A124477 -id:A124477 - cp's OEIS frontend

A198586 a(n) = (4^A001651(n+1) - 1)/3: numbers (4^k-1)/3 for k > 1, not multiples of 3.

5, 85, 341, 5461, 21845, 349525, 1398101, 22369621, 89478485, 1431655765, 5726623061, 91625968981, 366503875925, 5864062014805, 23456248059221, 375299968947541, 1501199875790165, 24019198012642645, 96076792050570581, 1537228672809129301

Offset: 1

T. D. Noe, Oct 30 2011

Numbers coprime to 6 producing 2 odd numbers in the Collatz iteration.
Numbers appearing in A198585 (sorted and duplicates removed). These numbers occur in A002450, numbers of the form (4^k-1)/3, for k = 2, 4, 5, 7, 8, 10, ... (note that k a multiple of 3 does not appear).
A124477 \ {0,1} is a subset: for these n, 3n+1 = 2^(p-3) with p > 3 prime, whence also n !== 0 (mod 3). - M. F. Hasler, Oct 16 2018
These are exactly the odd non-multiples of 3 such 3n+1 = 2^m for some m, i.e., n = (2^m-1)/3. This is possible iff m = 2k, so we get n = (4^k-1)/3. Then n == 0 (mod 3) <=> 4^k == 1 (mod 9) <=> k == 0 (mod 3) <=> k not in A001651. This yields the FORMULA. (Multiples of 3 are excluded because the original definition implied that the terms are in the Collatz-orbit of another odd number, i.e., of the form n = (3x+1)/2^r, which is impossible for x a multiple of 3.) - M. F. Hasler, Oct 16 2018
From Wolfdieter Lang, Jan 14 2022: (Start)
a(n) mod 8 = 5. As subsequence of A002450 for n >= 1.
{a(n) mod 6} == repeat{5, 1}. See the first comment, and the periodicity modulo 6 of A002450 for n >= 1.
{a(n) mod 72} == repeat{5, 13, 53, 61, 29, 37}. Proof by induction: First with the bisection formulas, a(1+2*k) = (4^(2+3*k) - 1)/3 and a(2+2*k) = (4^(3*k+4) - 1)/3, for k >= 0, then trisection, using (4^9 - 1)/3 = 873819 = 9*9709. (End)

T. D. Noe, Table of n, a(n) for n = 1..100
Index entries for linear recurrences with constant coefficients, signature (1,64,-64).

Cf. A001651, A002450, A124477, A198584.

[4^(3*n  div 2 + 1) div 3: n in [1..25]]; // Vincenzo Librandi, Oct 20 2018

e = 19; ex = Complement[Range[2,3*e], 3*Range[e]]; (4^ex - 1)/3
(* Second program: *)
Rest@ Map[(4^# - 1)/3 &, LinearRecurrence[{1, 1, -1}, {1, 2, 4}, 21]] (* Michael De Vlieger, Oct 17 2018 *)

is(n)=gcd(n,6)==1&&(n=3*n+1)>>valuation(n,2)==1 \\ M. F. Hasler, Oct 16 2018

A198586(n)=4^(3*n\2+1)\3 \\ M. F. Hasler, Oct 16 2018

Vec(x*(5 + 80*x - 64*x^2) / ((1 - x)*(1 - 8*x)*(1 + 8*x)) + O(x^20)) \\ Colin Barker, Jan 17 2020

a(n) = (4^A001651(n+1) - 1)/3. - M. F. Hasler, Oct 16 2018
From Colin Barker, Jan 17 2020: (Start)
G.f.: x*(5 + 80*x - 64*x^2) / ((1 - x)*(1 - 8*x)*(1 + 8*x)).
a(n) = a(n-1) + 64*a(n-2) - 64*a(n-3) for n>3.
a(n) = (-1 + (-8)^n + 3*8^n) / 3.
(End)

Definition corrected by M. F. Hasler, Oct 16 2018

A135659 a(n) = 24*n + 7.

