A177854 -id:A177854 - cp's OEIS frontend

A169818 Rank of n-th prime as defined in A177854.

0, 1, 1, 1, 2, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 3, 2, 2, 2, 2, 3, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 2, 2, 2, 1, 3, 2, 2, 3, 2, 3, 2, 2, 2, 3, 3, 3, 2, 3, 2, 3, 2, 3, 2, 2, 2, 3, 3, 2, 2, 2, 2, 2, 2, 3, 3, 2, 3, 3, 3, 3, 2, 2, 2, 2, 2, 2, 3, 3

Offset: 1

T. D. Noe, May 28 2010

nonn

This notion of rank is closely related to the Erdős-Selfridge classification of primes.

T. D. Noe, Table of n, a(n) for n=1..10000

For records see A177854.

rank[1]=0; rank[2]=0; rank[3]=1;
SetAttributes[rank,Listable];
rank[p_] := rank[p] = 1+Min[Max@@rank[First/@FactorInteger[p-1]], Max@@rank[First/@FactorInteger[p+1]]]; rank[Prime[Range[100]]]

a(A000720(A141453(n)))=1 n>1. [From R. J. Mathar, May 28 2010]

A239374 Smallest product of consecutive distinct prime factors of t = prime(n)^2 - 1 in ascending order that provides more than 1/3 factored parts for Brillhart-Lehmer-Selfridge primality test for prime(n).

Original entry on oeis.org

2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 6, 6

Offset: 2

Lei Zhou, Mar 17 2014

The first greater than 2 element of this sequence is a(99).

			n = 2: prime(2) = 3, 3^2 - 1 = 8 = 2^3, 2^3 > 3, 100% factorization.  So a(2) = 2.
n = 45: prime(45) = 197, 197^2 - 1 = 38808 = 2^3*3^2*7^2*11, 2^3 = 8,  log_197(8) = 0.3936 > 1/3, 39.36% factorization.  So a(45) = 2.
n = 99: prime(99) = 523, 523^2 - 1 = 273528 = 2^3*3^2*29*131, 2^3 = 8, log_523(8) = 0.3322 < 1/3, log_523(2^3*3^2) = 0.6832 > 1/3, 68.32% factorization. So a(99) = 6.

Lei Zhou, Table of n, a(n) for n = 2..10000

Cf. A000040, A177854.

Table[p = Prime[n]; ck = p^(1/3); sp = p^2 - 1; dp = sp; prod = 1; fp = Union[Transpose[FactorInteger[p + 1]][[1]], Transpose[FactorInteger[p - 1]][[1]]]; i = 0; While[i++; m = fp[[i]]; prod = prod*m; While[Divisible[sp, m], sp = sp/m]; (dp/sp) < ck]; prod, {n, 2, 100}]

cp's OEIS Frontend

A169818 Rank of n-th prime as defined in A177854.

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