A045309 -id:A045309 - cp's OEIS frontend

A034934 Numbers k such that (3*k + 1)/2 is prime.

1, 3, 7, 11, 15, 19, 27, 31, 35, 39, 47, 55, 59, 67, 71, 75, 87, 91, 99, 111, 115, 119, 127, 131, 151, 155, 159, 167, 171, 175, 179, 187, 195, 207, 211, 231, 235, 239, 255, 259, 267, 279, 287, 295, 299, 307, 311, 319, 327, 335, 339, 347, 371, 375, 379, 391

Offset: 1

Jud McCranie

nonn

Related to hyperperfect numbers of a certain form.

The formula by Jaroslav Krizek is explained as follows: If p = (3n+1)/2 is prime, then it is an integer, and p must be of the form p = 3m-1, i.e., p = A003627(k). On the other hand, if p = A003627(k), then all k < p are coprime to p, so we have B(p) = (Sum_{kM. F. Hasler, Nov 29 2010

			a(6) = 19 because for A003627(6) = 29, B(29) = A053818(29)/A023896(29) = 7714/406 = 19. Cf. A179871-A179891, A003627, A007645. - _Jaroslav Krizek_, Aug 01 2010

Amiram Eldar, Table of n, a(n) for n = 1..10000
Judson S. McCranie, A study of hyperperfect numbers, J. Int. Seq., Vol. 3 (2000), Article 00.1.3.

Cf. A003627, A007645, A038536, A045309, A175505, A179871-A179891.

[ n: n in [1..400 by 2] | IsPrime((3*n+1) div 2) ];

Select[Range[500], PrimeQ[(3# + 1)/2] &] (* Harvey P. Dale, Jan 15 2011 *)

is(n)=isprime((3*n+1)/2) \\ Charles R Greathouse IV, Feb 20 2017

a(n) = A175505(A003627(n)). - Jaroslav Krizek, Aug 01 2010

Corrected by Vincenzo Librandi, Mar 24 2010

A128287 Nonprime numbers k such that k divides A014137(k).

Original entry on oeis.org

1, 8, 133, 49378

Offset: 1

Alexander Adamchuk, Feb 23 2007

Prime p divides A014137(p) for p in A045309 (primes congruent to {0, 2} mod 3).

a(5) > 5000000. - Chai Wah Wu, Nov 13 2014

			1 is nonprime and divides A014137(1) = 2, so 1 is a term.
8 is nonprime and divides A014137(8) = 2056, so 8 is a term.

Cf. A014137, A000108, A045309, A045309.

s = 1; Do[s = s + (2n)!/n!/(n+1)!; If[ !PrimeQ[n] && Mod[s, n] == 0, Print[n]], {n, 1000}]

from _future_ import division
from sympy import isprime
A128287_list, x, s = [1], 1, 2
for i in range(2,10**5):
    x = x*(4*i-2)//(i+1)
    s += x
    if not (isprime(i) or s % i):
        A128287_list.append(i) # Chai Wah Wu, Nov 13 2014

One more term from Ryan Propper, Apr 02 2007

A169618 Table with T(n,k) = the number of ways to represent k as the sum of a square and a cube modulo n.

Original entry on oeis.org

1, 2, 2, 3, 3, 3, 6, 6, 2, 2, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 7, 11, 8, 12, 2, 6, 3, 12, 20, 4, 4, 12, 4, 4, 4, 15, 15, 6, 6, 6, 6, 6, 6, 15, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 18, 18, 6, 6, 18, 18, 6, 6, 18, 18, 6, 6, 13, 11, 18, 8, 20, 15, 6

Offset: 1

Franklin T. Adams-Watters, Dec 03 2009

The top left corner is T(1,0).
It appears that this table does not contain any 0's.
It appears that row n is constant iff n is squarefree, and no prime divisor of n is == 1 (mod 6). It is not hard to show that such rows are constant, since the cubes are equi-distributed in such moduli.

			The 6 ways to represent 0 (mod 4) are 0^2+0^3, 0^2+2^3, 1^2+3^3, 2^2+0^3, 2^2+2^3, and 3^2+3^3.

Cf. A022549, A002476, A045309.

al(n)=local(v);v=vector(n);for(i=0,n-1,for(j=0,n-1,v[(i^2+j^3)%n+1]++));v

A216061 Primes p such that p^3 + p + 1 is prime.

Original entry on oeis.org

2, 3, 5, 17, 29, 41, 53, 71, 83, 131, 179, 191, 239, 263, 311, 389, 491, 509, 557, 569, 593, 653, 701, 719, 743, 797, 821, 863, 887, 953, 971, 977, 1019, 1049, 1097, 1109, 1277, 1301, 1319, 1373, 1427, 1481, 1523, 1559, 1601, 1607, 1613, 1667, 1787, 1823

Offset: 1

César Eliud Lozada, Aug 31 2012

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Cf. A053182.

Subsequence of A045309.

