A052147 -id:A052147 - cp's OEIS frontend

A166010 a(n) = prime(n)^2-4.

0, 5, 21, 45, 117, 165, 285, 357, 525, 837, 957, 1365, 1677, 1845, 2205, 2805, 3477, 3717, 4485, 5037, 5325, 6237, 6885, 7917, 9405, 10197, 10605, 11445, 11877, 12765, 16125, 17157, 18765, 19317, 22197, 22797, 24645, 26565, 27885, 29925, 32037

Offset: 1

Vladimir Joseph Stephan Orlovsky, Oct 04 2009

Least common multiple of prime(n)-2 and prime(n)+2.

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A066830, A084921, A166011.

Cf. A049001, A084920, A182200; A182174.

[NthPrime(n)^2-4: n in [1..41]]; // Bruno Berselli, Apr 17 2012

f[n_]:=LCM[n-2,n+2]; lst={};Do[p=Prime[n];AppendTo[lst,f[p]],{n,5!}]; lst
Prime[Range[5!]]^2 - 4 (* Zak Seidov, Apr 17 2012 *)

a(n)=prime(n)^2-4 \\ Charles R Greathouse IV, Apr 17 2012

a(n) = A001248(n)-4 = A040976(n)*A052147(n). [Bruno Berselli, Apr 17 2012]

Definition rewritten by Bruno Berselli, Apr 17 2012

A321348 a(n) = Sum_{d|n} tau(d^n), where tau() is the number of divisors (A000005).

Original entry on oeis.org

1, 4, 5, 15, 7, 64, 9, 52, 30, 144, 13, 546, 15, 256, 289, 165, 19, 1140, 21, 1386, 529, 576, 25, 3848, 78, 784, 166, 2610, 31, 32768, 33, 486, 1225, 1296, 1369, 12321, 39, 1600, 1681, 10248, 43, 85184, 45, 6210, 6486, 2304, 49, 24250, 150, 7956

Offset: 1

Ilya Gutkovskiy, Nov 06 2018

nonn

a(n) is prime iff n is in A001359, which makes the sequence a supersequence of A006512. - Ivan N. Ianakiev, Nov 07 2018

Vincenzo Librandi, Table of n, a(n) for n = 1..5000

Cf. A000005, A007425, A035116, A061391.

Cf. A001359, A006512, A052147.

[&+[NumberOfDivisors(d^n): d in Divisors(n)]: n in [1..50]]; // Vincenzo Librandi, Nov 08 2018

with(numtheory): seq(coeff(series(add(tau(k^n)*x^k/(1-x^k),k=1..n),x,n+1), x, n), n = 1 .. 50); # Muniru A Asiru, Nov 25 2018

Table[Sum[DivisorSigma[0, d^n], {d, Divisors[n]}], {n, 50}]
a[n_] := Times @@ ((#[[2]] + 1) (n #[[2]] + 2)/2 & /@ FactorInteger[n]); a[1] = 1; Table[a[n], {n, 50}]

a(n) = sumdiv(n, d, numdiv(d^n)); \\ Michel Marcus, Nov 06 2018

from math import prod
from sympy import factorint
def A321348(n): return prod((e+1)*(n*e+2)>>1 for e in factorint(n).values()) # Chai Wah Wu, Dec 13 2022

a(n) = [x^n] Sum_{k>=1} tau(k^n)*x^k/(1 - x^k).
If n = Product (p_j^k_j) then a(n) = Product ((k_j + 1)*(n*k_j + 2)/2).
a(prime(n)) = prime(n) + 2 = A052147(n). - Michel Marcus, Nov 25 2018

A049236 a(n) is the number of distinct prime factors of prime(n) + 2.

Original entry on oeis.org

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

Offset: 1

Labos Elemer

			prime(27) = 103, prime(27) + 2 = 105 = 3*5*7 has 3 prime factors, so a(27) = 3.

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A000040, A001221, A001359, A052147, A064909, A064910, A064911.

Table[Length[FactorInteger[Prime[n] + 2]], {n, 1, 50}] (* G. C. Greubel, May 12 2017 *)

a(n) = omega(prime(n) + 2); \\ Amiram Eldar, Sep 16 2024

a(n) = A001221(A052147(n)). - Amiram Eldar, Sep 16 2024

A180172 a(n) = gcd(prime(n)+2, n).

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 11, 3, 1, 1, 1, 1, 1, 9, 1, 1, 3, 1, 1, 1, 1, 1, 3, 1, 1, 5, 1, 1, 1, 1, 1, 9, 1, 1, 13, 5, 1, 3, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 5, 11, 1, 1, 1, 1, 71, 1, 1, 1, 3, 1, 1, 3, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 15, 7, 1, 3, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1

Offset: 1

Zak Seidov, Aug 15 2010

nonn

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A003989, A052147, A084309, A084310, A180173.

