A096173 -id:A096173 - cp's OEIS frontend

A081256 Greatest prime factor of n^3 + 1.

2, 3, 7, 13, 7, 31, 43, 19, 73, 13, 37, 19, 157, 61, 211, 241, 13, 307, 7, 127, 421, 463, 13, 79, 601, 31, 37, 757, 271, 67, 19, 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: 1

Jan Fricke, Mar 14 2003

Record values appear to match the terms of A002383 for n>1. - Bill McEachen, Oct 18 2023

R. J. Mathar and T. D. Noe, Table of n, a(n) for n = 1..10000 (first 5000 terms from R. J. Mathar)
J. Buchmann, K. Győry, M. Mignotte, and N. Tzanakis, Lower bounds for P(x^3+k), an elementary approach, Publ. Math. Debrecen, Vol. 38, No. 1-2 (1991), pp. 145-163.

Cf. A081257, A081258, A096173, A096174.

Cf. A006530, A001093, A002383, A014442, A096172.

A081256 := proc(n)
    A006530(n^3+1) ;
end proc:
seq(A081256(n),n=1..20) ; # R. J. Mathar, Feb 13 2014

Table[Max[Transpose[FactorInteger[n^3 + 1]][[1]]], {n, 25}]

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

A081256(n)=vecmax(factor(n^3+1)[,1]) \\ It seems slightly slower to get the last element using ...[-1..-1][1]. - M. F. Hasler, Jun 15 2018

a(n) = A006530(A001093(n)). - M. F. Hasler, Jun 13 2018

a(n) >= 31 for n >= 70 (Buchmann et al., 1991). - Amiram Eldar, Oct 25 2024

More terms from Harvey P. Dale, Mar 22 2003

More terms from Hugo Pfoertner, Jun 20 2004

A105041 Positive integers k such that k^7 + 1 is semiprime.

Original entry on oeis.org

2, 10, 16, 18, 46, 52, 66, 72, 78, 106, 136, 148, 226, 228, 240, 262, 282, 330, 442, 508, 616, 630, 732, 750, 756, 768, 810, 828, 910, 936, 982, 1032, 1060, 1128, 1216, 1302, 1366, 1558, 1626, 1696, 1698, 1758, 1800, 1810, 1830, 1932, 1996, 2002, 2026, 2080

Offset: 1

Jonathan Vos Post, Apr 03 2005

We have the polynomial factorization n^7+1 = (n+1) * (n^6 - n^5 + n^4 - n^3 + n^2 - n + 1). Hence after the initial n=1 prime, the binomial can at best be semiprime and that only when both (n+1) and (n^6 - n^5 + n^4 - n^3 + n^2 - n + 1) are primes.

			n n^7+1 = (n+1) * (n^6 - n^5 + n^4 - n^3 + n^2 - n + 1).
2 129 = 3 * 43
10 10000001 = 11 * 909091
16 268435457 = 17 * 15790321
18 612220033 = 19 * 32222107
46 435817657217 = 47 * 9272716111

Robert Price, Table of n, a(n) for n = 1..1414

Cf. A001358, A085722, A096173, A186669, A104238, A103854, A105041, A105066, A105078, A105122, A105142, A105237, A104335, A104479, A104494, A104657, A105282.

IsSemiprime:=func< n | &+[ k[2]: k in Factorization(n) ] eq 2 >; [n: n in [1..2100] | IsSemiprime(n^7+1)]; // Vincenzo Librandi, Mar 12 2015

Select[Range[0,200000], PrimeQ[# + 1] && PrimeQ[(#^7 + 1)/(# + 1)] &] (* Robert Price, Mar 11 2015 *)
Select[Range[2500], Plus@@Last/@FactorInteger[#^7 + 1]==2 &] (* Vincenzo Librandi, Mar 12 2015 *)
Select[Range[2100],PrimeOmega[#^7+1]==2&] (* Harvey P. Dale, Jun 18 2019 *)

is(n)=isprime(n+1) && isprime((n^7+1)/(n+1)) \\ Charles R Greathouse IV, Aug 31 2021

a(n)^7 + 1 is semiprime. a(n)+1 is prime and a(n)^6 - a(n)^5 + a(n)^4 - a(n)^3 + a(n)^2 - a(n) + 1 is prime.

