A092402 -id:A092402 - cp's OEIS frontend

A048636 Primes of the form prime^3 + 2.

29, 127, 24391, 357913, 571789, 1442899, 5177719, 18191449, 30080233, 73560061, 80062993, 118370773, 127263529, 131872231, 318611989, 344472103, 440711083, 461889919, 590589721, 756058033, 865523179, 1095912793, 1298596573, 1341919729, 1524845953, 1697936059

Offset: 1

Enoch Haga

The first terms in the intersection with A092402, i.e., of the form p + 2^3, are (18191449, 1341919729, 2588282119, 3532642669, 16445197009, ...). - M. F. Hasler, Jan 13 2025

			a(2) = 127 = 5^3 + 2 and 5 is prime.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Vincenzo Librandi)

Cf. A048637.

Cf. A092402 (primes of the form p + 8), A321891 (union of the two); A188764 (primes of the form (product of distinct primes^3) + 2).

select(isprime, [ithprime(i)^3+2$i=1..300])[];  # Alois P. Heinz, Jan 13 2025

lst={};Do[s=Prime[n]^3;If[PrimeQ[p=s+2], AppendTo[lst, p]], {n, 6!}];lst (* Vladimir Joseph Stephan Orlovsky, Sep 26 2008 *)

forprime (p=2,1100,if(isprime(p^3+2),print1(p^3+2,", "))) \\ Hugo Pfoertner, Oct 30 2018

A092146 Primes of the form p + 10 where p is a prime.

Original entry on oeis.org

13, 17, 23, 29, 41, 47, 53, 71, 83, 89, 107, 113, 137, 149, 167, 173, 191, 233, 239, 251, 281, 293, 317, 347, 359, 383, 389, 419, 431, 443, 449, 467, 509, 557, 587, 617, 641, 653, 683, 701, 719, 743, 761, 797, 821, 839, 863, 887, 929, 947, 977, 1019, 1031

Offset: 1

Douglas Winston (douglas.winston(AT)srupc.com), Mar 31 2004

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A000040, A006512, A023203, A046132, A046117, A092402.

lst={};Do[p=Prime[n];If[PrimeQ[p=p+10],AppendTo[lst,p]],{n,6!}];lst (* Vladimir Joseph Stephan Orlovsky, Jan 29 2009 *)
Select[Prime[Range[200]]+10,PrimeQ] (* Harvey P. Dale, Aug 03 2018 *)

a(n) = 10 + A023203(n). - Alois P. Heinz, Feb 27 2020

A092216 Primes of the form p + 12 where p is a prime.

Original entry on oeis.org

17, 19, 23, 29, 31, 41, 43, 53, 59, 71, 73, 79, 83, 101, 109, 113, 139, 149, 151, 163, 179, 191, 193, 211, 223, 239, 241, 251, 263, 269, 281, 283, 293, 349, 359, 379, 401, 409, 421, 431, 433, 443, 461, 479, 491, 499, 503, 521, 569, 599, 613, 619, 631, 643, 653

Offset: 1

Douglas Winston (douglas.winston(AT)srupc.com), Apr 02 2004

nonn

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A000040, A006512, A046132, A046117, A092402, A092146.

f[n_]:=n+12; lst={};Do[r=Prime[n];p=f[r];If[PrimeQ[p],AppendTo[lst,p]],{n,6!}];lst (* Vladimir Joseph Stephan Orlovsky, Jul 04 2009 *)
Select[Prime[Range[5,120]],PrimeQ[#-12]&]  (* Harvey P. Dale, Feb 18 2011 *)

a(n) = 12 + A046133(n). - R. J. Mathar, Jun 21 2010

A156320 List of prime pairs of the form (p, p+8).

Original entry on oeis.org

3, 11, 5, 13, 11, 19, 23, 31, 29, 37, 53, 61, 59, 67, 71, 79, 89, 97, 101, 109, 131, 139, 149, 157, 173, 181, 191, 199, 233, 241, 263, 271, 269, 277, 359, 367, 389, 397, 401, 409, 431, 439, 449, 457, 479, 487, 491, 499, 563, 571, 569, 577, 593, 601, 599, 607, 653, 661, 683, 691, 701

Offset: 1

Vincenzo Librandi, Feb 08 2009

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Fernando Neres de Oliveira, On the Solvability of the Diophantine Equation p^x + (p + 8)^y = z^2 when p > 3 and p + 8 are Primes, Annals of Pure and Applied Mathematics (2018) Vol. 18, No. 1, 9-13.

