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Search: a105122 -id:a105122
Displaying 1-10 of 13 results found. page 1 2
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A105041 Positive integers k such that k^7 + 1 is semiprime. +10
17
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
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.
LINKS
Robert Price, Table of n, a(n) for n = 1..1414
FORMULA
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.
EXAMPLE
n n^7+1 = ((n+1) * (n^6 - n^5 + n^4 - n^3 + n^2 - n + 1).
2 129 = 3 x 43
10 10000001 = 11 * 909091
16 268435457 = 17 * 15790321
18 612220033 = 19 * 32222107
46 435817657217 = 47 * 9272716111
MATHEMATICA
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 *)
PROG
(Magma) 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
(PARI) is(n)=isprime(n+1) && isprime((n^7+1)/(n+1)) \\ Charles R Greathouse IV, Aug 31 2021
CROSSREFS
Cf. A001358, A085722, A096173, A186669, A104238, A103854, A105041, A105066, A105078, A105122, A105142, A105237, A104335, A104479, A104494, A104657, A105282.
KEYWORD
easy,nonn
AUTHOR
Jonathan Vos Post, Apr 03 2005
EXTENSIONS
More terms from R. J. Mathar, Dec 14 2009
STATUS
approved
A103854 Positive integers n such that n^6 + 1 is semiprime. +10
15
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
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.
LINKS
Robert Price, Table of n, a(n) for n = 1..1134
FORMULA
a(n)^6 + 1 is semiprime. (a(n)^2+1) is prime and (a(n)^4 - a(n)^2 + 1) is prime.
EXAMPLE
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
MATHEMATICA
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 *)
PROG
(PARI) is(n)=my(s=n^2); isprime(s+1) && isprime(s^2-s+1) \\ Charles R Greathouse IV, Aug 31 2021
CROSSREFS
Subsequence of A005574.
Cf. A001358, A001538, A085722, A096173, A186669, A104238, A103854, A105041, A105066, A105078, A105122, A105142, A105237, A104335, A104479, A104494, A104657, A105282.
KEYWORD
easy,nonn
AUTHOR
Jonathan Vos Post, Mar 31 2005
EXTENSIONS
More terms from Robert G. Wilson v, May 26 2006
STATUS
approved
A104494 Positive integers n such that n^17 + 1 is semiprime (A001358). +10
14
2, 58, 66, 166, 268, 270, 408, 600, 672, 808, 822, 970, 1050, 1090, 1150, 1200, 1212, 1380, 1578, 1752, 1912, 1950, 1986, 2016, 2038, 2292, 2340, 2548, 2590, 2656, 2718, 2800, 2856, 3162, 3300, 3342, 3738, 4138, 4152, 4228, 4270, 4272, 4362, 4782, 5080, 5166 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
LINKS
Robert Price, Table of n, a(n) for n = 1..1000
FORMULA
a(n)^17 + 1 is semiprime (A001358).
EXAMPLE
2^17 + 1 = 131073 = 3 * 43691,
58^17 + 1 = 951208868148684143308060622849 = 59 * 16122184205909900734034925811,
66^17 + 1 = 8555529718761317069203003539457 = 67 * 127694473414348015958253784171,
1050^17 + 1 = 2292018317801032401637344360351562500000000000000001 = 1051 * 2180797638250268698037435166842590390104662226451.
MATHEMATICA
Select[Range[1000000], PrimeQ[# + 1] && PrimeQ[(#^17 + 1)/(# + 1)] &] (* Robert Price, Mar 10 2015 *)
Select[Range[5200], PrimeOmega[#^17+1]==2&] (* Harvey P. Dale, Mar 07 2017 *)
PROG
(PARI) for(n=1, 3000, if(!ispseudoprime(n^17+1), forprime(p=1, 10^4, if((n^17+1)%p==0, if(ispseudoprime((n^17+1)/p), print1(n, ", ")); break)))) \\ Derek Orr, Mar 09 2015
(Magma) IsSemiprime:=func< n | &+[ k[2]: k in Factorization(n) ] eq 2 >; [n: n in [1..1200]|IsSemiprime(n^17+1)]; // Vincenzo Librandi, Mar 10 2015
CROSSREFS
Cf. A001358, A006313, A103854, A104238, A104335, A105041, A105066, A105078, A105122, A105142, A105237, A104479.
KEYWORD
easy,nonn
AUTHOR
Jonathan Vos Post, Apr 19 2005
EXTENSIONS
a(14)-a(46) from Robert Price, Mar 09 2015
STATUS
approved
A104335 Positive integers n such that n^14 + 1 is semiprime (A001358). +10
13
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
x^14+1 has factors (1 + x^2) (1 - x^2 + x^4 - x^6 + x^8 - x^10 + x^12).
