A125136 -id:A125136 - cp's OEIS frontend

A125135 Triangle read by rows in which row n lists prime factors of p^p - 1 where p = prime(n).

3, 2, 13, 2, 2, 11, 71, 2, 3, 29, 4733, 2, 5, 15797, 1806113, 2, 2, 3, 53, 264031, 1803647, 2, 2, 2, 2, 10949, 1749233, 2699538733, 2, 3, 3, 109912203092239643840221, 2, 11, 461, 1289, 831603031789, 1920647391913

Offset: 1

N. J. A. Sloane, Jan 21 2007

			Triangle begins:
3;
2, 13;
2, 2, 11, 71;
2, 3, 29, 4733;
2, 5, 15797, 1806113;
2, 2, 3, 53, 264031, 1803647;
2, 2, 2, 2, 10949, 1749233, 2699538733;
2, 3, 3, 109912203092239643840221;
2, 11, 461, 1289, 831603031789, 1920647391913;
2, 2, 7, 59, 16763, 84449, 2428577, 14111459, 58320973, 549334763;
...
n=4: p=7, 7^7-1 = 823542 = 2*3*29*4733 gives row 4.

Sam Wagstaff, Factorizations of p^p - 1 for most p < 180

Cf. A006486, A088730, A125136, A212552, A214812.

for p in [ n : n in [1..100] | IsPrime(n) ] do "\nDoing p =", p; n := p^p -1; Factorisation(n); end for; // John Cannon

T:= n-> (p-> sort(map(i-> i[1]$i[2], ifactors(p^p-1)[2]))[])(ithprime(n)):
seq(T(n), n=1..10);  # Alois P. Heinz, May 20 2022

A125137 a(n) = p^p + 1, where p = prime(n).

Original entry on oeis.org

5, 28, 3126, 823544, 285311670612, 302875106592254, 827240261886336764178, 1978419655660313589123980, 20880467999847912034355032910568, 2567686153161211134561828214731016126483470, 17069174130723235958610643029059314756044734432, 10555134955777783414078330085995832946127396083370199442518

Offset: 1

N. J. A. Sloane, Jan 21 2007

nonn

Vincenzo Librandi, Table of n, a(n) for n = 1..77

See A125136 for factorizations. Cf. A088730, A125135.

Cf. A104131, A114846.

[p^p+1: p in PrimesUpTo(40)]; // Vincenzo Librandi, Mar 27 2014

Table[Prime[n]^Prime[n] + 1, {n , 20}] (* Vincenzo Librandi, Mar 27 2014 *)

a(n) = A051674(n)+1. - R. J. Mathar, Apr 23 2007

A352400 a(n) is the left Aurifeuillian factor of p^p + 1 for A002145(n), where A002145 lists the primes congruent to 3 (mod 4).

Original entry on oeis.org

1, 113, 58367, 113631466919, 348275601426959, 8403855868042458448127, 7248206084007410402911299180581471, 105318477338066161993242388018074119617, 830220061043693789623432394289631761145130727636121

Offset: 1

Patrick A. Thomas, Jun 08 2022

nonn

For prime factorizations of p^p + 1 see A125136.

			105318477338066161993242388018074119617 is the smaller Aurifeuillian factor of 47^47 + 1, and 47 is the 8th term of A002145, so it is a(8).

Patrick A. Thomas, Table of n, a(n) for n = 1..65
Calculators, Aurifeuillian LMs
Wikipedia, Aurifeuillean factorization.

Cf. A002145, A125136, A230377, A352711, A352732, A352401.

If R is (p^p+1)/(p+1), where p == 3 (mod 4) and p > 7, then an approximation of the left Aurifeuillian factor of R is (1/e) * sqrt(R/(1+z)), where z =
   2/(3p) + 28/(45p^2) + 1706/(2835p^3) if p=1,79,109,121,151  or 169 (mod 210),
   2/(3p) + 28/(45p^2) +   86/(2835p^3) if p=19,31,61,139,181  or 199 (mod 210),
   2/(3p) -  8/(45p^2) +  194/(2835p^3) if p=37,43,67,127,163  or 193 (mod 210),
   2/(3p) -  8/(45p^2) - 1426/(2835p^3) if p=13,73,97,103,157  or 187 (mod 210),
  -2/(3p) -  8/(45p^2) + 1426/(2835p^3) if p=23,53,107,113,137 or 197 (mod 210),
  -2/(3p) -  8/(45p^2) -  194/(2835p^3) if p=17,47,83,143,167  or 173 (mod 210),
  -2/(3p) + 28/(45p^2) -   86/(2835p^3) if p=11,29,71,149,179  or 191 (mod 210),
  -2/(3p) + 28/(45p^2) - 1706/(2835p^3) if p=41,59,89,101,131  or 209 (mod 210).

A352401 The right Aurifeuillian factor of p^p + 1 for primes p congruent to 3 (mod 4).

Original entry on oeis.org

7, 911, 407353, 870542161121, 2498077661567473, 63472256064447557254913, 54382651771205224279713471565249817, 767102704711961850109296220485687497279, 6066304600323886604542912453739225327712511596287519

Offset: 1

Patrick A. Thomas, Jun 08 2022

nonn

For prime factorizations of p^p + 1 see A125136.

			870542161121 is the larger Aurifeuillian factor of 19^19 + 1, and 19 is the 4th term of A002145, so a(4) = 870542161121.

Patrick A. Thomas, Table of n, a(n) for n = 1..65
Calculators, Aurifeuillian LMs

Cf. A002145, A125136, A230379, A352711, A352732, A352400.

cp's OEIS Frontend

A125135 Triangle read by rows in which row n lists prime factors of p^p - 1 where p = prime(n).

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A125137 a(n) = p^p + 1, where p = prime(n).

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A352400 a(n) is the left Aurifeuillian factor of p^p + 1 for A002145(n), where A002145 lists the primes congruent to 3 (mod 4).

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Formula

A352401 The right Aurifeuillian factor of p^p + 1 for primes p congruent to 3 (mod 4).

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