Cf. A023202, A092402.

Flatten[Select[{#, # + 8} &/@Prime[Range[1000]], PrimeQ[Last[#]]&]] (* Vincenzo Librandi, Nov 01 2012 *)

from sympy import isprime, primerange
for pn in primerange(1,300):
    if isprime(pn+8):
        print(pn, pn+8)
# Stefano Spezia, Dec 06 2018

a(2n+1) = A023202(n+1). a(2n+2) = A092402(n+1). [R. J. Mathar, Feb 09 2009]

A206768 a(n) = smallest number k such that sigma(k-n) = sigma(k) - n, with k > n+1.

Original entry on oeis.org

3, 5, 5, 7, 7, 11, 81, 11, 11, 13, 13, 17, 4431, 17, 17, 19, 19, 23, 25, 23, 23, 29

Offset: 1

Paolo P. Lava, Jan 10 2013

This sequence begins
3, 5, 5, 7, 7, 11, 81, 11, 11, 13, 13, 17, 4431, 17, 17, 19, 19, 23, 25, 23, 23, 29, ?, 29, ?, 29, 29, 31, 31, 37, ?, 37, 51, 37, 37, 41, 81, 41, 41, 43, 43, 47, ?, 47, 47, 53, ?, 53, 3364, 53, 53, 59, ?, 59, ?, 59, 59, 61, 61, 67, ?, 67, ?, 67, 67, 71, ?, 71, 71, 73, 73, 79, 91, 79, ?, 79, 79, 83, ?, 83, 83, 89, ?, 89, ?, 89, 89, 101, ?, 97, ?, 97, 125, 97, 97, 101, ?, 101, 101, 103, 103, 107... where the other missing terms (designated by "?") are > 10^6, if they exist.
For a given n, n being even, among the integers k satisfying the property sigma(k-n) = sigma(k)-n, we will find prime numbers p, such that p and p-n are primes. This is because in that case sigma(p-n) = (p-n)+1 = (p+1)-n = sigma(p)-n. For instance, when n is even, for n=2 to 14, a(n) is the first term of A006512, A046132, A046117, A092402, A092146, A092216, A098933. If we restrict to composite numbers, then see A084293. - Michel Marcus, Feb 16 2013
For the missing terms mentioned in first comment, a(n) is > 10^7. - Michel Marcus, Sep 21 2013

			a(13) = 4431 because 4431 is the minimum number for which sigma(4431-13) = sigma(4418)= 6771 and sigma(4431) - 13 = 6784 -13 = 6771.
a(19) = 25 because 25 is the minimum number for which sigma(25-19) = sigma(6) = 12 and sigma(25) - 19 = 31 -19 = 12.

Cf. A015886.

A206768:=proc(q)
local k,n;
for n from 1 to q do
  for k from n+1 to q do
  if sigma(-n+k)=sigma(k)-n then print(k); break; fi;
od; od; end:
A206768(1000000000);

A307563 Numbers k such that both 6k - 1 and 6k + 7 are prime.

Original entry on oeis.org

1, 2, 4, 5, 9, 10, 12, 15, 17, 22, 25, 29, 32, 39, 44, 45, 60, 65, 67, 72, 75, 80, 82, 94, 95, 99, 100, 109, 114, 117, 120, 124, 127, 137, 152, 155, 164, 169, 172, 177, 185, 194, 199, 204, 205, 214, 215, 220, 229, 240, 242, 247, 254, 260, 262, 267, 269, 270, 289, 304, 312, 330, 334, 347, 355, 359, 369, 374, 379, 389

Offset: 1

Sally Myers Moite, Apr 14 2019

nonn

There are 140 such numbers between 1 and 1000.
These numbers correspond to all the prime pairs which differ by 8 except 3 and 11.
Numbers in this sequence are those which are not 6cd - c - d - 1, 6cd + c - d, 6cd - c + d or 6cd + c + d - 1, that is, they are not (6c - 1)d - c - 1, (6c - 1)d + c, (6c + 1)d - c or (6c + 1)d + c - 1.

			a(4) = 5, so 6(5) - 1 = 29 and 6(5) + 7 = 37 are both prime.

