8, 12, 14, 15, 17, 20, 21, 24, 27, 33, 38, 41, 50, 54, 57, 59, 62, 66, 69, 71, 75, 77, 78, 80, 89, 90, 99, 101, 105, 110, 111, 117, 119, 131, 138, 141, 143, 147, 150, 153, 155, 161, 162, 164, 167, 168, 173, 176, 188, 189, 192, 194, 203, 206, 209, 215, 218

Offset: 1

Hugo Pfoertner, Jun 22 2004

nonn

Numbers k > 1 such that k^2 + k + 1 is a prime. - Vincenzo Librandi, Nov 16 2010

Therefore essentially the same as A002384. - Georg Fischer, Oct 06 2018

			a(5) = 8 because (8^3-1)/(8-1) = 511/7 = 73 is prime.

Hugo Pfoertner, Table of n, a(n) for n = 1..10000

Cf. A096174 (n^3+1)/(n+1) is prime, A081257 largest prime factor of n^3-1, A096175 n^3-1 is an odd semiprime.
Cf. A028491, A004061. - Daniel McCandless (dkmccandless(AT)gmail.com), Aug 31 2009
Cf. A002384.

[n: n in [2..220] |IsPrime((n^3-1) div (n -1))]; // Vincenzo Librandi, Oct 07 2018

Select[Range[2,550],PrimeQ[(#^3-1)/(#-1)]&] (* Harvey P. Dale, Sep 10 2009 *)

is(n)=isprime(n^2+n+1) \\ Charles R Greathouse IV, Jun 05 2017

3 and 5 added by Daniel McCandless (dkmccandless(AT)gmail.com), Aug 31 2009

Corrected terms, including many previously omitted terms, from Harvey P. Dale, Sep 10 2009

A129295 Numbers m such that m^3 - 1 has no divisors d with 1 < d < m - 1.

Original entry on oeis.org

3, 4, 6, 8, 12, 14, 20, 24, 38, 54, 62, 80, 90, 110, 138, 150, 164, 168, 192, 194, 272, 278, 314, 332, 348, 398, 402, 434, 500, 572, 642, 644, 720, 728, 762, 798, 812, 860, 864, 878, 920, 992, 1020, 1022, 1070, 1092, 1098, 1118, 1130, 1182, 1202, 1230, 1260, 1308

Offset: 1

Reinhard Zumkeller, Apr 09 2007

nonn

Numbers m such that A129294(m) = #{1,m-1} = 2.

Essentially the same as A096175. Note that m^3 - 1 = (m - 1)*(m^2 + m + 1), so m - 1 must be prime. For m > 4, the smallest divisor > 1 of m^2 + m + 1 is no larger than sqrt(m^2 + m + 1) < m + 1 unless m^2 + m + 1 is also prime. Also note that gcd(m, m^2 + m 1 ) = gcd(m - 1, m^2 + m + 1) = 1, so m^2 + m + 1 must also be prime, making m^3 - 1 a semiprime. - Jianing Song, Aug 01 2018

			{1,11,157,1727} is the set of divisors of 12^3 - 1, therefore 12 is a term, since A129294(12) = #{1,11} = 2.

Cf. A096175, A129293, A129294.

a(n) = A096175(n-2) for n > 2. - Jianing Song, Aug 01 2018

A242262 Semiprimes of the form k^3 - 1.

Original entry on oeis.org

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

A261435 Numbers k such that k^5-1 is a semiprime.

Original entry on oeis.org

12, 24, 30, 44, 62, 68, 74, 110, 164, 198, 308, 492, 572, 594, 662, 728, 770, 824, 854, 860, 864, 942, 954, 968, 1152, 1154, 1284, 1382, 1452, 1668, 1694, 1748, 1760, 1788, 1914, 2090, 2252, 2274, 2340, 2382, 2438, 2448, 2648, 2658, 2664, 2690, 2714, 2790

Offset: 1

Vincenzo Librandi, Aug 20 2015

Numbers k such that k-1 and k^4+k^3+k^2+k+1 are both prime.

			a(1) = 12 because 12^5-1 = 248831 = 11*22621.

Cf. numbers k such that k^p-1 is a semiprime, where p is prime: A096175(p=3), this sequence (p=5), A261436 (p=7), A261460 (p=11).

Cf. A104238.

IsSemiprime:=func; [n: n in [2..5000] | IsSemiprime(s) where s is n^5- 1];

Mathematica

Select[Range[5000], PrimeOmega[#^5 - 1] == 2 &]

PARI

isok(n)=bigomega(n^5-1)==2 \\ Anders Hellström, Aug 20 2015

A268186 Numbers n such that n^2 + 2, n^2 - 2, n + 2 and n - 2 are all semiprimes.