More terms from R. J. Mathar, Dec 14 2009

A103854 Positive integers n such that n^6 + 1 is semiprime.

Original entry on oeis.org

2, 4, 10, 36, 56, 94, 126, 224, 260, 270, 300, 350, 686, 716, 780, 1036, 1070, 1080, 1156, 1174, 1210, 1394, 1416, 1434, 1440, 1460, 1524, 1550, 1576, 1616, 1654, 1660, 1700, 1756, 1860, 1980, 2054, 2084, 2096, 2116, 2224, 2454, 2600, 2664, 2770, 2864

Offset: 1

Jonathan Vos Post, Mar 31 2005

n^6+1 can only be prime when n = 1, n^6+1 = 2. This is because the sum of cubes formula gives the polynomial factorization n^6+1 = (n^2+1) * (n^4 - n^2 + 1). Hence n^6+1 can only be semiprime when both (n^2+1) and (n^4 - n^2 + 1) are primes.

			n n^6+1 = (n^2+1) * (n^4 - n^2 + 1)
2 65 = 5 * 13
4 4097 = 17 * 241
10 1000001 = 101 * 9901
36 2176782337 = 1297 * 1678321
56 30840979457 = 3137 * 9831361
94 689869781057 = 8837 * 78066061
126 4001504141377 = 15877 * 252031501
224 126324651851777 = 50177 * 2517580801

Robert Price, Table of n, a(n) for n = 1..1134

Subsequence of A005574.

Cf. A001358, A001538, A085722, A096173, A186669, A104238, A103854, A105041, A105066, A105078, A105122, A105142, A105237, A104335, A104479, A104494, A104657, A105282.

semiprimeQ[n_] := Plus @@ Last /@ FactorInteger[n] == 2; Select[ 2Range@1526, semiprimeQ[ #^6 + 1] &] (* Robert G. Wilson v, May 26 2006 *)
Select[Range[200000], PrimeQ[#^2 + 1] && PrimeQ[(#^6 + 1)/(#^2 + 1)] &] (* Robert Price, Mar 11 2015 *)

is(n)=my(s=n^2); isprime(s+1) && isprime(s^2-s+1) \\ Charles R Greathouse IV, Aug 31 2021

a(n)^6 + 1 is semiprime. (a(n)^2+1) is prime and (a(n)^4 - a(n)^2 + 1) is prime.

More terms from Robert G. Wilson v, May 26 2006

A105122 Positive integers n such that n^11 + 1 is semiprime.

Original entry on oeis.org

2, 6, 12, 232, 262, 280, 330, 430, 508, 772, 786, 852, 1012, 1522, 1566, 1626, 1810, 2346, 2556, 2676, 3658, 3888, 3910, 4018, 4048, 4258, 4830, 5188, 5322, 5478, 5848, 6090, 6366, 6568, 7018, 7458, 7602, 7606, 7822, 8178, 8928, 9420, 9618, 9676, 10398

Offset: 1

Jonathan Vos Post, Apr 08 2005

Since n^11 + 1 = (n+1) * (n^10 - n^9 + n^8 - n^7 + n^6 - n^5 + n^4 - n^3 + n^2 - n + 1), n^11 + 1 can be prime only if both (n+1) and (n^10 - n^9 + n^8 - n^7 + n^6 - n^5 + n^4 - n^3 + n^2 - n + 1) are prime.

			2^11+1 = 2049 = 3 * 683,
6^11+1 = 362797057 = 7 * 51828151,
1012^11+1 = 1140212079231804336089593374834689 = 1013 * 1125579545144920371263172137053.

Robert Price, Table of n, a(n) for n = 1..4303

Cf. A001358 (semiprimes), A085722, A096173, A186669, A104238, A103854, A105041, A105066, A105078, A105122, A105142, A105237, A104335, A104479, A104494, A104657, A105282.