LINKS
Robert Price, Table of n, a(n) for n = 1..1114
EXAMPLE
4^14 + 1 = 268435457 = 17 * 15790321,
74^14 + 1 = 147653612273582215982104577 = 5477 * 26958848324553992328301,
1140^14 + 1 = 6261349103849104148619671961600000000000001 = 1299601 * 4817901112610027345792802530622860401.
MATHEMATICA
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 *)
CROSSREFS
Cf. A001358, A085722, A096173, A186669, A104238, A103854, A105041, A105066, A105078, A105122, A105142, A105237, A104335, A104479, A104494, A104657, A105282.
KEYWORD
easy,nonn
AUTHOR
Jonathan Vos Post, Apr 17 2005
EXTENSIONS
More terms from Robert G. Wilson v, Apr 18 2005
STATUS
approved
A104479 Positive integers n such that n^16 + 1 is semiprime (A001358). +10
13
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
n^16 + 1 is an irreducible polynomial over Z and thus can be either prime (A006313) or semiprime.
LINKS
Robert Price, Table of n, a(n) for n = 1..102
FORMULA
a(n)^16 + 1 is semiprime (A001358).
EXAMPLE
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.
MATHEMATICA
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 *)
PROG
(Magma) 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
CROSSREFS
Cf. A006313, A001358, A085722, A096173, A186669, A104238, A103854, A105041, A105066, A105078, A105122, A105142, A105237, A104335, A104479, A104494, A104657, A105282.
KEYWORD
easy,nonn
AUTHOR
Jonathan Vos Post, Apr 18 2005
EXTENSIONS
More terms from Vincenzo Librandi, Dec 21 2010
Corrected (adding 202, 208, and 220) by Harvey P. Dale, Aug 21 2011
STATUS
approved
A105078 Positive integers n such that n^10 + 1 is semiprime. +10
13
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
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.
LINKS
Robert Price, Table of n, a(n) for n = 1..1105
EXAMPLE
4^10+1 = 1048577 = 17 * 61681,
16^10+1 = 1099511627777 = 257 * 4278255361,
1010^10+1 = 1104622125411204510010000000001 = 1020101 * 1082855644108970101989901.
MATHEMATICA
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 *)
CROSSREFS
Cf. A001358, A085722, A096173, A186669, A104238, A103854, A105041, A105066, A105078, A105122, A105142, A105237, A104335, A104479, A104494, A104657, A105282.
KEYWORD
easy,nonn
AUTHOR
Jonathan Vos Post, Apr 06 2005
EXTENSIONS
More terms from Robert G. Wilson v, Apr 08 2005
STATUS
approved
A105142 Positive integers n such that n^12 + 1 is semiprime. +10
13
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
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.
LINKS
Robert Price, Table of n, a(n) for n = 1..1515
EXAMPLE
2^12+1 = 4097 = 17 * 241,
6^12+1 = 2176782337 = 1297 * 1678321,
34^12+1 = 2386420683693101057 = 1336337 * 1785792568561,
1136^12+1 = 4618915067251126036363854530631172097 = 1665379926017 * 2773490297975392253706241.
MATHEMATICA
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 *)
CROSSREFS
Cf. A001358 (semiprimes), A085722, A096173, A186669, A104238, A103854, A105041, A105066, A105078, A105122, A105142, A105237, A104335, A104479, A104494, A104657, A105282.
KEYWORD
easy,nonn
AUTHOR
Jonathan Vos Post, Apr 09 2005
EXTENSIONS
a(16)-a(46) from Robert G. Wilson v, Feb 10 2006
STATUS
approved
A105237 Positive integers n such that n^13 + 1 is semiprime. +10
13
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
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.
LINKS
Robert Price, Table of n, a(n) for n = 1..1036
EXAMPLE
2^13+1 = 8193 = 3 * 2731,
22^13+1 = 282810057883082753 = 23 * 12296089473177511,
1030^13+1 = 1468533713451564313811276230000000000001 = 1031 * 1424377995588326201562828545101842871.
MATHEMATICA
Select[Range[0, 300000], PrimeQ[# + 1] && PrimeQ[(#^13 + 1)/(# + 1)] &] (* Robert Price, Mar 11 2015 *)
PROG
(Magma) 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
CROSSREFS
Cf. A001358, A085722, A096173, A186669, A104238, A103854, A105041, A105066, A105078, A105122, A105142, A105237, A104335, A104479, A104494, A104657, A105282.
