A000225 -id:A000225 - cp's OEIS frontend

A224521 Numbers a(n) with property a(n) + a(n+5) = 2^(n+5) - 1 = A000225(n+5).

0, 1, 3, 7, 15, 31, 62, 124, 248, 496, 992, 1985, 3971, 7943, 15887, 31775, 63550, 127100, 254200, 508400, 1016800, 2033601, 4067203, 8134407, 16268815, 32537631, 65075262, 130150524, 260301048, 520602096, 1041204192, 2082408385, 4164816771, 8329633543

Offset: 0

Arie Bos, Apr 09 2013

This is the case k=5 of a(n) + a(n+k) = 2^(n+k) - 1 = A000225(n+k). The sequences A000975, A077854, A153234 and A224520 correspond to cases k=1,2,3 and 4, respectively.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-2,0,0,-1,3,-2).

Cf. A000975, A077854, A153234, A224520.

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( x/((1-x)*(1-2*x)*(1+x^5)) )); // G. C. Greubel, Jun 06 2019

CoefficientList[Series[x/((1-x)*(1-2*x)*(1+x^5)), {x,0,40}], x] (* G. C. Greubel, Oct 11 2017 *)
LinearRecurrence[{3,-2,0,0,-1,3,-2},{0,1,3,7,15,31,62},40] (* Harvey P. Dale, Apr 29 2020 *)

my(x='x+O('x^40)); concat([0], Vec(x/((1-x)*(1-2*x)*(1+x^5)))) \\ G. C. Greubel, Oct 11 2017

print([2**(n+5)//33 for n in range(31)]) # Karl V. Keller, Jr., Jul 03 2021

(x/((1-x)*(1-2*x)*(1+x^5))).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, Jun 06 2019

a(n) + a(n+5) = 2^(n+5) - 1.
From Joerg Arndt, Apr 09 2013: (Start)
G.f.: x/((1-x)*(1+x)*(1-2*x)*(1-x+x^2-x^3+x^4)).
a(n) = +3*a(n-1) -2*a(n-2) -1*a(n-5) +3*a(n-6) -2*a(n-7). (End)
a(n) = floor(2^(n+5)/33). - Karl V. Keller, Jr., Jul 03 2021

A260758 Least k > 0 such that M(n)^2 + 2k is prime, where M(n) = 2^n - 1 = A000225(n).

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 10, 5, 1, 3, 14, 11, 4, 5, 4, 5, 8, 30, 2, 6, 1, 8, 29, 12, 29, 30, 11, 2, 4, 5, 19, 29, 2, 9, 7, 11, 4, 74, 16, 24, 8, 18, 10, 30, 56, 15, 35, 24, 4, 35, 19, 111, 19, 18, 1, 57, 8, 20, 14, 2, 2, 48, 29, 26, 92, 24, 19, 155, 2, 78, 35, 56, 113, 33, 70, 32, 7

Offset: 0

M. F. Hasler, Jul 30 2015

nonn

			M(0)^2 + 2*1 = 0 + 2 = 2 is prime, thus a(0)=1.
M(1)^2 + 2*1 = 1 + 2 = 3 is prime, thus a(1)=1.
M(2) = 2^2-1 = 3 and 3*3 + 2k = 11 is a prime for k=1, thus a(2) = 1.
M(3) = 2^3-1 = 7 and 7*7 + 2k = 53 is a prime for k=2 but not for k=1, thus a(3) = 2.
M(4) = 2^4-1 = 15 and 15*15 + 2k = 227 is a prime for k=1, thus a(4) = 1.

Cf. A260757.

a(n)=for(k=1,9e9,ispseudoprime((2^n-1)^2+2*k)&&return(k))

A282534 Integers that are powers of Mersenne numbers A000225 (i.e., of the form (2^n - 1)^m).

Original entry on oeis.org

1, 3, 7, 9, 15, 27, 31, 49, 63, 81, 127, 225, 243, 255, 343, 511, 729, 961, 1023, 2047, 2187, 2401, 3375, 3969, 4095, 6561, 8191, 16129, 16383, 16807, 19683, 29791, 32767, 50625, 59049, 65025, 65535, 117649, 131071, 177147, 250047, 261121, 262143, 524287, 531441, 759375

Offset: 1

Andrew Ivashenko, Feb 18 2017

nonn

The cardinality of the set of subsets in a multiset excluding empty subsets.

			3 = (2^2-1), 7 = (2^3-1), 9 = (2^2-1)^2, 81 = (2^2-1)^4, 1070599167 = (2^10-1)^3.

