A053705 -id:A053705 - cp's OEIS frontend

A067025 Exponents of prime-powers corresponding to terms of A053705(n)+2.

2, 2, 2, 2, 4, 2, 5, 2, 6, 2, 2, 2, 2, 4, 2, 2, 3, 2, 2, 2, 4, 2, 2, 2, 9, 4, 3, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 5, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 4, 2, 2, 7, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 6, 2, 2, 2, 2, 2, 2, 2

Offset: 1

Labos Elemer, Dec 29 2001

nonn

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

Cf. A025475, A053705, A067023.

lista(pmax) = {my(f); forprime(p = 2, pmax, f = factor(p+2); if(#f~ == 1 && f[1,2] > 1, print1(f[1,2], ", ")));} \\ Amiram Eldar, Jul 18 2024

More terms from Amiram Eldar, Jul 18 2024

A067024 Smallest prime p such that p+2 has exactly n distinct prime factors.

Original entry on oeis.org

2, 13, 103, 1153, 15013, 255253, 4849843, 111546433, 4360010653, 100280245063, 5245694198743, 152125131763603, 7149881192889433, 421842990380476663, 16294579238595022363, 1106494163767990292293, 74135108972455349583763, 4632891063696575353839163, 278970415063349480483707693, 24012274383139350058948392193

Offset: 1

Labos Elemer, Dec 29 2001

nonn

			For n = 1,...,7 the factors of 2+a(n) are as follows: 2*2, 3*5, 3*5*7, 3*5*7*11, 3*5*7*11*13, 3*5*7*11*13*17, 3*5*7*11*13*17*19; i.e., a(n) = A002110(n+1)/2 which is prime for n = 2,...,7.

Sean A. Irvine, Table of n, a(n) for n = 1..58 (terms 1..38 from Michael S. Branicky)
Michael S. Branicky, Python program
Sean A. Irvine, Java program (github)

Cf. A000040, A001221, A002110, A053705, A067023.

Python
```
# see linked program
```

a(n) = Min_{p in A000040 ; A001221(p+2) = n}.

a(8)-a(15) from Donovan Johnson, Jan 21 2009

a(16) and beyond from Michael S. Branicky, Feb 07 2023

A053704 Prime powers p^w (w >= 2) such that p^w-2 is prime.

Original entry on oeis.org

4, 9, 25, 49, 81, 169, 243, 361, 729, 841, 1369, 1849, 2209, 2401, 3721, 5041, 6859, 7921, 10609, 11449, 14641, 16129, 17161, 19321, 19683, 28561, 29791, 29929, 36481, 44521, 49729, 50653, 54289, 57121, 66049, 85849, 97969, 113569, 128881

Offset: 1

Labos Elemer, Feb 14 2000

nonn

Terms k of A025475 such that k - 2 is prime.

			4 = 2^2 is a term since 4-2 = 2 is prime.
243 = 3^5 is a term because 241 is prime.

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

Cf. A025475, A053705.

Select[Range[130000],!PrimeQ[#]&&PrimePowerQ[#]&&PrimeQ[#-2]&] (* Harvey P. Dale, Oct 07 2020 *)
seq[max_] := Module[{s = {}, p = 2}, While[p^2 <= max, s = Join[s, Select[p^Range[2, Floor[Log[p, max]]], PrimeQ[# - 2] &]]; p = NextPrime[p]]; Union[s]]; seq[150000] (* Amiram Eldar, Aug 27 2024 *)

a(n) = A053705(n) + 2. - Amiram Eldar, Aug 27 2024

Definition clarified by Harvey P. Dale, Oct 07 2020

cp's OEIS Frontend

A067025 Exponents of prime-powers corresponding to terms of A053705(n)+2.

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A067024 Smallest prime p such that p+2 has exactly n distinct prime factors.

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A053704 Prime powers p^w (w >= 2) such that p^w-2 is prime.

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