A058764 -id:A058764 - cp's OEIS frontend

A082505 a(n) = sum of (n-1)-th row terms of triangle A134059.

0, 1, 6, 12, 24, 48, 96, 192, 384, 768, 1536, 3072, 6144, 12288, 24576, 49152, 98304, 196608, 393216, 786432, 1572864, 3145728, 6291456, 12582912, 25165824, 50331648, 100663296, 201326592, 402653184, 805306368, 1610612736, 3221225472

Offset: 0

Labos Elemer, Apr 28 2003

a(n) is the least number x such that gcd(2^x, x-phi(x)) = 2^n. If cototient is replaced by totient, analogous values are different: A053576.

			G.f. = x + 6*x^2 + 12*x^3 + 24*x^4 + 48*x^5 + 96*x^6 + 192*x^7 + 384*x^8 + ...

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (2).

Cf. A000010, A007283, A009195, A050339, A134059.

Essentially the same as A003945 (and perhaps also A058764).

[0, 1] cat [ &+[ 3*Binomial(n,k): k in [0..n] ]: n in [1..30] ]; // Klaus Brockhaus, Dec 02 2009

0,1,seq(3*2^(n-1), n=2..40); # G. C. Greubel, Apr 27 2021

{0}~Join~Map[Total, {{1}}~Join~Table[3 Binomial[n, k], {n, 30}, {k, 0, n}]] (* Michael De Vlieger, Jul 03 2016, after Harvey P. Dale at A134059 *)
Table[3*2^(n-1) -(3/2)*Boole[n==0] -2*Boole[n==1], {n,0,40}] (* G. C. Greubel, Apr 27 2021 *)
Join[{0,1},NestList[2#&,6,30]] (* Harvey P. Dale, Jan 22 2024 *)

{a(n) = local(A); if( n<1, 0, A = vector(n); A[1] = 1; for( k=2, n, A[k] = (-6*k + 16) * A[k-1] + 2 * sum( j=1, k-1, A[j] * A[k-j])); A[n])} /* Michael Somos, Jul 23 2011 */

a(n)=if(n<2,n,3<<(n-1)) \\ Charles R Greathouse IV, Jun 16 2012

[0,1]+[3*2^(n-1) for n in (2..40)] # G. C. Greubel, Apr 27 2021

a(n) = A007283(n-1) for n>1, with a(0) = 0 and a(1) = 1.
G.f.: x * (1 + 4*x) / (1 - 2*x) = x / (1 - 6*x / (1 + 4*x)). - Michael Somos, Jun 15 2012
Starting (1, 6, 12, 24, 48, ...) = binomial transform of [1, 5, 1, 5, 1, 5, ...]. - Gary W. Adamson, Nov 18 2007
a(n+1) = Sum_{k=0..n} A109466(n,k)*A144706(k). - Philippe Deléham, Oct 30 2008
a(n) = (-6*n + 16) * a(n-1) + 2 * Sum_{k=1..n-1} a(k) * a(n-k) if n>1. - Michael Somos, Jul 23 2011
E.g.f.: (-3 - 4*x + 3*exp(2*x))/2. - Ilya Gutkovskiy, Jul 04 2016
a(n) = 3*2^(n-1) - (3/2)*[n=0] - 2*[n=1]. - G. C. Greubel, Apr 27 2021

More terms from Klaus Brockhaus, Dec 02 2009

A091629 Product of digits associated with A091628(n). Essentially the same as A007283.

Original entry on oeis.org

6, 12, 24, 48, 96, 192, 384, 768, 1536, 3072, 6144, 12288, 24576, 49152, 98304, 196608, 393216, 786432, 1572864, 3145728, 6291456, 12582912, 25165824, 50331648, 100663296, 201326592, 402653184, 805306368, 1610612736, 3221225472

Offset: 1

Enoch Haga, Jan 24 2004

Sequence arising in Farideh Firoozbakht's solution to Prime Puzzle 251 - 23 is the only pointer prime (A089823) not containing digit "1".

The monotonic increasing value of successive product of digits strongly suggests that in successive n the digit 1 must be present.

G. C. Greubel, Table of n, a(n) for n = 1..1000
Tanya Khovanova, Recursive Sequences
Carlos Rivera, Puzzle 251, Pointer primes, The Prime Puzzles and Problems Connection.
Index entries for linear recurrences with constant coefficients, signature (2).

Sequences of the form (2*m+1)*2^n: A000079 (m=0), A007283 (m=1), A020714 (m=2), A005009 (m=3), A005010 (m=4), A005015 (m=5), A005029 (m=6), A110286 (m=7), A110287 (m=8), A110288 (m=9), A175805 (m=10), A248646 (m=11), A164161 (m=12), A175806 (m=13), A257548 (m=15).
Cf. A089823, A091628, A091630, A091631, A091632.
Similar to A003945, A042950, A058764, A087009, A081808.