Original entry on oeis.org

12, 53, 84, 204, 207, 251, 379, 413, 456, 471, 483, 631, 687, 705, 765, 783, 1079, 1135, 1140, 1167, 1269, 1335, 1347, 1395, 1475, 1515, 1587, 1641, 1709, 1767, 1851, 1855, 1943, 1959, 2049, 2157, 2319, 2325, 2575, 2843, 2865, 3099, 3153, 3225, 3267, 3601, 3779

Offset: 1

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Author

K. D. Bajpai, Jan 28 2016

Keywords

nonn
less

Examples

12 appears in the sequence because: 12^2 + 2 = 146 = 2*73 12^2 - 2 = 142 = 2*71 12 + 2 = 14 = 2*7 12 - 2 = 10 = 2*5 are all semiprimes.

Links

K. D. Bajpai, Table of n, a(n) for n = 1..3379

Crossrefs

Subsequence of A105571.

Cf. A001358, A086005, A096173, A096175, A105571, A124936, A237037, A268043.

Programs

Magma

IsSemiprime:=func;[ n : n in [2..10000] | IsSemiprime(n^2 + 2) and IsSemiprime(n^2 - 2) and IsSemiprime(n + 2) and IsSemiprime(n - 2)];

Maple

with(numtheory): select(n -> (bigomega(n^2 + 2)=2 and bigomega(n^2 - 2)=2 and bigomega(n + 2)=2 and bigomega(n - 2)=2), [seq(n, n=1..10000)]);

Mathematica

Select[Range[10000], PrimeOmega[#^2 + 2] == PrimeOmega[#^2 - 2] == PrimeOmega[# + 2] == PrimeOmega[# - 2] == 2 &]

PARI

for(n = 1, 10000,if(bigomega(n^2 + 2) == 2 && bigomega(n^2 - 2) == 2 && bigomega(n + 2) == 2 && bigomega(n - 2) == 2, print1(n, ", ")))

A317138 Numbers k such that (2k)^3 - 1 is a semiprime.

Original entry on oeis.org

3, 4, 6, 7, 10, 12, 19, 27, 31, 40, 45, 55, 69, 75, 82, 84, 96, 97, 136, 139, 157, 166, 174, 199, 201, 217, 250, 286, 321, 322, 360, 364, 381, 399, 406, 430, 432, 439, 460, 496, 510, 511, 535, 546, 549, 559, 565, 591, 601, 615, 630, 654, 717, 720, 724, 727, 742

Offset: 1

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Jianing Song, Aug 01 2018

Keywords

nonn

Comments

Numbers k such that 2k - 1 and 4k^2 + 2k + 1 are both prime.

Examples

From _K. D. Bajpai_, Nov 16 2019: (Start) a(3) = 6 is a term because (2*6)^3 - 1 = 1727 = 11*157, which is a semiprime. a(4) = 7 is a term because (2*7)^3 - 1 = 2743 = 13*211, which is a semiprime. 9 is not in the sequence because (2*9)^3 - 1 = 5831 = 7*7*7*17, which is not semiprime. (End)

Links

K. D. Bajpai, Table of n, a(n) for n = 1..9443

Crossrefs

Cf. A096175, A242262.

Cf. A237037 (numbers k such that (2k)^3 + 1 is a semiprime).

Programs

Magma

IsSemiprime:=func; [n: n in [2..800] | IsSemiprime(s) where s is (2*n)^3-1]; // Vincenzo Librandi, Aug 04 2018

Maple

issp:= n-> not isprime(n) and numtheory[bigomega](n)=2: select( n-> issp((2*n)^3-1), [seq(n, n=1..200)]); # K. D. Bajpai, Nov 16 2019

Mathematica

Select[Range@ 750, PrimeOmega[(2 #)^3 - 1] == 2 &] (* Michael De Vlieger, Aug 02 2018 *)

PARI

for(k=1,500,if(bigomega((2*k)^3-1)==2,print1(k,", ")))

Formula

a(n) = A096175(n)/2 = (1/2)*(A242262(n) + 1)^(1/3).

Showing 1-8 of 8 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.
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cp's OEIS Frontend

A081257 a(n) is the greatest prime factor of (n^3 - 1).

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A268043 Numbers k such that k^3 - 1 and k^3 + 1 are both semiprimes.

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A096176 Numbers k such that (k^3-1)/(k-1) is prime.

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A129295 Numbers m such that m^3 - 1 has no divisors d with 1 < d < m - 1.

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A242262 Semiprimes of the form k^3 - 1.

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Mathematica

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A261435 Numbers k such that k^5-1 is a semiprime.

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Magma

Mathematica

PARI

A268186 Numbers n such that n^2 + 2, n^2 - 2, n + 2 and n - 2 are all semiprimes.

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