Original entry on oeis.org

7, 31, 55, 79, 103, 127, 151, 175, 199, 223, 247, 271, 295, 319, 343, 367, 391, 415, 439, 463, 487, 511, 535, 559, 583, 607, 631, 655, 679, 703, 727, 751, 775, 799, 823, 847, 871, 895, 919, 943, 967, 991, 1015, 1039, 1063, 1087, 1111, 1135, 1159, 1183, 1207

Offset: 0

Artur Jasinski, Nov 25 2007

Conjecture: All Mersenne Primes (A000668) > 3 are in this sequence.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A107006, A124477, A135657, A135982.

Table[24n + 7, {n, 0, 100}]
LinearRecurrence[{2,-1},{7,31},60] (* Harvey P. Dale, Jul 14 2013 *)

From Colin Barker, Apr 02 2012: (Start)
a(n) = 2*a(n-1) - a(n-2).
G.f.: (7+17*x)/(1-x)^2. (End)
E.g.f.: (7 + 24*x)*exp(x). - G. C. Greubel, Oct 25 2016

Offset changed to 0 by Omar E. Pol, Oct 25 2016

A135982 a(n) = 2^(24n+7)-1.

Original entry on oeis.org

127, 2147483647, 36028797018963967, 604462909807314587353087, 10141204801825835211973625643007, 170141183460469231731687303715884105727

Offset: 0

Artur Jasinski, Dec 09 2007

Index entries for linear recurrences with constant coefficients, signature (16777217,-16777216).

Cf. A107006, A124477, A135657.

Mathematica
```
Table[2^(24n + 7) - 1, {n, 0, 20}]
```

a(n)=2^(24*n+7)-1 \\ Charles R Greathouse IV, Oct 16 2015

a(n) = 2^A135659(n) - 1.

G.f.: ( 127+16777088*x ) / ( (16777216*x-1)*(x-1) ). - R. J. Mathar, Apr 02 2012

A139480 a(n) = A000043(n) - 3.

Original entry on oeis.org

0, 2, 4, 10, 14, 16, 28, 58, 86, 104, 124, 518, 604, 1276, 2200, 2278, 3214, 4250, 4420, 9686, 9938, 11210, 19934, 21698, 23206, 44494, 86240, 110500, 132046, 216088, 756836, 859430, 1257784, 1398266, 2976218, 3021374, 6972590, 13466914, 20996008, 24036580, 25964948

Offset: 2

Artur Jasinski, Apr 22 2008

nonn

2^a(n)-1 is divisible by 3. For (2^a(n)-1)/3 see A124477.

For a(n)/2 see A139481.

Amiram Eldar, Table of n, a(n) for n = 2..48

Cf. A000043, A000668, A124477, A139479, A139481.

MersennePrimeExponent[Range[2, 48]] - 3 (* Amiram Eldar, Oct 17 2024 *)

Definition corrected by Omar E. Pol, May 23 2008
Edited by N. J. A. Sloane, May 23 2008
a(40)-a(42) from Amiram Eldar, Oct 17 2024

A145044 Primes p (A000043) such that 2^p-1 is prime (A000668) and congruent to 271 mod 6!.

Original entry on oeis.org

13, 61, 2281, 3217, 23209, 44497, 132049, 13466917, 30402457, 42643801

Offset: 1

Artur Jasinski, Sep 30 2008

Mersenne numbers (with the exception of the first two) are congruent to 31, 127, 271, 607 mod 6!. This sequence is a subsequence of A000043.

Cf. A000043, A000668, A124477, A139484, A145038, A112633, A145041, A145042, A145044, A145045, A145046.

Select[MersennePrimeExponent[Range[47]], PowerMod[2, #, 6!] == 272 &] (* Amiram Eldar, Mar 22 2020 *)

a(10) from Amiram Eldar, Mar 22 2020

A145045 Primes p (A000043) such that 2^p-1 is prime (A000668) and congruent to 607 mod 6!

Original entry on oeis.org

107, 86243, 756839, 25964951, 37156667

Offset: 1

Artur Jasinski, Sep 30 2008

nonn

Mersenne numbers (with the exception of the first two) are congruent to 31, 127, 271, 607 mod 6!. This sequence is a subset of A000043.