Select[ Range[10721], PrimeQ[ # + 1] && PrimeQ[(#^11 + 1)/(# + 1)] &] (* Robert G. Wilson v, Apr 09 2005 *)

More terms from Robert G. Wilson v, Apr 09 2005

A237040 Semiprimes of the form k^3 + 1.

Original entry on oeis.org

9, 65, 217, 4097, 5833, 10649, 21953, 74089, 195113, 216001, 343001, 373249, 474553, 1000001, 1061209, 1191017, 1404929, 3241793, 3796417, 4251529, 6859001, 9261001, 12487169, 21952001, 29791001, 35937001, 43614209, 45882713, 55742969, 62099137, 89915393, 94818817, 117649001

Offset: 1

Jonathan Sondow, Feb 02 2014

nonn

k^3 + 1 is a term iff k + 1 and k^2 - k + 1 are both prime.
Is the sequence infinite? This is an analog of Landau's 4th problem, namely, are there infinitely many primes of the form k^2 + 1?
In other words: are there infinitely many primes p such that p^2 - 3*p + 3 is also prime? - Charles R Greathouse IV, Jul 02 2017

			9 = 3*3 = 2^3 + 1 is the first semiprime of the form n^3 + 1, so a(1) = 9.

Vincenzo Librandi, Table of n, a(n) for n = 1..1400
Eric Weisstein's World of Mathematics, Semiprime
Wikipedia, Semiprime
Wikipedia, Landau's problems

Cf. A001358, A002383, A002496, A046315, A081256, A096173, A096174, A237037, A237038, A237039.

Cf. A242262 (semiprimes of the form k^3 - 1).

IsSemiprime:= func; [s: n in [1..500] | IsSemiprime(s) where s is n^3 + 1]; // Vincenzo Librandi, Jul 02 2017

L = Select[Range[500], PrimeQ[# + 1] && PrimeQ[#^2 - # + 1] &]; L^3 + 1
Select[Range[50]^3 + 1, PrimeOmega[#] == 2 &] (* Zak Seidov, Jun 26 2017 *)

lista(nn) = for (n=1, nn, if (bigomega(sp=n^3+1) == 2, print1(sp, ", "));); \\ Michel Marcus, Jun 27 2017

list(lim)=my(v=List(),n,t); forprime(p=3,sqrtnint(lim\1-1,3)+1, if(isprime(t=p^2-3*p+3), listput(v,t*p))); Vec(v) \\ Charles R Greathouse IV, Jul 02 2017

a(n) = A096173(n)^3 + 1 = 8*A237037(n)^3 + 1.

A104335 Positive integers n such that n^14 + 1 is semiprime (A001358).

Original entry on oeis.org

4, 74, 94, 116, 270, 464, 556, 654, 1140, 1156, 1246, 1306, 1736, 2464, 2470, 2604, 2804, 2836, 2900, 3054, 3890, 4006, 4056, 4330, 4736, 4780, 5016, 5294, 5340, 5486, 5700, 5834, 6434, 7114, 7304, 8626, 8880, 9164, 9546, 9744, 9980, 10086, 10166

Offset: 1

Jonathan Vos Post, Apr 17 2005

x^14+1 has factors (1 + x^2) (1 - x^2 + x^4 - x^6 + x^8 - x^10 + x^12).

			4^14 + 1 = 268435457 = 17 * 15790321,
74^14 + 1 = 147653612273582215982104577 = 5477 * 26958848324553992328301,
1140^14 + 1 = 6261349103849104148619671961600000000000001 = 1299601 * 4817901112610027345792802530622860401.

Robert Price, Table of n, a(n) for n = 1..1114

Cf. A001358, A085722, A096173, A186669, A104238, A103854, A105041, A105066, A105078, A105122, A105142, A105237, A104335, A104479, A104494, A104657, A105282.

Select[ Range[2, 10422, 2], PrimeQ[ #^2 + 1] && PrimeQ[ #^12 - #^10 + #^8 - #^6 + #^4 - #^2 + 1] &] (*Robert G. Wilson v, Apr 18 2005 *)
Select[Range[2,10200,2],PrimeOmega[#^14+1]==2&] (* Harvey P. Dale, Oct 16 2011 *)

More terms from Robert G. Wilson v, Apr 18 2005

A104479 Positive integers n such that n^16 + 1 is semiprime (A001358).

