Marina Ibrishimova - cp's OEIS frontend

A276290 Products of odd primes p and q such that either p or q is in the trajectory of (p*q)+1 under the Collatz 3x+1 map (A014682).

25, 35, 55, 65, 77, 85, 95, 115, 133, 143, 145, 155, 161, 185, 203, 205, 209, 215, 217, 235, 253, 259, 265, 287, 295, 305, 329, 341, 355, 365, 371, 391, 395, 403, 407, 415, 427, 437, 445

Offset: 1

Marina Ibrishimova, Aug 27 2016

nonn

Conjecture: If n is the product of two odd primes p and q and p is equal to 3, then neither p nor q is in the trajectory of (p*q)+1 under the Collatz 3x+1 map (A014682). - Marina Ibrishimova, Aug 29 2016

If there were any multiples of three present in this sequence, then there would also be nontrivial cycles among Collatz-trajectories. It has been empirically checked that for the first 2^22 = 4194304 primes from p=2 to p=71378569, 3*p certainly is not included in this sequence. - Antti Karttunen, Aug 30 2016

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Index entries for sequences related to 3x+1 (or Collatz) problem

Cf. A014682, A065091, A276260.

Subsequence of A046315.

function isitCollatzProduct(p,q){var n=p*q;var cur=n+1;while(cur!=p&&cur!=q&&cur!=2){if(cur%2!=0){cur=3*cur+1}else{cur=cur/2}}if(cur==p||cur==q){return cur}else{return 0}}

Select[Range[9, 450, 2], And[PrimeOmega@ # == 2, Function[w, Total@ Boole@ Map[MemberQ[NestWhileList[If[EvenQ@ #, #/2, 3 # + 1] &, Times @@ w + 1, # > 1 &], #] &, w] > 0]@ Flatten@ Apply[Table[#1, {#2}] &, FactorInteger@ #, {1}]] &] (* Michael De Vlieger, Aug 28 2016 *)

has(p,q)=my(t=p*q+1); while(t>2, t=if(t%2,3*t+1,t/2); if(t==p || t==q, return(1))); 0
list(lim)=forprime(p=3,lim\3, forprime(q=3,min(lim\p,p), if(has(p,q), listput(v,p*q)))); Set(v) \\ Charles R Greathouse IV, Aug 27 2016

Terms corrected by Charles R Greathouse IV, Aug 27 2016

A276260 Odd primes p such that p is in the trajectory of p+1 under the Collatz 3x+1 map (A014682).

Original entry on oeis.org

5, 13, 17, 53, 61, 107, 251, 283, 1367

Offset: 1

Marina Ibrishimova, Aug 26 2016

a(10) > 10^7 if it exists. - Felix Fröhlich, Aug 26 2016
a(10) > 10^9 if it exists. - Charles R Greathouse IV, Aug 26 2016
a(10) > 10^12 if it exists. - Charles R Greathouse IV, Sep 07 2016

Index entries for sequences related to 3x+1 (or Collatz) problem

Cf. A014682, A276290.

function isit_collatz_prime(p)
{
    var cur = p+1;
    while(cur != p && cur != 2)
    {
       if(cur%2!=0)
       {
           cur = 3*cur + 1;
       }else
       {
        cur = cur/2;
       }
    }
    if(cur === p ){return "p is a Collatz prime";}
    else {return "p is not a Collatz prime";}
}

Select[Prime@ Range[2, 10^5], MemberQ[NestWhileList[If[EvenQ[#], #/2, 3 # + 1] &, # + 1, # > 1 &], #] &] (* Michael De Vlieger, Aug 26 2016 *)

next_collatz_iteration(n) = if(n%2==1, return(3*n+1), return(n/2))
is(n) = if(n%2==1 && ispseudoprime(n), my(k=n+1); while(k > 1, k=next_collatz_iteration(k); if(k==n, return(1)))); 0 \\ Felix Fröhlich, Aug 26 2016

has(n)=my(k=n+1); k>>=valuation(k,2); while(k>1, k+=(k+1)>>1; k>>=valuation(k,2); if(k==n, return(1))); 0
forprime(p=3,1e9, if(has(p), print1(p", "))) \\ Charles R Greathouse IV, Aug 26 2016

A269452 phi(A157352(n)), n >= 1, where phi is Euler's totient function A000010, and A157352 gives the products of two distinct safe primes.

Original entry on oeis.org

24, 40, 60, 88, 132, 184, 220, 232, 276, 348, 328, 460, 424, 492, 580, 636, 664, 712, 820, 1012, 904, 996, 1060, 1068, 1048, 1276, 1356, 1384, 1432, 1660, 1572, 1804, 1528, 1780, 1864, 1912, 2076, 2332, 2260, 2148, 2008, 2292, 2668, 2248, 2620, 2344, 2796, 2868, 3012, 2872, 3460, 3652, 3772, 3372

Offset: 1

Marina Ibrishimova, Feb 27 2016

nonn

phi(p*q) = (p-1)(q-1) where p, q are distinct safe primes.