[3*2^n : n in [1..40]]; // Wesley Ivan Hurt, Jul 17 2020

3*2^Range[1, 60] (* Vladimir Joseph Stephan Orlovsky, Jun 09 2011 *)

[3*2^n for n in range(1,51)] # G. C. Greubel, Jan 05 2023

a(n) = 3 * 2^n = product of digits of A091628(n).
From Philippe Deléham, Nov 23 2008: (Start)
a(n) = 6*2^(n-1).
a(n) = 2*a(n-1), with a(1) = 6.
G.f.: 6*x/(1-2*x). (End)
E.g.f.: 3*(exp(2*x) - 1). - G. C. Greubel, Jan 05 2023

Edited and extended by Ray Chandler, Feb 07 2004

A047233 Numbers that are congruent to {0, 4} mod 6.

Original entry on oeis.org

0, 4, 6, 10, 12, 16, 18, 22, 24, 28, 30, 34, 36, 40, 42, 46, 48, 52, 54, 58, 60, 64, 66, 70, 72, 76, 78, 82, 84, 88, 90, 94, 96, 100, 102, 106, 108, 112, 114, 118, 120, 124, 126, 130, 132, 136, 138, 142, 144, 148, 150, 154, 156, 160, 162, 166, 168, 172, 174, 178, 180, 184, 186, 190, 192, 196, 198

Offset: 1

N. J. A. Sloane

Apart from initial term(s), dimension of the space of weight 2*n cusp forms for Gamma_0(17).

Nonnegative k such that k*(k + 2)/6 is an integer. - Bruno Berselli, Mar 06 2018

Bruno Berselli, Table of n, a(n) for n = 1..10000
William A. Stein, Dimensions of the spaces S_k(Gamma_0(N)).
William A. Stein, The modular forms database.
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A047241: (6*n - (-1)^n - 5)/2.

Cf. A342819.

Flatten[{#,#+4}&/@(6Range[0,30])] (* Harvey P. Dale, Jul 07 2013 *)

forstep(n=0,200,[4,2],print1(n", ")) \\ Charles R Greathouse IV, Oct 17 2011

From Bruno Berselli, Jun 24 2010: (Start)
G.f.: 2*x^2*(2 + x)/((1 + x)*(1 - x)^2).
a(n) = a(n-1) + a(n-2) - a(n-3) for n>3.
a(n) = (6*n + (-1)^n - 5)/2. (End)
a(n) = 6*n - a(n-1) - 8 for n>1, a(1)=0. - Vincenzo Librandi, Aug 05 2010
a(n+1) = Sum_{k>=0} A030308(n,k)*A058764(k+1). - Philippe Deléham, Oct 17 2011
Sum_{n>=2} (-1)^n/a(n) = log(3)/4 - sqrt(3)*Pi/36. - Amiram Eldar, Dec 13 2021
E.g.f.: 2 + ((6*x -5)*exp(x) + exp(-x))/2. - David Lovler, Aug 25 2022

A087009 Least m such that omega(m) + Omega(m) = n, or 0 if no such m exists.

Original entry on oeis.org

1, 0, 2, 4, 6, 12, 24, 48, 96, 192, 384, 768, 1536, 3072, 6144, 12288, 24576, 49152, 98304, 196608, 393216, 786432, 1572864, 3145728, 6291456, 12582912, 25165824, 50331648, 100663296, 201326592, 402653184, 805306368, 1610612736, 3221225472

Offset: 0

Lekraj Beedassy, Oct 13 2003

Index entries for linear recurrences with constant coefficients, signature (2).

First occurrence of n in A080256.

Join[{1, 0, 2, 4}, LinearRecurrence[{2}, {6}, 40]] (* Jean-François Alcover, Mar 07 2020 *)

a(n) = {m = 1; while (omega(m) + bigomega(m) != n, m++); m} \\ Michel Marcus, Oct 23 2013

For n > 3, a(n) = 2^(n-3)*3. - Ray Chandler, Nov 01 2003
a(n) = A058764(n-2). - Philippe Deléham, Oct 17 2011
G.f.: (2*x^4-2*x^2+2*x-1)/(2*x-1). - Colin Barker, Oct 23 2012

Corrected and extended by Ray Chandler, Nov 01 2003

A152764 Bisection of A138144.

Original entry on oeis.org

1, 111, 11011, 1100011, 110000011, 11000000011, 1100000000011, 110000000000011, 11000000000000011, 1100000000000000011, 110000000000000000011, 11000000000000000000011

Offset: 1

Omar E. Pol, Dec 14 2008

			n ...... a(n)
1 ....... 1
2 ...... 111
3 ..... 11011
4 .... 1100011
5 ... 110000011

Index entries for linear recurrences with constant coefficients, signature (101,-100).

Cf. A135577, A138120, A138146, A152756, A152758, A058764.