KEYWORD
easy,nonn
AUTHOR
Jonathan Vos Post, Apr 12 2005
EXTENSIONS
a(19)-a(24) from Vincenzo Librandi, Dec 21 2010
STATUS
approved
A104657 Positive integers n such that n^19 + 1 is semiprime (A001358). +10
12
2, 10, 28, 106, 190, 292, 556, 756, 858, 906, 1012, 1030, 1032, 1060, 1372, 1450, 1488, 1720, 1722, 1758, 1782, 1822, 1972, 2356, 2436, 2446, 2620, 2748, 2788, 2998, 3186, 3300, 3318, 3360, 3466, 3510, 3822, 3852, 4138, 4326, 4506, 4908, 5236, 5518, 5782 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
We have the polynomial factorization: n^19 + 1 = (n + 1) * (n^18 - n^17 + n^16 - n^15 + n^14 - n^13 + 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^18 - n^17 + n^16 - n^15 + n^14 - n^13 + 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.
LINKS
Robert Price, Table of n, a(n) for n = 1..1000
FORMULA
a(n)^19 + 1 is semiprime (A001358).
EXAMPLE
2^19 + 1 = 524289 = 3 * 174763,
10^19 + 1 = 10000000000000000001 = 11 * 909090909090909091,
1012^19 + 1 = 125438178100868833265294241234853844232270960601988910249 = 1013 * 1238284087866424810121364671617510801898035149081825373.
MATHEMATICA
Select[Range[1000000], PrimeQ[# + 1] && PrimeQ[(#^19 + 1)/(# + 1)] &] (* Robert Price, Mar 10 2015 *)
Select[Range[5800], PrimeOmega[#^19+1]==2&] (* Harvey P. Dale, Feb 15 2019 *)
PROG
(Magma) IsSemiprime:=func< n | &+[ k[2]: k in Factorization(n) ] eq 2 >; [n: n in [1..1100]|IsSemiprime(n^19+1)]; // Vincenzo Librandi, Mar 10 2015
CROSSREFS
Cf. A001358, A006313, A103854, A104238, A104335, A105041, A105066, A105078, A105122, A105142, A105237, A104479, A104494.
KEYWORD
easy,nonn
AUTHOR
Jonathan Vos Post, Apr 21 2005
EXTENSIONS
a(12)-a(45) from Robert Price, Mar 09 2015
STATUS
approved
A105282 Positive integers n such that n^20 + 1 is semiprime (A001358). +10
11
2, 4, 46, 154, 266, 472, 748, 1434, 1738, 2058, 2204, 2222, 2428, 2478, 2510, 2866, 3132, 3288, 3576, 3688, 3756, 4142, 4506, 4940, 5164, 6252, 6330, 6786, 7180, 7300, 7338, 7416, 7628, 7806, 9270, 9312, 10044, 10722, 10860, 12126, 12422, 12668, 12998, 13350 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
We have the polynomial factorization: n^20 + 1 = (n^4 + 1) * (n^16 - n^12 + n^8 - n^4 + 1). Hence after the initial n=1 prime, the binomial can never be prime. It can be semiprime iff n^4+1 is prime and (n^16 - n^12 + n^8 - n^4 + 1) is prime.
LINKS
Robert Price, Table of n, a(n) for n = 1..1405
FORMULA
a(n)^20 + 1 is semiprime (A001358).
EXAMPLE
2^20 + 1 = 1048577 = 17 * 61681,
4^20 + 1 = 1099511627777 = 257 * 4278255361,
46^20 + 1 = 1799519816997495209117766334283777 = 4477457 * 401906666439788301510827761,
1434^20 + 1 =
1352019721694375552250489804528860551814233886722212960509362177 =
4228599998737 * 319732233386510278346888399489424537759394853595121.
MATHEMATICA
Select[Range[1000000], PrimeQ[#^4 + 1] && PrimeQ[(#^20 + 1)/(#^4 + 1)] &] (* Robert Price, Mar 09 2015 *)
PROG
(Magma)IsSemiprime:=func< n | &+[ k[2]: k in Factorization(n) ] eq 2 >; [n: n in [1..1000] | IsSemiprime(n^20+1)] // Vincenzo Librandi, Dec 21 2010
CROSSREFS
Cf. A000040, A001358, A006313, A103854, A104238, A104335, A105041, A105066, A105078, A105122, A105142, A105237, A104479, A104494, A104657.
KEYWORD
easy,nonn
AUTHOR
Jonathan Vos Post, Apr 25 2005
EXTENSIONS
a(9)-a(44) from Robert Price, Mar 09 2015
STATUS
approved
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Last modified August 29 11:15 EDT 2024. Contains 375512 sequences. (Running on oeis4.)