Cf. A000043, A000668, A124477, A139484, A145038, A112633, A145041, A145042, A145044, A145045, A145046.

p = {2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377, 6972593, 13466917, 20996011, 24036583, 25964951, 30402457, 32582657, 37156667, 43112609}; a = {}; Do[If[Mod[2^p[[n]] - 1, 6! ] == 607, AppendTo[a, p[[n]]]], {n, 1, Length[p]}]; a (*Artur Jasinski*)

Comment rewritten by Harvey P. Dale, Sep 02 2023

A139481 a(n) = A139480(n)/2.

Original entry on oeis.org

0, 1, 2, 5, 7, 8, 14, 29, 43, 52, 62, 259, 302, 638, 1100, 1139, 1607, 2125, 2210, 4843, 4969, 5605, 9967, 10849, 11603, 22247, 43120, 55250, 66023, 108044, 378418, 429715, 628892, 699133, 1488109, 1510687, 3486295, 6733457, 10498004, 12018290

Offset: 2

Artur Jasinski, Apr 22 2008

Amiram Eldar, Table of n, a(n) for n = 2..48

Cf. A000043, A000668, A124477, A139479, A139480, A146768.

(MersennePrimeExponent[Range[2, 48]] - 3)/2 (* Amiram Eldar, Oct 17 2024 *)

a(n) = A146768(n-1)-1. - R. J. Mathar, Mar 30 2011

Edited by N. J. A. Sloane, May 23 2008

A121290 a(n) = (2^prime(n) - 8)/24 for n>=2.

Original entry on oeis.org

0, 1, 5, 85, 341, 5461, 21845, 349525, 22369621, 89478485, 5726623061, 91625968981, 366503875925, 5864062014805, 375299968947541, 24019198012642645, 96076792050570581, 6148914691236517205

Offset: 1

Lekraj Beedassy, Aug 24 2006

Previous name was: (2^(p-3) - 1)/3, where p is an odd prime, i.e., p = A065091.

Cf. A000043, A000668, A124477, A139479, A139480.

Table[(2^Prime[n] - 8)/24, {n, 2, 100}] (* Artur Jasinski *)

New name from Joerg Arndt, Jul 21 2017

A135983 a(n)=2^A107006(n)-1.

Original entry on oeis.org

127, 2147483647, 604462909807314587353087, 10141204801825835211973625643007, 170141183460469231731687303715884105727, 2854495385411919762116571938898990272765493247

Offset: 1

Artur Jasinski, Dec 09 2007

nonn

A107006(n) are successive primes of the form 24n+7.

Cf. A107006, A124477, A135657, A135982.

p = Select[24*Range[0, 20] + 7, PrimeQ]; 2^p - 1

A139483 Numbers n such that 24n+7 is prime.

Original entry on oeis.org

0, 1, 3, 4, 5, 6, 8, 9, 11, 15, 18, 19, 20, 25, 26, 30, 31, 34, 38, 40, 41, 43, 44, 45, 51, 53, 54, 55, 58, 59, 60, 61, 64, 65, 69, 73, 74, 76, 78, 81, 83, 89, 93, 95, 96, 99, 104, 106, 110, 111, 113, 115, 116, 120, 128, 136, 138, 139, 141, 144, 146, 148, 149, 150, 151

Offset: 1

Artur Jasinski, Apr 23 2008

Cf. A107006, A124477, A139479.

a = {}; Do[If[PrimeQ[24 n + 7], AppendTo[a, n]], {n, 0, 200}]; a
Select[Range[0,200],PrimeQ[24#+7]&] (* Harvey P. Dale, Sep 02 2015 *)

is(n)=isprime(24*n+7) \\ Charles R Greathouse IV, Jun 06 2017

cp's OEIS Frontend

A198586 a(n) = (4^A001651(n+1) - 1)/3: numbers (4^k-1)/3 for k > 1, not multiples of 3.

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A135659 a(n) = 24*n + 7.

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A135982 a(n) = 2^(24n+7)-1.

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A139480 a(n) = A000043(n) - 3.

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A145044 Primes p (A000043) such that 2^p-1 is prime (A000668) and congruent to 271 mod 6!.

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A145045 Primes p (A000043) such that 2^p-1 is prime (A000668) and congruent to 607 mod 6!

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A139481 a(n) = A139480(n)/2.

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A121290 a(n) = (2^prime(n) - 8)/24 for n>=2.

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