Giovanni Resta, Table of n, a(n) for n = 1..10000

Cf. A000225, A056652.

mx = 10^6; Union@ Flatten@ {1, #^Range[Log[#, mx]] & /@ (2^ Range[2, Log2@ mx] -1)} (* Giovanni Resta, Mar 08 2017 *)

ismn(n) = n++; n == 2^valuation(n,2);
isok(n) = ismn(n) || (ispower(n,,&m) && ismn(m)); \\ Michel Marcus, Feb 18 2017

More terms from Michel Marcus, Feb 18 2017

A295584 Odd numbers that are not a product of Mersenne numbers (A000225).

Original entry on oeis.org

5, 11, 13, 17, 19, 23, 25, 29, 33, 35, 37, 39, 41, 43, 47, 51, 53, 55, 57, 59, 61, 65, 67, 69, 71, 73, 75, 77, 79, 83, 85, 87, 89, 91, 95, 97, 99, 101, 103, 107, 109, 111, 113, 115, 117, 119, 121, 123, 125, 129, 131, 133, 137, 139, 141, 143, 145, 149, 151, 153

Offset: 1

N. J. A. Sloane, Dec 15 2017

nonn

Numbers m such that no commutative ring has m units.

Robert Israel, Table of n, a(n) for n = 1..10000
S. Chebolu and K. Lockridge, How Many Units Can a Commutative Ring Have?, Amer. Math. Monthly, 124 (2017), 960-965; arXiv, arXiv:1701.02341 [math.AC], 2017.

Odd numbers not in A282572; complement of A296241.

Cf. A000225.

N:= 1000: # to get all terms <= N
P:= {1}:
for k from 2 do
  m:= 2^k-1;
  if m > N then break fi;
  P:= map(p -> seq(p*m^j, j=0..floor(log[m](N/p))), P);
od:
sort(convert({seq(i,i=1..N,2)} minus P, list)); # Robert Israel, Dec 15 2017

nn == 1000;
P = {1};
For[k = 2, True, k++,
   m = 2^k - 1;
   If[m > nn, Break[]
];
P = (Function[p, Table[p m^j, {j, 0, Log[m, nn/p]}]] /@ P) // Flatten];
Range[1, nn, 2] ~Complement~ P (* Jean-François Alcover, Sep 18 2018, after Robert Israel *)

A318164 a(n) = A000225(n)^A000217(n-1), n > 0.

Original entry on oeis.org

1, 3, 343, 11390625, 819628286980801, 977480813971145474830595007, 151313661355466579537756144585602921111718527, 24161564501550368558430041444810830996032029256261885166168212890625

Offset: 1

Muniru A Asiru, Aug 19 2018

nonn

			For n = 3, a(3) = 7^3 = 343.

Bijan Davvaz, Polygroup theory and related systems, World Scientific Publishing Co. Plc. Ltd., New Jersey, (2013), p. 157 (for a(3) = 7^3 = 343).

Muniru A Asiru, Table of n, a(n) for n = 1..19

Cf. A000225, A000217.

GAP
```
List([1..8],n->(2^n-1)^(n*(n-1)/2));
```

SetDefaultRealField(RealField(100)); [Round((2^n-1)^(n*(n-1)/2)): n in [1..8]]; // G. C. Greubel, Oct 19 2018

a:=n->(2^n-1)^(n*(n-1)/2): seq(a(n),n=1..8);

a[n_] := (2^n-1)^(n*(n-1)/2); Array[a, 8] (* Stefano Spezia, Sep 02 2018 *)

a(n) = (2^n-1)^(n*(n-1)/2); \\ Michel Marcus, Aug 21 2018

a(n) = (2^n - 1)^(n*(n - 1)/2), n > 0.

A336104 Number of permutations of the prime indices of A000225(n) = 2^n - 1 with at least one non-singleton run.

Original entry on oeis.org

0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 24, 0, 0, 0, 0, 0, 96, 0, 120, 6, 0, 0, 720, 0, 0, 0, 0, 0, 720, 0, 0, 0, 0, 0, 322560, 0, 0, 0, 5040, 0, 4320, 0, 0, 0, 0, 0, 362880, 0, 0

Offset: 1

Gus Wiseman, Sep 03 2020

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The a(21) = 6 permutations of {4, 4, 31, 68}:
  (4,4,31,68)
  (4,4,68,31)
  (31,4,4,68)
  (31,68,4,4)
  (68,4,4,31)
  (68,31,4,4)

A335432 is the anti-run version.
A335459 is the version for factorial numbers.
A336105 counts all permutations of this multiset.
A336107 is not restricted to predecessors of powers of 2.
A003242 counts anti-run compositions.
A005649 counts anti-run patterns.
A008480 counts permutations of prime indices.
A325534 counts separable partitions, ranked by A335433.
A325535 counts inseparable partitions, ranked by A335448.
A333489 ranks anti-run compositions.
A335433 lists numbers whose prime indices have an anti-run permutation.
A335448 lists numbers whose prime indices have no anti-run permutation.
A335452 counts anti-run permutations of prime indices.
A335489 counts strict permutations of prime indices.
Cf. A056239, A106351, A112798, A114938, A292884, A336102.
The numbers 2^n - 1: A000225, A001265, A001348, A046051, A046800, A046801, A049093, A325610, A325611, A325612, A325625.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[Select[Permutations[primeMS[2^n-1]],MatchQ[#,{_,x_,x_,_}]&]],{n,30}]

a(n) = A336107(2^n - 1).

a(n) = A336105(n) - A335432(n).