Original entry on oeis.org

3, 4, 9, 12, 14, 16, 18, 20, 26, 29, 40, 41, 48, 58, 70, 73, 81, 87, 92, 96, 104, 111, 113, 114, 118, 122, 130, 140, 142, 144, 146, 150, 157, 162, 164, 167, 168, 172, 173, 184, 187, 192, 194, 195, 199, 200, 202, 208, 220, 230, 232, 244, 253, 256, 266, 278, 292, 295, 298

Offset: 1

Jonathan Vos Post, Apr 18 2005

n^16 + 1 is an irreducible polynomial over Z and thus can be either prime (A006313) or semiprime.

			3^16 + 1 = 43046722 = 2 * 21523361,
4^16 + 1 = 4294967297 = 641 * 6 700417,
9^16 + 1 = 1853020188851842 = 2 * 926510094425921,
12^16 + 1 = 184884258895036417 = 153953 * 1200913648289,
200^16 + 1 = 6553600000000000000000000000000000001 =
162123499503471553 * 40423504427621041217.

Robert Price, Table of n, a(n) for n = 1..102

Cf. A006313, A001358, A085722, A096173, A186669, A104238, A103854, A105041, A105066, A105078, A105122, A105142, A105237, A104335, A104479, A104494, A104657, A105282.

IsSemiprime:=func< n | &+[ k[2]: k in Factorization(n) ] eq 2 >; [n: n in [2..300]|IsSemiprime(n^16+1)] // Vincenzo Librandi, Dec 21 2010

Select[Range[300],PrimeOmega[#^16+1]==2&] (* Harvey P. Dale, Aug 21 2011 *)
Select[Range[1000], 2 == Total[Transpose[FactorInteger[#^16 + 1]][[2]]] &] (* Robert Price, Mar 11 2015 *)

a(n)^16 + 1 is semiprime (A001358).

More terms from Vincenzo Librandi, Dec 21 2010

Corrected (adding 202, 208, and 220) by Harvey P. Dale, Aug 21 2011

A105078 Positive integers n such that n^10 + 1 is semiprime.

Original entry on oeis.org

4, 16, 26, 54, 110, 120, 126, 260, 314, 420, 444, 470, 570, 646, 714, 890, 946, 1010, 1294, 1306, 1394, 1640, 1674, 1794, 1920, 1964, 2116, 2174, 2360, 2430, 2624, 2666, 2884, 2924, 3094, 3106, 3174, 3220, 3504, 3686, 3826, 3884, 3924, 4046, 4540, 4700

Offset: 1

Jonathan Vos Post, Apr 06 2005

We have the polynomial factorization: n^10+1 = (n^2+1) * (n^8 - n^6 + n^4 - n^2 + 1) Hence after the initial n=1 prime the binomial can only be semiprime if n^2 + 1 is prime and n^8 - n^6 + n^4 - n^2 + 1 is prime.

			4^10+1 = 1048577 = 17 * 61681,
16^10+1 = 1099511627777 = 257 * 4278255361,
1010^10+1 = 1104622125411204510010000000001 = 1020101 * 1082855644108970101989901.

Robert Price, Table of n, a(n) for n = 1..1105

Cf. A001358, A085722, A096173, A186669, A104238, A103854, A105041, A105066, A105078, A105122, A105142, A105237, A104335, A104479, A104494, A104657, A105282.

Select[ Range[5000], PrimeQ[ #^2 + 1] && PrimeQ[(#^10 + 1)/(#^2 + 1)] &] (* Robert G. Wilson v, Apr 08 2005 *)
Select[Range[4700], PrimeOmega[#^10+1]==2&] (* Harvey P. Dale, Jan 13 2013 *)

More terms from Robert G. Wilson v, Apr 08 2005

A105142 Positive integers n such that n^12 + 1 is semiprime.