Robert Israel, Table of n, a(n) for n = 1..10000
Sally M. Moite, “Primeless” Sieves for Primes and for Prime Pairs Which Differ by 2m, vixra:1903.0353 (2019).

The primes are A023202, A092402, A031926.
Similar sequences for twin primes are A002822, A067611, for "cousin" primes A056956, A186243.
Intersection of A024898 and A153218.
Cf. also A307561, A307562.

select(t -> isprime(6*t-1) and isprime(6*t+7), [$1..500]); # Robert Israel, May 27 2019

isok(n) = isprime(6*n-1) && isprime(6*n+7); \\ Michel Marcus, Apr 16 2019

A276835 Numerator of a modified exponentiated von Mangoldt function defined recursively.

Original entry on oeis.org

1, 2, 3, 2, 5, 1, 7, 1, 3, 1, 11, 1, 13, 1, 1, 6, 17, 3, 19, 4, 1, 3, 23, 90, 5, 5, 3, 3, 29, 12, 31, 112, 3, 105, 1, 50, 37, 5, 1, 27, 41, 81, 43, 10, 1, 105, 47, 539, 7, 77, 15, 4, 265, 2, 3, 520, 3, 351, 59, 945

Offset: 1

Mats Granvik, Sep 20 2016

Conjecture: For n>3: If and only if the ratio A276835(n)/A276836(n) is equal to n then n is equal to the greater of the twin primes A006512.
Justification: Whenever n is equal to the greater of the twin primes then in the recurrence that defines the table t(n,k) at k=1 the Product_{i=1..n-1} t(n,k+i)=1, and Product_{i=1..n-1} t(n-2,k+i) = 1 because by definition of a prime the only divisors are 1 (at n=k in table t(n,k)) and the prime itself (at k=1 in the table t(n,k)) and thereby n/Product_{i=1..n-1}t(n,k+i)/Product_{i=1..n-1}t(n-2,k+i) = n. Since the exponentiated von Mangoldt function is the unique arithmetic function such that when multiplied over the divisors, is equal to n, and since the exponentiated von Mangoldt function is equal to n at prime numbers only, and since at n not equal to the greater of the twin primes the modified recurrence for the exponentiated von Mangoldt function by recursion messes with the output so much that the output cannot possibly be equal to n at any other numbers than at n equal to the greater of the twin primes.
Setting x = 1 gives ratios A276835(n)/A276836(n) equal to n when n is equal to the greater of the twin primes A006512.
Setting x = 2 gives ratios A276835(n)/A276836(n) equal to n when n is equal to A046132.
Setting x = 3 gives ratios A276835(n)/A276836(n) equal to n when n is equal to A046117.
Setting x = 4 gives ratios A276835(n)/A276836(n) equal to n when n is equal to A092402, and so on.

			The ratio A276835/A276836 starts: 1, 2, 3, 2, 5, 1/2, 7, 1/3, 3, 1/4, 11/3, 1/5, 13,...
The greater twin primes A006512 start: 5,7,13,... where the ratio is equal to n.

Robert G. Wilson v, Table of n, a(n) for n = 1..1000

Cf. A276836, A006512, A014963.

Clear[t, x]; (*setting x=1 gives ratio equal to n when n is the greater of the twin primes, x=2 gives ratio equal to n when n is the greater of the cousin primes and so on.*) x = 1; nn = 60; t[1, 1] = 1; t[n_, k_] := t[n, k] = If[k == 1, n/Product[t[n - 2*x, k + i], {i, 1, n - 2*x}]/Product[t[n, k + i], {i, 1, n - 1}], If[Mod[n, k] == 0, t[n/k, 1], 1], 1]; Monitor[a = Table[t[n, 1], {n, 1, nn}];, n]; Numerator[a] (* Mats Granvik, Sep 20 2016, Sep 29 2016 *)

From Mats Granvik, Sep 20 2016, Sep 29 2016: (Start)
Recurrence for the ratio A276835(n)/A276836(n):
Let:
x = 1;
T(1, 1) = 1;
T(n, k) = If k = 1 then n/Product_{i=1..n-2*x}(T(n-2*x, k + i))/Product_{i=1..n-1}(T(n, k + i)) else if Mod(n, k) = 0 then T(n/k, 1) else 1 else 1.
Then A276835(n)/A276836(n) = T(n,1). (End)

A321891 Prime numbers of the form p^3 + q, where p and q are primes.