[GCD(n,NthPrime(n) +2): n in [1..110]]; // G. C. Greubel, Mar 12 2023

Mathematica
```
Table[GCD[n,Prime[n]+2],{n,200}]
```

[gcd(nth_prime(n) + 2, n) for n in range(1,111)] # G. C. Greubel, Mar 12 2023

A329727 Numbers k such that k^3 +- 2 and k +- 2 are prime.

Original entry on oeis.org

129, 1491, 1875, 2709, 5655, 6969, 10335, 14325, 14421, 17319, 26559, 35109, 37509, 43719, 50229, 52629, 101871, 102795, 104325, 105501, 120429, 127599, 132699, 136395, 137829, 157521, 172425, 173685, 179481, 186189, 191829, 211371, 219681, 221199, 229215, 234195

Offset: 1

K. D. Bajpai, Nov 19 2019

nonn

All terms in this sequence are divisible by 3.

			a(1) = 129:
  129^3 + 2 = 2146691;
  129^3 - 2 = 2146687;
  129   + 2 =     131;
  129   - 2 =     127; all four results are prime.
a(2) = 1491:
  1491^3 + 2 = 3314613773;
  1491^3 - 2 = 3314613769;
  1491   + 2 =       1493;
  1491   - 2 =       1489; all four results are prime.

Daniel Starodubtsev, Table of n, a(n) for n = 1..10000

Intersection of A038599, A067200, and A087679.

Cf. A040976, A052147, A090121, A268043, A268186.

[k:k in [1..250000]|forall{m:m in [-2,2]|IsPrime(k+m) and IsPrime(k^3+m)}]; // Marius A. Burtea, Nov 20 2019

Select[Range[500000], PrimeQ[#^3 + 2] && PrimeQ[#^3 - 2] && PrimeQ[# + 2] && PrimeQ[# - 2] &]

isok(k) = isprime(k-2) && isprime(k+2) && isprime(k^3-2) && isprime(k^3+2); \\ Michel Marcus, Nov 24 2019

list(lim)=my(v=List(),p=127,k); forprime(q=131,lim+2,if(q-p==4 && isprime((k=p+2)^3-2) && isprime(k^3+2), listput(v,k)); p=q); Vec(v) \\ Charles R Greathouse IV, May 06 2020

A049563 a(n) = ((prime(n)-1)! + 1) mod (prime(n) + 2).

Original entry on oeis.org

2, 3, 4, 1, 7, 1, 10, 1, 1, 16, 1, 1, 22, 1, 1, 1, 31, 1, 1, 37, 1, 1, 1, 1, 1, 52, 1, 55, 1, 1, 1, 1, 70, 1, 76, 1, 1, 1, 1, 1, 91, 1, 97, 1, 100, 1, 1, 1, 115, 1, 1, 121, 1, 1, 1, 1, 136, 1, 1, 142, 1, 1, 1, 157, 1, 1, 1, 1, 175, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 211, 1, 217, 1, 1, 1, 1, 1

Offset: 1

Labos Elemer

nonn

Residue of (prime(n)-1)!+1 modulo prime(n)+2.

			a(3) = 4 since prime(3) = 5, and 4! + 1 = 25 gives residue 4 when divided by prime(3) + 2 = 7.

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

Cf. A052147, A060371.

[(Factorial(p-1)+1) mod (p+2): p in PrimesUpTo(500)]; // Bruno Berselli, Apr 10 2015

Table[Mod[(Prime[k] - 1)! + 1, Prime[k] + 2], {k, 1, 200}]

a(n) = ((prime(n)-1)! + 1) % (prime(n) + 2); \\ Michel Marcus, May 28 2018

[Mod(factorial(p-1)+1,p+2) for p in primes(500)] # Bruno Berselli, Apr 10 2015

a(n) = A060371(n) mod A052147(n). - Amiram Eldar, Mar 13 2025

A132705 For an integer n with prime factorization (p_1)(p_2)(p_3)* ... (p_k), a(n) = (p_1+2)(p_2+2)(p_3+2) ... *(p_k+2).

Original entry on oeis.org

2, 3, 4, 5, 16, 7, 20, 9, 64, 25, 28, 13, 80, 15, 36, 35, 256, 19, 100, 21, 112, 45, 52, 25, 320, 49, 60, 125, 144, 31, 140, 33, 128, 65, 76, 63, 400, 49, 84, 75, 448, 43, 180, 45, 208, 175, 100, 49, 1280, 81, 196, 95, 240, 55, 500, 91, 576, 105, 124, 60

Offset: 0

Jonathan Vos Post, Nov 16 2007

a(0)=2 and a(1)=3 by convention. For an integer n with prime factorization prime(i_1)*prime(i_2)*prime(i_3)* ... *prime(i_k), a(n) = A052147(i_1)*A052147(i_2)*A052147(i_3)* ... *A052147(i_k). This sequence is to p+2 as A064478 is to p+1 for primes p.

