A051647 -id:A051647 - cp's OEIS frontend

A381212 a(n) is the smallest element of the set of bases and exponents (including exponents = 1) in the prime factorization of n.

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

Offset: 2

Paolo Xausa, Feb 19 2025

The corresponding largest elements are given by A081812.
The positions of terms > 1 are given by A001694.
Records of a(n) = 2, 3, 4, 5,.. appear at n=4=2^2, 27=3^3, 625=5^4, 3125=5^5, 117649=7^6, 823543=7^7 ,... (subsequence A051647).- R. J. Mathar, Mar 05 2025

			a(36) = 2 because 36 = 2^2*3^2, the set of these bases and exponents is {2, 3} and its smallest element is 2.
a(31500) = 1 because 31500 = 2^2*3^2*5^3*7^1, the set of these bases and exponents is {1, 2, 3, 5, 7} and its smallest element is 1.

Paolo Xausa, Table of n, a(n) for n = 2..10000

Cf. A001694, A081812, A288636, A331048, A381201, A381202, A381203, A381204, A381205, A381214.

A381212 := proc(n)
    local a,pe;
    a := n ;
    for pe in ifactors(n)[2] do
        a := min(a,op(1,pe),op(2,pe)) ;
    end do:
    a ;
end proc:
seq(A381212(n),n=2..100) ; # R. J. Mathar, Mar 05 2025

A381212[n_] := Min[Flatten[FactorInteger[n]]];
Array[A381212, 100, 2]

a(n) = my(f=factor(n)); vecmin(setunion(Set(f[,1]), Set(f[,2]))); \\ Michel Marcus, Feb 20 2025

A051887 Minimal and special 2k-Germain primes, where 2k is in A002110 (primorial numbers).

Original entry on oeis.org

2, 2, 2, 2, 2, 5, 17, 11, 11, 11, 2, 23, 7, 43, 19, 3, 5, 2, 7, 3, 61, 53, 2, 41, 47, 2, 5, 7, 31, 2, 47, 13, 113, 7, 137, 103, 43, 41, 97, 3, 173, 97, 41, 13, 97, 59, 29, 53, 3, 107, 127, 197, 3, 487, 433, 31, 281, 587, 7, 89, 41, 47, 193, 239, 41, 7, 31, 67

Offset: 1

Labos Elemer, Dec 15 1999

nonn

a(n) is the minimal prime p such that primorial(n)*p + 1 is also prime.
While p is in A005384, the primorial(n)*p + 1 primes are in A051902 (primorial-safe primes).
Analogous to or subset of A051686, where the even numbers are 2, 6, ..., A002110(n), ...

			a(25) = 47 because primorial(25)*47 + 1 is also prime and minimal with this property: primorial(25)*47 + 1 = 47*2305567963945518424753102147331756070 + 1 = 108361694305439365963395800924592535291 is a minimal prime. The first 6 terms (2,2,2,2,2,5) correspond to first entries in A005384, A007693, A051645, A051647, A051653, A051654, respectively.

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

Cf. A002110, A005384, A005385, A051686, A007693, A051886, A051888, A051902.

Table[p = 2; While[! PrimeQ[Product[Prime@ i, {i, n}] p + 1], p = NextPrime@ p]; p, {n, 68}] (* Michael De Vlieger, Jun 29 2017 *)

a(n) = {my(p = 2, r = vecprod(primes(n))); while(!isprime(p * r + 1), p = nextprime(p+1)); p;} \\ Amiram Eldar, Feb 25 2025

a(n) = (A051902(n)-1)/A002110(n). - Amiram Eldar, Feb 25 2025

More terms from Michael De Vlieger, Jun 29 2017

A051648 Primes of form 210*p + 1 where p is a prime.

Original entry on oeis.org

421, 631, 1051, 1471, 2311, 2731, 3571, 4831, 6091, 9871, 11131, 12391, 14071, 15331, 17431, 18691, 21211, 28771, 29191, 32971, 34231, 37591, 38011, 40111, 40531, 46831, 48091, 52711, 56911, 58171, 59011, 65731, 66571, 72871, 73291, 74131, 75391, 84211, 87991

Offset: 1

Labos Elemer

Generalization of A005385. Can be called 210-safe primes.

A002110(4)*p + 1 = 210*p + 1 (prime).

			631 is in the sequence because 631 = 210*p + 1, where p=3.

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

Cf. A005384, A005385, A007693, A002110.

Select[Table[210n+1,{n,Prime[Range[100]]}],PrimeQ] (* Harvey P. Dale, Dec 25 2016 *)

isok(k) = isprime(k) && k % 210 == 1 && isprime((k-1)/210); \\ Amiram Eldar, Feb 24 2025

a(n) = 210 * A051647(n) + 1. - Amiram Eldar, Feb 24 2025

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A381212 a(n) is the smallest element of the set of bases and exponents (including exponents = 1) in the prime factorization of n.

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A051887 Minimal and special 2k-Germain primes, where 2k is in A002110 (primorial numbers).

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A051648 Primes of form 210*p + 1 where p is a prime.

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