Original entry on oeis.org

11, 13, 19, 29, 31, 37, 61, 67, 79, 97, 109, 127, 139, 157, 181, 199, 241, 271, 277, 367, 397, 409, 439, 457, 487, 499, 571, 577, 601, 607, 661, 691, 709, 727, 751, 769, 829, 919, 937, 991, 1021, 1039, 1069, 1117, 1171, 1201, 1231, 1237, 1291, 1297, 1327, 1381

Offset: 1

Pierandrea Formusa, Nov 20 2018

nonn

For reasons of parity, either p or q must be equal to 2, so this actually is the union of (mostly) "primes of the form p + 8" (A092402) and (rarely) "primes of the form p^3 + 2" (A048636 = 29, 127, 24391, 357913, ...). - M. F. Hasler, Jan 13 2025

Except for 13, these primes are the minimum or maximum prime numbers of the respective decade. - Davide Rotondo, Jan 31 2025

			37 is prime and 37 = 2^3 + 29, where 2 and 29 are primes, therefore 37 is a term.

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

Union of A048636 and A092402. - Michel Marcus, Nov 21 2018

N:= 2000: # to get terms <= N
A1:= select(t -> isprime(t) and isprime(t-8), {11,seq(i,i=13 ..N,6)}):
v:= floor((N-2)^(1/3)):
B:= select(t -> isprime(t) and isprime(t^3+2), {3,seq(i,i=5..v,6)}):
sort(convert(A1 union map(t -> t^3+2,B), list)); # Robert Israel, Mar 05 2020

nmax=4; Select[Union[Prime[Range[nmax]]^3 + 2, Prime[Range[Prime[nmax]^3]] + 8], PrimeQ] (* Amiram Eldar, Nov 21 2018 *)

include "globals.mzn";
int: n = 2;
int: max_val = 1200000;
array[1..n+1] of var 2..max_val: x;
% primes between 2..max_valset of int:
prime = 2..max_val diff { i | i in 2..max_val, j in 2..ceil(sqrt(i)) where i mod j = 0} ;
set of int: primes; primes = prime union {2};
solve satisfy;
constraint all_different(x) /\ x[1] in primes /\ x[2] in primes /\ x[3] in primes /\
pow(x[1], 3)+pow(x[2], 1)= x[3] ;
output [ show(x)]

list(lim)=my(v=List()); forprime(p=3,sqrtnint((lim\=1)-2,3), if(isprime(p^3+2), listput(v,p^3+2))); forprime(p=11,lim+8, if(isprime(p-8), listput(v,p))); Set(v) \\ Charles R Greathouse IV, Jan 13 2025

select( {is_A321891(n)=isprime(n)&& (isprime(n-8)|| (ispower(n-2, 3, &n)&&isprime(n)))}, [1..1234]) \\ M. F. Hasler, Jan 13 2025

More terms from Amiram Eldar, Nov 21 2018

A098933 Primes of the form p+14, where p is a prime.

Original entry on oeis.org

17, 19, 31, 37, 43, 61, 67, 73, 97, 103, 127, 151, 163, 181, 193, 211, 241, 271, 277, 283, 307, 331, 367, 373, 397, 433, 457, 463, 523, 571, 577, 601, 607, 613, 631, 661, 673, 691, 733, 757, 787, 811, 823, 853, 877, 967, 991, 997, 1033, 1063, 1117, 1123, 1201

Offset: 1

Douglas Winston (douglas.winston(AT)srupc.com), Oct 20 2004

Cf. A000040 A006512 A067829 A094743 A046132 A067830 A046117 A067831 A092402 A092146 A092216.

isok(n) = isprime(n) && isprime(n - 14) \\ Michel Marcus, Jul 17 2013

cp's OEIS Frontend

A048636 Primes of the form prime^3 + 2.

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A092146 Primes of the form p + 10 where p is a prime.

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A092216 Primes of the form p + 12 where p is a prime.

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A156320 List of prime pairs of the form (p, p+8).

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A206768 a(n) = smallest number k such that sigma(k-n) = sigma(k) - n, with k > n+1.

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A307563 Numbers k such that both 6k - 1 and 6k + 7 are prime.

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A276835 Numerator of a modified exponentiated von Mangoldt function defined recursively.

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A321891 Prime numbers of the form p^3 + q, where p and q are primes.

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