2^(a(n)/2) == 1 (mod A157352(n)). For the reference see a comment on A269454. - Wolfdieter Lang, Mar 31 2016

Vincenzo Librandi, Table of n, a(n) for n = 1..1500

Cf. A000010, A157352, A269454.

EulerPhi /@ Select[Select[Range@ 4000, PrimeNu@ # == 2 &], Times @@ Map[If[PrimeQ[(# - 1)/2], #, 0] &, Map[First, FactorInteger@ #]] == # &] (* Michael De Vlieger, Feb 28 2016 *)

a(n) = phi(A157352(n)), n >= 1.

More terms from Michael De Vlieger, Feb 28 2016

A269454 Safe primes that are not congruent to -1 mod 8.

Original entry on oeis.org

5, 11, 59, 83, 107, 179, 227, 347, 467, 563, 587, 1019, 1187, 1283, 1307, 1523, 1619, 1907, 2027, 2099, 2459, 2579, 2819, 2963, 3203, 3467, 3779, 3803, 3947, 4139, 4259, 4283, 4547, 4787, 5099, 5387, 5483, 5507, 5939, 6659, 6779, 6827, 6899, 7187, 7523

Offset: 1

Marina Ibrishimova, Feb 27 2016

nonn

For safe primes see A005385.
Conjecture: If p and q are two distinct safe primes not congruent to -1 mod 8 then the order of 2 mod p*q is phi(p*q)/2. For phi see A000010.
Note: The order of 2 mod p*q is the smallest positive integer k such that 2^k = 1 mod p*q. See Rosen's definition of the order of an integer on p.334. Also, k is smaller than or equal to phi(p*q)/2 for all products of distinct odd primes p and q. See Cohen's Prop. 1.4.2 on p. 25.
2^(phi(p*q)/2) == 1 (mod p*q) for all distinct odd primes p and q. See Nagell's corollary to Theorem 64, p. 106, with a = 2 and n = p*q. - Wolfdieter Lang, Mar 31 2016

Henri Cohen, Graduate Texts In Mathematics: A Course in Computational Algebraic Number Theory, Springer, 2000, p. 25
Trygve Nagell, Introduction to Number Theory, Chelsea, 1964, p. 106.
Kenneth H. Rosen, Elementary Number Theory And Its Applications, AT&T Laboratories, 2005, p. 334

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

Cf. A005385, A007522.

[ p: p in PrimesUpTo(8000) | IsPrime((p-1) div 2) and not p mod 8 eq 7]; // Vincenzo Librandi, Feb 28 2016

Select[Prime@ Range@ 1000, And[PrimeQ[(# - 1)/2], MemberQ[Range[0, 6], Mod[#, 8]]] &] (* Michael De Vlieger, Feb 28 2016 *)

lista(nn) = {forprime(p=3, nn, if (((p % 8) != 7) && isprime((p-1)/2), print1(p, ", ")););} \\ Michel Marcus, Mar 24 2016

A005385 without its intersection with A007522.

More terms from Vincenzo Librandi, Feb 28 2016

A269453 The order of 2 mod m when m is the product of two distinct safe primes.

Original entry on oeis.org

12, 20, 30, 44, 33, 92, 110, 116, 69, 174, 164, 230, 212, 246, 290, 318, 332, 356, 410, 253, 452, 249, 530, 534, 524, 638, 678, 692, 716, 830, 393, 902, 764, 890, 932, 956, 1038, 1166, 1130, 537, 1004, 573, 1334, 1124, 1310, 1172, 1398, 717, 753, 1436, 1730, 913, 1886, 1686, 1790

Offset: 1

Marina Ibrishimova, Feb 27 2016

nonn

The smallest positive integer k for which 2^k == 1 (mod m) where m = p*q with p, q distinct safe primes.

Cf. A002326, A157352, A269452.

MultiplicativeOrder[2, #] & /@ Select[Select[Range@ 4200, PrimeNu@ # == 2 &], Times @@ Map[If[PrimeQ[(# - 1)/2], #, 0] &, Map[First, FactorInteger@ #]] == # &] (* Michael De Vlieger, Feb 28 2016 *)

a(n) is the order of 2 modulo A157352(n). - Wolfdieter Lang, Mar 31 2016

More terms from Michael De Vlieger, Feb 28 2016

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User: Marina Ibrishimova

A276290 Products of odd primes p and q such that either p or q is in the trajectory of (p*q)+1 under the Collatz 3x+1 map (A014682).

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A276260 Odd primes p such that p is in the trajectory of p+1 under the Collatz 3x+1 map (A014682).

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A269452 phi(A157352(n)), n >= 1, where phi is Euler's totient function A000010, and A157352 gives the products of two distinct safe primes.

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A269454 Safe primes that are not congruent to -1 mod 8.

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A269453 The order of 2 mod m when m is the product of two distinct safe primes.

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