A057613 Numbers that are both lucky numbers (A000959) and of form 2^k-1 (A000225).

Original entry on oeis.org

1, 3, 7, 15, 31, 63, 127, 511, 1023, 4095, 8191, 131071, 524287, 2097151, 4194303, 8388607, 33554431, 67108863, 8589934591

Offset: 1

Naohiro Nomoto, Oct 09 2000

a(20) >= 17179869183 = 2^34 - 1, if it exists. - Kevin P. Thompson, Nov 24 2021

			a(7) = 127 since A000959(27) = A000225(7) = 127 and it is the 7th such number with this property.

Cf. A000959, A000225.

More terms from Mark Weston (mweston(AT)uvic.ca), Oct 16 2001
a(18) from Giovanni Resta, May 10 2020
a(19) from Kevin P. Thompson, Nov 24 2021

A063529 a(n) = M(2^n-1), where M() is A029834, a discrete version of the Mangoldt function: if n is prime then floor(log(n)) else 0 and 2^n-1 is A000225.

Original entry on oeis.org

0, 1, 1, 0, 3, 0, 4, 0, 0, 0, 0, 0, 9, 0, 0, 0, 11, 0, 13, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 21, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 42, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 61, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Jason Earls, Aug 01 2001

Cf. A029832, A029833, A053821, A062590, A029834, A000225.

j=[]; for(n=1,150,j=concat(j, if(isprime(2^n-1),floor(log(2^n-1)),))); j

A157968 2^n - 1 (A000225) where n is nonprime number (A141468).

Original entry on oeis.org

0, 1, 15, 63, 127, 255, 511, 1023, 4095, 16383, 32767, 65535, 262143, 1048575, 2097151, 4194303, 16777215, 33554431, 67108863, 134217727, 268435455, 1073741823, 4294967295

Offset: 0

Jani Melik, Mar 10 2009

nonn

a(0) = 0 since 2^0 - 1 = 0 and 0 is nonprime, a(1) = 1 since 2^1 - 1 = 1 and 1 is nonprime, a(2) = 15 since 2^4 - 1 = 15 and 4 is nonprime, ...

A000668

A261514 Indices of Mersenne numbers (A000225) of the form x^2 + xy + y^2.

Original entry on oeis.org

0, 1, 2, 3, 5, 6, 7, 9, 13, 14, 15, 17, 18, 19, 21, 25, 26, 27, 31, 37, 38, 39, 42, 45, 49, 51, 54, 57, 61, 62, 63, 65, 67, 74, 75, 78, 81, 85, 89, 93, 98, 101, 103, 107, 111, 114, 117, 122, 125, 126, 127, 133, 134, 135, 139, 147, 153, 162, 171, 183, 186, 189, 195

Offset: 1

Altug Alkan, Nov 18 2015

nonn

Inspired by intersection of nice and core sequences. Initial Mersenne numbers of the form x^2 + xy + y^2 are 0, 1, 3, 7, 31, 63, 127, 511, 8191, 16383.

			a(4) = 3 because 2^3 - 1 = 1^2 + 1*2 + 2^2 = 7.
a(5) = 5 because 2^5 - 1 = 1^2 + 1*5 + 5^2 = 31.

Cf. A000225, A003136.

is(n)=#bnfisintnorm(bnfinit(z^2+z+1), 2^n-1);
for(n=0, 200, if(n==0 || is(n), print1(n, ", ")))

cp's OEIS Frontend

A224521 Numbers a(n) with property a(n) + a(n+5) = 2^(n+5) - 1 = A000225(n+5).

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A260758 Least k > 0 such that M(n)^2 + 2k is prime, where M(n) = 2^n - 1 = A000225(n).

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A282534 Integers that are powers of Mersenne numbers A000225 (i.e., of the form (2^n - 1)^m).

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A295584 Odd numbers that are not a product of Mersenne numbers (A000225).

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Maple

Mathematica

A318164 a(n) = A000225(n)^A000217(n-1), n > 0.

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A336104 Number of permutations of the prime indices of A000225(n) = 2^n - 1 with at least one non-singleton run.

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Mathematica

Formula

A057613 Numbers that are both lucky numbers (A000959) and of form 2^k-1 (A000225).

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Extensions

A063529 a(n) = M(2^n-1), where M() is A029834, a discrete version of the Mangoldt function: if n is prime then floor(log(n)) else 0 and 2^n-1 is A000225.

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