LinearRecurrence[{101,-100},{1,111,11011,1100011},20] (* Harvey P. Dale, Nov 26 2019 *)

Vec(-x*(10*x-1)*(10*x+1)^2/((x-1)*(100*x-1)) + O(x^100)) \\ Colin Barker, Sep 16 2013

From Colin Barker, Sep 16 2013: (Start)
a(n) = 11+11*10^(2*n-3) for n>2.
a(n) = 101*a(n-1)-100*a(n-2) for n>4.
G.f.: -x*(10*x-1)*(10*x+1)^2 / ((x-1)*(100*x-1)). (End)

A053578 Values of cototient function for A053577.

Original entry on oeis.org

1, 1, 2, 1, 4, 1, 4, 1, 8, 1, 8, 8, 1, 1, 1, 16, 16, 1, 1, 16, 1, 1, 1, 1, 32, 1, 32, 1, 1, 32, 32, 1, 1, 1, 1, 1, 1, 64, 1, 1, 1, 1, 1, 64, 1, 64, 1, 64, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 128, 1, 1, 1, 1, 1, 128, 1, 1, 1, 1, 1, 128, 1, 128, 128, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Labos Elemer, Jan 18 2000

nonn

Except for 2^0 = 1, there are only finitely many values of k such that cototient(k) = 2^m for fixed m.

			For p prime, cototient(p) = 1. Smallest values for which cototient(x) = 2^w are A058764(w) = A007283(w-1) = 3*2^(w-1) = 6, 12, 24, 48, 96, 192, .., 49152 for w = 2, 3, 4, 5, 6, ..., 15. [Corrected by _M. F. Hasler_, Nov 10 2016]

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

Cf. A051953, A053577, A058764, A007283.

Select[Table[k - EulerPhi[k], {k, 1, 400}], # == 2^IntegerExponent[#, 2] &] (* Amiram Eldar, Jun 09 2024 *)

lista(kmax) = {my(c); for(k = 2, kmax, c = k - eulerphi(k); if(c >> valuation(c, 2) == 1, print1(c, ", ")));} \\ Amiram Eldar, Jun 09 2024

Edited and corrected by M. F. Hasler, Nov 10 2016

A114958 a(n) = 62^(n+1) - 5(n+1) - 4.

Original entry on oeis.org

3, 10, 29, 72, 163, 350, 729, 1492, 3023, 6090, 12229, 24512, 49083, 98230, 196529, 393132, 786343, 1572770, 3145629, 6291352, 12582803, 25165710, 50331529, 100663172, 201326463, 402653050, 805306229, 1610612592, 3221225323

Offset: 0

Creighton Dement, Feb 21 2006

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-5,2).

Cf. A098011, A042950, A110164, A101229, A058764, A087009, A111286, A007283.

[6*2^(n+1) - 5*(n+1) - 4: n in [0..30] ]; // Vincenzo Librandi, May 18 2011

Vec((3 - 2*x + 4*x^2) / ((1 - x)^2*(1 - 2*x)) + O(x^40)) \\ Colin Barker, Apr 30 2019

From Colin Barker, Apr 30 2019: (Start)
G.f.: (3 - 2*x + 4*x^2) / ((1 - x)^2*(1 - 2*x)).
a(n) = 4*a(n-1) - 5*a(n-2) + 2*a(n-3) for n>2.
(End)

A072073 Number of solutions to cototient(x) = A051953(x) = 2^n.

Original entry on oeis.org

1, 2, 3, 3, 4, 4, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10

Offset: 1

Labos Elemer, Jun 13 2002

nonn

a(n) increases at A000043(n).

Since A051953(p) = 1 for p prime, and given that there are an infinite number of primes, we disregard a(0) = oo. - Michael De Vlieger, Mar 25 2020

			InvCototient(2^0) has an infinite number of entries, so 2^0=1 is left out.
n=14: 2^14=16384, InvCototient(16384) = {24576,28672,31744,32512,32764,32768}, so a(14)=6;

Cf. A051953, A058764, A000043, A063740, A000079.

Length /@ Most@ Split@ DeleteCases[Select[Array[# - EulerPhi[#] &, 10^6], IntegerQ@ Log2@ # &], 1] (* Michael De Vlieger, Mar 25 2020 *)

a(n) = A063740(A000079(n)). - Ridouane Oudra, Jun 02 2024

cp's OEIS Frontend

A082505 a(n) = sum of (n-1)-th row terms of triangle A134059.

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A091629 Product of digits associated with A091628(n). Essentially the same as A007283.

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A047233 Numbers that are congruent to {0, 4} mod 6.

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A087009 Least m such that omega(m) + Omega(m) = n, or 0 if no such m exists.

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A152764 Bisection of A138144.

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A053578 Values of cototient function for A053577.

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A114958 a(n) = 6*2^(n+1) - 5*(n+1) - 4.

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A114958 a(n) = 62^(n+1) - 5(n+1) - 4.