A082539 -id:A082539 - cp's OEIS frontend

A084196 Number of primes q

0, 0, 1, 1, 3, 0, 2, 1, 5, 2, 2, 0, 3, 1, 6, 3, 6, 0, 1, 7, 0, 3, 6, 4, 1, 2, 2, 6, 0, 3, 3, 5, 2, 3, 3, 3, 0, 1, 9, 2, 9, 1, 8, 0, 3, 3, 1, 4, 6, 0, 3, 11, 0, 8, 2, 8, 6, 3, 0, 2, 1, 5, 3, 7, 0, 2, 1, 0, 5, 1, 2, 13, 2, 0, 3, 10, 3, 0, 2, 0, 11, 0, 11, 2, 5, 5, 6, 0, 4, 2, 6, 13, 2, 5, 2, 13, 4, 4, 1, 0, 1, 4

Offset: 1

Reinhard Zumkeller, May 18 2003

nonn

a(A049084(A082539(n)))=0, a(A049084(A084197(n)))>0, a(A049084(A084198(n)))=1;

			n=5, prime(5)=11: (11+1) mod (q+1) = 0 for 3 primes q<11: 2, 3,
and 5, therefore a(5)=3.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A084198, A084200.

a084196 n = a084196_list !! (n-1)
a084196_list = f [] a000040_list where
   f ps' (p:ps) = length [q | q <- ps', mod (p + 1) (q + 1) == 0] :
                  f (p : ps') ps where
-- Reinhard Zumkeller, Jan 06 2014

Table[Count[Mod[p+1,Prime[Range[PrimePi[p]-1]]+1],0],{p,Prime[Range[110]]}] (* Harvey P. Dale, Aug 11 2023 *)

A084200 LeastCommonMultiple{q+1: q prime, q < prime(n), q+1 divides prime(n)+1}.

Original entry on oeis.org

1, 1, 3, 4, 12, 1, 6, 4, 24, 6, 8, 1, 42, 4, 24, 18, 60, 1, 4, 72, 1, 40, 84, 90, 14, 6, 8, 108, 1, 114, 32, 132, 6, 140, 30, 152, 1, 4, 168, 6, 180, 14, 96, 1, 18, 40, 4, 224, 228, 1, 18, 240, 1, 252, 6, 264, 270, 136, 1, 6, 4, 294, 308, 312, 1, 6, 4, 1, 348, 14, 6, 360, 8, 1

Offset: 1

Reinhard Zumkeller, May 18 2003

nonn

Matthew House, Table of n, a(n) for n = 1..10000

Cf. A084196, A082539, A084197.

from math import lcm
from sympy import sieve
def a(n):
    pn = sieve[n]
    return lcm(*[q+1 for q in sieve.primerange(2, pn) if (pn+1)%(q+1)==0])
print([a(n) for n in range(1, 100)]) # Michael S. Branicky, Aug 13 2024

Definition and terms corrected by Matthew House, Aug 13 2024

A084197 Primes p such that there exists at least one prime q

Original entry on oeis.org

5, 7, 11, 17, 19, 23, 29, 31, 41, 43, 47, 53, 59, 67, 71, 79, 83, 89, 97, 101, 103, 107, 113, 127, 131, 137, 139, 149, 151, 163, 167, 173, 179, 181, 191, 197, 199, 211, 223, 227, 233, 239, 251, 257, 263, 269, 271, 281, 283, 293, 307, 311, 317, 331, 347, 349, 353

Offset: 1

Reinhard Zumkeller, May 18 2003

nonn

A084196(A049084(a(n)))>0.

Matthew House, Table of n, a(n) for n = 1..10000

Cf. A082539, A084198, A084200.

Corrected by T. D. Noe, Oct 25 2006

A164623 Primes p such that p(p-1)/2-5 and p(p-1)/2+5 are also prime numbers.

