A073501 -id:A073501 - cp's OEIS frontend

A102326 Primes p such that the largest prime divisor of p^4+1 is less than p.

10181, 14051, 18979, 25253, 57173, 58013, 60101, 62497, 65951, 66541, 69457, 75931, 82241, 82261, 84229, 87721, 88339, 88819, 91499, 92333, 95917, 99523, 105557, 107747, 109229, 118493, 118927, 137339, 146291, 155399, 157019

Offset: 1

Labos Elemer, Jan 05 2005

nonn

Primes in A309562. - Robert Israel, Aug 09 2019

			p = 10181, 1+p^4 = 10743894862923122 = 2*17*1657*4657*5113*8009, so the largest prime factor is 8009 < p = 10181.

Robert Israel, Table of n, a(n) for n = 1..1000

Cf. A000040, A065091, A073501, A309562.

filter:= proc(p) max(numtheory:-factorset(p^4+1)) < p end proc:
select(filter, [seq(ithprime(i),i=1..20000)]); # Robert Israel, Aug 09 2019

<Ray Chandler, Jan 08 2005 *)
Select[Prime[Range[15000]],FactorInteger[#^4+1][[-1,1]]<#&] (* Harvey P. Dale, Feb 27 2017 *)

isok(p) = isprime(p) && (vecmax(factor(p^4+1)[,1]) < p); \\ Michel Marcus, Jul 09 2018

Extended by Ray Chandler, Jan 08 2005

A091490 Primes p such that all prime divisors of p^2 + p + 1 are less than p.

Original entry on oeis.org

67, 79, 137, 149, 163, 181, 191, 211, 229, 263, 269, 277, 313, 373, 431, 439, 499, 521, 571, 631, 653, 787, 809, 811, 821, 823, 919, 947, 971, 991, 997, 1049, 1069, 1087, 1109, 1129, 1153, 1231, 1237, 1283, 1291, 1367, 1429, 1451, 1459, 1471, 1493, 1511

Offset: 1

R. K. Guy and Robert G. Wilson v, Jan 14 2004

nonn

R. J. Mathar, Table of n, a(n) for n = 1..4725
Robin Chapman, Comments on the Somon Davis paper
Simon Davis, A Proof of the Odd Perfect Number Conjecture, arXiv:hep-th/0401052 [But see the comments on this paper by Robin Chapman]

Cf. A073501.

PrimeFactors[ n_Integer ] := Flatten[ Table[ #[ [ 1 ] ], {1} ] & /@ FactorInteger[ n ] ]; Prime[ Select[ Range[ 242 ], Prime[ # ] > PrimeFactors[ Prime[ # ]^2 + Prime[ # ] + 1 ][ [ -1 ] ] & ] ]

A102325 Primes p such that the largest prime divisor of p^3 + 1 is less than p.

Original entry on oeis.org

17, 19, 23, 31, 101, 103, 173, 179, 257, 263, 293, 353, 373, 431, 467, 491, 521, 563, 593, 619, 641, 677, 719, 739, 773, 821, 829, 857, 859, 863, 881, 929, 941, 953, 1031, 1051, 1087, 1091, 1109, 1129, 1229, 1297, 1327, 1399, 1433, 1487, 1489, 1499, 1583

Offset: 1

Labos Elemer, Jan 05 2005

nonn

			p = 17, 1 + p^3 = 1 + 4913 = 2*3*3*3*7*13, so the largest prime factor is 13 < p = 17.

Giovanni Resta, Table of n, a(n) for n = 1..10000

Cf. A000040, A065091, A073501, A102326-A102330.

Mathematica
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<Ray Chandler, Jan 08 2005 *)
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A102327 Primes p such that the largest prime factor of p^5 + 1 is less than p.

Original entry on oeis.org

1753, 2357, 7103, 9749, 13441, 16453, 21467, 22739, 25153, 28409, 29059, 33247, 33347, 36781, 42853, 51427, 57751, 58453, 62347, 65777, 66593, 69119, 72923, 78643, 80407, 83591, 85619, 89909, 91411, 99409, 101209, 101363, 113171, 124337

Offset: 1

Labos Elemer, Jan 05 2005

nonn

			p = 1753, 1 + p^5 = 16554252702583994 = 2*41*151*691*877*1361*1621, so the largest prime factor is 1621 < p = 1753.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000040, A006530, A065091, A073501.

Select[Prime[Range[15000]], Max[PrimeFactorList[1 + #^5]] < # &] (* Ray Chandler, Jan 08 2005 *)
Select[Prime[Range[12000]],FactorInteger[#^5+1][[-1,1]]<#&]  (* Harvey P. Dale, Mar 14 2011 *)

isok(p)= isprime(p) && (vecmax(factor(p^5+1)[,1]) < p); \\ Michel Marcus, Jul 11 2018

Solutions to {A006530(1 + p^5) < p} where p is a prime.

Extended by Ray Chandler, Jan 08 2005

A102328 Primes p such that the largest prime divisor of p^6 + 1 is less than p.

Original entry on oeis.org

30977, 69127, 104681, 109807, 114671, 141637, 146057, 160319, 160639, 170371, 171169, 176087, 211723, 216119, 217081, 319321, 381673, 389083, 390151, 416219, 437401, 484609, 492257, 525571, 564713, 565241, 574127, 591601, 612173, 621259

Offset: 1

Labos Elemer, Jan 05 2005

nonn

			p = 30977, p^6 + 1 = 883560179055825771003237890 = 2*5*13*37*61*113*181*13921*18517*22189*25741, so the largest prime factor is 25741 < p = 30977.

Amiram Eldar, Table of n, a(n) for n = 1..1000

Cf. A000040, A006530, A065091, A073501, A102327.

Select[Prime[Range[60000]], Max[PrimeFactorList[1 + #^6]] < # &] (* Ray Chandler, Jan 08 2005 *)

is(k) = isprime(k) && vecmax(factor(k^6+1)[, 1]) < k; \\ Amiram Eldar, Jun 21 2024

Solutions to {A006530(p^6+1) < p} where p is a prime number.

Extended by Ray Chandler, Jan 08 2005

A102329 a(n) is the least prime p such that the largest prime divisor of 1+p^n is less than p.

Original entry on oeis.org

3, 7, 17, 10181, 1753, 30977, 1507853

Offset: 1

Labos Elemer, Jan 05 2005

nonn

First term of sequences A065091, A073501, A102325-A102328.
These are the smallest p primes such that 1+p^n are p-smooth, i.e. with all prime-divisors less than p. The same holds for A102325-A102328. For the majority of p's, 1+p^n appear to be not p-smooth, i.e. have large enough greatest prime divisors.
a(8) > 4.5*10^6. - Giovanni Resta, May 02 2017

Cf. A000040, A065091, A073501.

a(7)=1507853 from Ray Chandler, Jan 08 2005

cp's OEIS Frontend

A102326 Primes p such that the largest prime divisor of p^4+1 is less than p.

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A091490 Primes p such that all prime divisors of p^2 + p + 1 are less than p.

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A102325 Primes p such that the largest prime divisor of p^3 + 1 is less than p.

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A102327 Primes p such that the largest prime factor of p^5 + 1 is less than p.

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A102328 Primes p such that the largest prime divisor of p^6 + 1 is less than p.

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Mathematica

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A102329 a(n) is the least prime p such that the largest prime divisor of 1+p^n is less than p.

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