[p: p in PrimesUpTo(2000) | IsPrime(p^3+p+1)]; // Bruno Berselli, Sep 01 2012

A := {}; for n to 1000 do p := ithprime(n); if isprime(p^3+p+1) then A := `union`(A, {p}) end if end do; A := A

Select[Prime[Range[400]], PrimeQ[#^3 + # + 1] &] (* Bruno Berselli, Sep 01 2012 *)

A308470 a(n) = (gcd(phi(n), 4*n^2 - 1) - 1)/2, where phi is A000010, Euler's totient function.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 2, 0, 1, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 7, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 4, 7, 0, 1, 0, 0, 2, 0, 0, 0, 1, 3, 2, 0, 0, 1, 0, 2, 0, 4

Offset: 1

Juri-Stepan Gerasimov, May 29 2019

nonn

2*a(n) + 1 = gcd(phi(2*n), (2*n - 1)*(2*n + 1)).
a(A000040(n)) = A099618(n).
Records occur at n = 1, 7, 22, 62, 172, 213, 372, 427, 473, ...

			a(7) = 1 because (gcd(phi(7), 4*7^2 - 1) - 1)/2 = (gcd(6, 195) - 1)/2 = (3 - 1)/2 = 1.

Cf. A000010, A000040, A002476, A045309, A099618.

[(Gcd(EulerPhi(n),4*n^2-1)-1)/2: n in [1..95]];

Table[(GCD[EulerPhi[n], 4n^2 - 1] - 1)/2, {n, 100}] (* Alonso del Arte, May 30 2019 *)

from math import gcd
def A000010(n):
    if n == 1:
        return 1
    d, m = 1, 0
    while d < n:
        if gcd(d,n) == 1:
            m = m+1
        d = d+1
    return m
n = 0
while n < 30:
    n = n+1
    print(n,(gcd(A000010(n),4*n**2-1)-1)//2) # A.H.M. Smeets, Aug 18 2019

a(A000040(n)) = A099618(n).
a(A002476(n)) = 1.
a(A045309(n)) = 0.

A370268 Intersection of A189715 and A370267.

Original entry on oeis.org

1, 4, 6, 7, 9, 10, 15, 16, 22, 24, 25, 28, 31, 33, 36, 40, 42, 49, 54, 55, 58, 60, 63, 64, 70, 73, 79, 81, 87, 88, 90, 96, 97, 100, 103, 105, 106, 112, 118, 121, 124, 127, 132, 135, 144, 145, 150, 151, 154, 159, 160, 166, 168, 169, 175, 177, 186, 193, 196, 198, 199, 202, 214, 216, 217, 220, 223, 225, 231, 232, 240, 241, 247

Offset: 1

Peter Munn, Feb 13 2024

A189715 and A370267 are closely related in that they may be generated by the same process, but starting from numbers of the form 6m+1 and 8m+1 respectively - see A370267 for details.
Independent definition: numbers with an even number of prime factors not of the form 3m+1 and an even number of prime factors not of the form 8m+-1 (counting repetitions).
The sequence starts with the first 72 nonzero numbers of the form x^2 + 6y^2 (see A002481). After the absence of 0, this sequence next differs from A002481 by including 247, 391, 442, ... . From these early intermittent differences, the densities of the two sequences diverge progressively, driven by the absence from A002481 of many of the squarefree composite numbers that are present here though their prime factors are not. (Both sequences are closed under multiplication.) Asymptotic densities are 1/4 and 0 respectively.
Likewise, if we list the even terms halved, we find a similar relationship to the nonzero terms of A002480. The first 66 terms match, then we find we have generated intermittent extra terms: 221, 299, 323, ... .
Numbers whose squarefree part is congruent to {1,7} mod 24, {10,22} mod 48, {15,33} mod 72, or {6,42} mod 144. (Each congruence describes a coset of A334832 under A059897(.,.) as described in A334832. This sequence corresponds to the subgroup of the quotient group generated by {6,7,10}.)

Intersection of A189715 and A370267.
A002481\{0}, A334832 are subsequences.
Cf. A002480, A042999, A045309, A059897.

isok(k) = {c = core(k); c%24 == 1 || c%24 == 7 || c%48 == 10 || c%48 == 22 || c%72 == 15 || c%72 == 33 || c%144 == 6 || c%144 == 42}

{a(n) : n >= 1} = {A059897(i,j*k) : i in A334832, j in {1,7}, k in {1,6,10,15}}.

cp's OEIS Frontend

A034934 Numbers k such that (3*k + 1)/2 is prime.

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A128287 Nonprime numbers k such that k divides A014137(k).

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A169618 Table with T(n,k) = the number of ways to represent k as the sum of a square and a cube modulo n.

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A216061 Primes p such that p^3 + p + 1 is prime.

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A308470 a(n) = (gcd(phi(n), 4*n^2 - 1) - 1)/2, where phi is A000010, Euler's totient function.

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A370268 Intersection of A189715 and A370267.

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