If a(1) were 1 rather than 3, the sequence would be completely multiplicative with a(p) = p + 2. - Charles R Greathouse IV, Sep 02 2009

Cf. A000040, A003958, A003959, A064476, A064479, A052147, A064478.

from math import prod
from sympy import factorint
def A132705(n): return prod((p+2)**e for p,e in factorint(n).items()) if n!=1 else 3 # Chai Wah Wu, Mar 26 2025

A216945 Numbers k such that k-2, k^2-2, k^3-2, k^4-2 and k^5-2 are all prime.

Original entry on oeis.org

15331, 289311, 487899, 798385, 1685775, 1790991, 1885261, 1920619, 1967925, 2304805, 2479735, 3049201, 3114439, 3175039, 3692065, 4095531, 4653649, 5606349, 5708235, 6113745, 6143235, 6697425, 7028035, 7461601, 8671585, 8997121, 9260131, 10084915, 10239529

Offset: 1

Michel Lagneau, Sep 20 2012

nonn

k^6-2 is also prime for k = 1685775, 4095531, 4653649, 5606349, 13219339, 13326069, 18439561, ...

Sequence is infinite under Schinzel's Hypothesis H. a(n) >> n log^5 n. - Charles R Greathouse IV, Sep 20 2012

Cf. A052147, A028870, A038599, A154831, A154833, A154935.

Select[Range[20000000], And@@PrimeQ/@(Table[n^i-2, {i, 1, 5}]/.n->#)&]
Select[Prime[Range[680000]]+2,AllTrue[#^Range[2,5]-2,PrimeQ]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Sep 11 2020 *)

Sequence is A052147 intersection A028870 intersection A038599 intersection A154831 intersection A154833.

A334971 a(n) is the least prime p such that p+2 is divisible by n-th prime.

Original entry on oeis.org

2, 7, 3, 5, 31, 11, 83, 17, 67, 317, 29, 109, 367, 41, 139, 157, 293, 59, 199, 211, 71, 709, 911, 443, 677, 503, 101, 2459, 107, 337, 379, 653, 409, 137, 743, 149, 1097, 487, 499, 863, 1609, 179, 571, 191, 983, 197, 631, 1559, 6581, 227, 1163, 1193, 239, 751

Offset: 1

Zak Seidov, May 18 2020

nonn

			a(1) = 2 because 2+2=4 is divisible by 2 (1st prime),
a(2) = 7 because 7+2=9 is divisible by 3 (2nd prime),
a(3) = 3 because 3+2=5 is divisible by 5 (3rd prime).

Zak Seidov, Table of n, a(n) for n = 1..10000

Cf. A052147.

a(n) = my(p=2); while ((p+2) % prime(n), p=nextprime(p+1)); p; \\ Michel Marcus, May 18 2020

A384488 Numbers k having a divisor d such that d - k/d is prime.

Original entry on oeis.org

3, 4, 6, 8, 10, 12, 14, 15, 18, 20, 24, 26, 28, 30, 32, 35, 36, 38, 40, 42, 44, 48, 50, 54, 60, 62, 63, 66, 68, 70, 72, 74, 78, 80, 84, 86, 88, 90, 92, 96, 98, 99, 102, 104, 108, 110, 114, 120, 122, 126, 128, 130, 132, 138, 140, 143, 144, 146, 150, 152, 154, 158, 162, 164, 168, 170, 174, 176, 180

Offset: 1

Juri-Stepan Gerasimov, May 30 2025

nonn

Presumably, all odd terms are in A000466.

			a(6) = 12 is a term because 12 = 1*12 with 12 - 1 = 11 prime.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000466, A005408, A355643. Includes A005563 and 2 * A052147.

[k: k in [1..180] | not #[d: d in Divisors (k) | IsPrime(d-(k div d))] eq 0];

filter:= k -> ormap(d -> d^2 > k and isprime(d - k/d), numtheory:-divisors(k)):
select(filter, [$1..200]); # Robert Israel, Jun 30 2025

A384488Q[k_] := AnyTrue[Divisors[k], PrimeQ[# - k/#] &];
Select[Range[200], A384488Q] (* Paolo Xausa, Jun 30 2025 *)

isok(k) = fordiv(k, d, if (isprime(d - k/d), return(1))); \\ Michel Marcus, Jun 01 2025

cp's OEIS Frontend

A166010 a(n) = prime(n)^2-4.

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A321348 a(n) = Sum_{d|n} tau(d^n), where tau() is the number of divisors (A000005).

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A049236 a(n) is the number of distinct prime factors of prime(n) + 2.

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A180172 a(n) = gcd(prime(n)+2, n).

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A329727 Numbers k such that k^3 +- 2 and k +- 2 are prime.

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Magma

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A049563 a(n) = ((prime(n)-1)! + 1) mod (prime(n) + 2).

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Magma

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A132705 For an integer n with prime factorization (p_1)*(p_2)*(p_3)* ... *(p_k), a(n) = (p_1+2)*(p_2+2)*(p_3+2)* ... *(p_k+2).

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A132705 For an integer n with prime factorization (p_1)(p_2)(p_3)* ... (p_k), a(n) = (p_1+2)(p_2+2)(p_3+2) ... *(p_k+2).