Original entry on oeis.org

13, 157, 673, 1069, 1117, 1153, 1213, 1597, 2029, 2089, 2437, 2713, 2833, 3613, 4057, 4909, 5653, 6337, 6529, 7549, 7993, 8053, 9613, 10789, 11497, 11689, 12073, 12373, 13309, 13669, 13789, 14173, 15289, 15937, 16249, 18097, 18637, 19249, 19993

Offset: 1

Vladimir Joseph Stephan Orlovsky, Aug 17 2009

Primes A000040(k) such that A008837(k)+-5 are also prime numbers.

			13 is in the sequence because 13*6-5=73 and 13*6+5=83 are both prime.

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

Cf. A008846, A020882, A068228, A068229, A082539, A086519, A107159, A158708, A139494, A164620, A164621, A164622, A023203.

Select[Prime[Range[2300]], PrimeQ[# (# - 1)/2 - 5] && PrimeQ[# (# - 1)/2 + 5] &]

forprime(p=2,10^6,my(b=binomial(p,2));if(isprime(b-5)&isprime(b+5),print1(p,", "))); /* Joerg Arndt, Apr 10 2013 */

Edited by R. J. Mathar, Aug 20 2009

Mathematica code adapted to the definition by Bruno Berselli, Apr 10 2013

A300984 Numbers whose sum of squarefree divisors and sum of nonsquarefree divisors are both squarefree numbers.

Original entry on oeis.org

676, 1352, 2704, 5408, 5476, 8788, 10816, 10952, 14884, 21316, 21632, 21904, 29768, 35152, 42632, 43264, 43808, 59536, 70304, 85264, 86528, 95048, 114244, 119072, 140608, 148996, 170528, 173056, 175232, 190096, 202612, 209764, 228488, 238144, 262088, 281216

Offset: 1

Michel Lagneau, Mar 17 2018

nonn

Conjecture: a(n) is of the form a(n) = 2^i*p^j with i, j integers and p prime. This has been verified for n up to 10^7.
Observation: For n < = 10^7, p belongs to the set E = {13, 37, 61, 73, 109, 157, 181, 193, 229, 277, 313, 373, 397, 409, 421, 433, 457, 541, 601, 613, 661, 673, 709, 733, 757, 769, 829, 853, 877, 997, 1009, 1021, 1033, 1069, 1093, 1117, 1129, 1153, 1201, 1213, 1237, 1297, 1381, 1429, 1453, 1489}. We observe that E minus {181, 433, 601, 769, 853, 1021, 1429} belongs to A082539.
Generalization: For n <= 10^m with m > 7, it is conjectured that a majority of primes p where a(n) = 2^i*p^j are in A082539. For example, with m = 7, 84% of the primes p are in A082539.

			676 is in the sequence because A048250(676) = 42 = 2*3*7 and A162296(676) = 1239 = 3*7*59 are both squarefree numbers.

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

Cf. A013929, A048250, A082539, A162296.

lst={};Do[If[SquareFreeQ[Total[Select[Divisors[n],SquareFreeQ]]]&& SquareFreeQ[DivisorSigma[1,n]-Total[Select[Divisors[n],SquareFreeQ]]],AppendTo[lst,n]],{n,300000}];lst

isok(n) = my(sd = sumdiv(n,d,d*issquarefree(d))); issquarefree(sd) && issquarefree(sigma(n) - sd); \\ Michel Marcus, Mar 17 2018

cp's OEIS Frontend

A084196 Number of primes q

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Mathematica

A084200 LeastCommonMultiple{q+1: q prime, q < prime(n), q+1 divides prime(n)+1}.

Views

Author

Keywords

Links

Crossrefs

Programs

Python

Extensions

A084197 Primes p such that there exists at least one prime q

Views

Author

Keywords

Comments

Links

Crossrefs

Extensions

A164623 Primes p such that p*(p-1)/2-5 and p*(p-1)/2+5 are also prime numbers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A300984 Numbers whose sum of squarefree divisors and sum of nonsquarefree divisors are both squarefree numbers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A164623 Primes p such that p(p-1)/2-5 and p(p-1)/2+5 are also prime numbers.