Original entry on oeis.org

2, 6, 34, 46, 142, 174, 204, 238, 312, 466, 550, 616, 690, 730, 1136, 1280, 1302, 1330, 1486, 1586, 1610, 1638, 1644, 1652, 1688, 1706, 1772, 1934, 1952, 1972, 2040, 2102, 2108, 2142, 2192, 2238, 2250, 2376, 2400, 2554, 2612, 2646, 3006, 3094, 3550, 3642

Offset: 1

Jonathan Vos Post, Apr 09 2005

Since n^12 + 1 = (n^4+1) * (n^8 - n^4 + 1), n^12 + 1 can be semiprime only if both n^4 + 1 and n^8 - n^4 + 1 are prime.

			2^12+1 = 4097 = 17 * 241,
6^12+1 = 2176782337 = 1297 * 1678321,
34^12+1 = 2386420683693101057 = 1336337 * 1785792568561,
1136^12+1 = 4618915067251126036363854530631172097 = 1665379926017 * 2773490297975392253706241.

Robert Price, Table of n, a(n) for n = 1..1515

Cf. A001358 (semiprimes), A085722, A096173, A186669, A104238, A103854, A105041, A105066, A105078, A105122, A105142, A105237, A104335, A104479, A104494, A104657, A105282.

Select[ Range@3691, PrimeQ[ #^4 + 1] && PrimeQ[(#^12 + 1)/(#^4 + 1)] &] (* Robert G. Wilson v *)
Select[Range[4000],PrimeOmega[#^12+1]==2&] (* Harvey P. Dale, Jan 24 2013 *)

a(16)-a(46) from Robert G. Wilson v, Feb 10 2006

A105237 Positive integers n such that n^13 + 1 is semiprime.

Original entry on oeis.org

2, 22, 108, 126, 180, 256, 336, 490, 630, 652, 660, 682, 708, 760, 828, 862, 882, 1030, 1038, 1128, 1162, 1216, 1318, 1450, 1612, 1930, 1950, 2010, 2236, 2268, 2380, 2436, 2658, 2752, 2800, 2962, 2998, 3036, 3048, 3318, 3672, 3922, 4152, 4396, 4506, 4816

Offset: 1

Jonathan Vos Post, Apr 12 2005

We have the polynomial factorization: n^13+1 = (n+1) * (n^12 - n^11 + n^10 - n^9 + n^8 - n^7 + n^6 - n^5 + n^4 - n^3 + n^2 - n + 1) Hence after the initial n=1 prime, the binomial can never be prime. It can be semiprime iff n+1 is prime and n^12 - n^11 + n^10 - n^9 + n^8 - n^7 + n^6 - n^5 + n^4 - n^3 + n^2 - n + 1 is prime.

			2^13+1 = 8193 = 3 * 2731,
22^13+1 = 282810057883082753 = 23 * 12296089473177511,
1030^13+1 = 1468533713451564313811276230000000000001 = 1031 * 1424377995588326201562828545101842871.

Robert Price, Table of n, a(n) for n = 1..1036

Cf. A001358, A085722, A096173, A186669, A104238, A103854, A105041, A105066, A105078, A105122, A105142, A105237, A104335, A104479, A104494, A104657, A105282.

IsSemiprime:=func< n | &+[ k[2]: k in Factorization(n) ] eq 2 >; [n: n in [1..1600]|IsSemiprime(n^13+1)] // Vincenzo Librandi, Dec 21 2010

Select[Range[0, 300000], PrimeQ[# + 1] && PrimeQ[(#^13 + 1)/(# + 1)] &] (* Robert Price, Mar 11 2015 *)

a(19)-a(24) from Vincenzo Librandi, Dec 21 2010

cp's OEIS Frontend

A081256 Greatest prime factor of n^3 + 1.

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A105041 Positive integers k such that k^7 + 1 is semiprime.

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A103854 Positive integers n such that n^6 + 1 is semiprime.

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A105122 Positive integers n such that n^11 + 1 is semiprime.

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A237040 Semiprimes of the form k^3 + 1.

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A104335 Positive integers n such that n^14 + 1 is semiprime (A001358).

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A104479 Positive integers n such that n^16 + 1 is semiprime (A001358).

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