A104564 -id:A104564 - cp's OEIS frontend

A104659 Number of distinct prime divisors of 44...441 (with n 4s).

1, 2, 1, 2, 2, 2, 2, 4, 2, 1, 5, 3, 2, 6, 3, 3, 3, 3, 2, 4, 4, 4, 4, 4, 4, 5, 1, 2, 6, 4, 4, 6, 4, 4, 4, 5, 4, 8, 4, 4, 7, 3, 2, 7, 3, 7, 4, 6, 3, 4, 6, 2, 6, 1, 4, 7, 2, 5, 4, 4, 4, 6, 4, 2, 3, 6, 3, 5, 4, 3, 11, 5, 4, 4, 5, 7, 3, 4, 3, 5, 4, 4, 3, 3, 6, 8, 3, 4, 4, 2, 6, 6, 1, 7, 8, 4, 4, 7, 4, 6, 6, 4, 4, 5, 6

Offset: 1

Parthasarathy Nambi, Apr 21 2005, extended Aug 08 2010

There are very few primes in this sequence. 41 appears as the smallest prime divisor frequently. There are many semiprimes.
41 is prime.
4441 is prime.
44444 444441 is prime.
4444 444444 444444 444444 444441 is prime.
4444444444444444444444444444444444444444444444444444441 is prime.
Computed using www.alpertron.com.ar/ECM.HTM

			The number of distinct prime divisors of 441 is 2.
The number of distinct prime divisors of 44444444444444444444444444444441 is four.

Amiram Eldar, Table of n, a(n) for n = 1..199
Dario Alpern, Factorization using the Elliptic Curve Method
Makoto Kamada, Factorizations of 44...441.

Cf. A001221, A104564, A104517, A104484, A173768.

f[n_] := Length@ FactorInteger[(4*10^(n + 1) - 31)/9]; Array[f, 105] (* Robert G. Wilson v, Aug 09 2010 *)
PrimeNu/@Rest[FromDigits/@Table[PadLeft[{1},n,4],{n,110}]] (* Harvey P. Dale, Mar 16 2012 *)

a(n) = A001221(A173768(n+1)). - Amiram Eldar, Jan 24 2020

a(32) - a(105) from Robert G. Wilson v, Aug 09 2010

A104889 Number of distinct prime divisors of 44...447 (with n 4s).

Original entry on oeis.org

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

Offset: 1

Parthasarathy Nambi, Apr 24 2005

nonn

Also number of distinct prime factors of (10^(n + 1) - 1)*4/9 + 3. - Stefan Steinerberger, Feb 21 2006

			Number of distinct prime divisors of 47 is 1 (prime).
Number of distinct prime divisors of 447 is 2.
Number of distinct prime divisors of 4447 is 1 (prime).

Amiram Eldar, Table of n, a(n) for n = 1..201

Cf. A001221, A104564, A173772.

Table[Length[FactorInteger[(10^(n + 1) - 1)*4/9 + 3]], {n, 1, 40}] (* Stefan Steinerberger, Feb 21 2006 *)

a(n) = A001221(A173772(n+1)). - Amiram Eldar, Jan 27 2020

More terms from Stefan Steinerberger, Feb 21 2006

Offset corrected and more terms added by Amiram Eldar, Jan 27 2020

A104890 Number of distinct prime divisors of 66...661 (with n 6s).

Original entry on oeis.org

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

Offset: 1

Parthasarathy Nambi, Apr 24 2005

Also number of distinct prime factors of 6/9*(10^(n + 1) - 1) - 5. - Stefan Steinerberger, Feb 26 2006

			Number of distinct prime divisors of 61 is 1 (prime).
Number of distinct prime divisors of 661 is 1 (prime).
Number of distinct prime divisors of 6661 is 1 (prime).

Amiram Eldar, Table of n, a(n) for n = 1..199

Cf. A001221, A104564, A173805.

Table[Length[FactorInteger[6/9*(10^(n + 1) - 1) - 5]], {n, 1, 50}] (* Stefan Steinerberger, Feb 26 2006 *)
Table[PrimeNu[FromDigits[PadLeft[{1},n,6]]],{n,2,90}] (* Harvey P. Dale, Jun 06 2021 *)

a(n) = A001221(A173805(n+1)). - Amiram Eldar, Jan 24 2020

More terms from Stefan Steinerberger, Feb 26 2006

Offset corrected and more terms added by Amiram Eldar, Jan 24 2020

A105972 Number of distinct prime divisors of 88...881 (with n 8's).

Original entry on oeis.org

0, 1, 1, 2, 4, 3, 2, 2, 2, 2, 3, 3, 3, 3, 2, 2, 3, 8, 1, 3, 2, 2, 5, 2, 4, 6, 2, 2, 5, 3, 4, 4, 4, 2, 5, 5, 5, 4, 6, 3, 5, 3, 2, 5, 4, 4, 7, 4, 4, 6, 3, 5, 5, 3, 5, 6, 5, 5, 5, 4, 3, 5, 4, 2, 7, 3, 3, 8, 3, 5, 6, 7, 3, 6, 3, 3, 7, 3, 1, 5, 5, 4, 8, 4, 3, 4, 4

Offset: 0

Parthasarathy Nambi, Apr 28 2005

			The number of distinct prime divisors of 81 is 1.
The number of distinct prime divisors of 881 is 1 (prime).
The number of distinct prime divisors of 8881 is 2.

Amiram Eldar, Table of n, a(n) for n = 0..203

Cf. A001221, A104564, A104517, A104659, A173810.

Table[PrimeNu[(8*10^(n+1) - 71)/9], {n,0,50}] (* G. C. Greubel, May 16 2017 *)
PrimeNu/@Table[10 FromDigits[PadRight[{},n,8]]+1,{n,0,90}] (* Harvey P. Dale, Apr 25 2019 *)

a(n) = omega((8*10^(n+1)-71)/9); \\ Michel Marcus, Jan 27 2014

a(n) = A001221(A173810(n+1)). - Michel Marcus, Jan 27 2014

More terms from Michel Marcus, Jan 27 2014

A104563 A floretion-generated sequence relating to centered square numbers.

Original entry on oeis.org

0, 1, 3, 5, 8, 13, 19, 25, 32, 41, 51, 61, 72, 85, 99, 113, 128, 145, 163, 181, 200, 221, 243, 265, 288, 313, 339, 365, 392, 421, 451, 481, 512, 545, 579, 613, 648, 685, 723, 761, 800, 841, 883, 925, 968, 1013, 1059, 1105, 1152, 1201, 1251

Offset: 0

Creighton Dement, Mar 15 2005

Floretion Algebra Multiplication Program, FAMP Code: a(n) = 1vesrokseq[A*B] with A = - .5'i - .5i' + .5'ii' + .5e, B = + .5'ii' - .5'jj' + .5'kk' + .5e. RokType: Y[sqa.Findk()] = Y[sqa.Findk()] + Math.signum(Y[sqa.Findk()])*p (internal program code). Note: many slight variations of the "RokType" already exist, such that it has become difficult to assign them all names.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-4,4,-3,1).

Cf. A001844, A004525, A104564, A098180, A059100, A010000, A047599, A007877.

LinearRecurrence[{3, -4, 4, -3, 1}, {0, 1, 3, 5, 8}, 60] (* Amiram Eldar, Dec 14 2024 *)

concat(0, Vec(x*(1 + x)*(1 - x + x^2) / ((1 - x)^3*(1 + x^2)) + O(x^40))) \\ Colin Barker, Apr 29 2019

G.f.: x*(1 + x^3)/((1 + x^2)*(1 - x)^3).
FAMP result: 2*a(n) + 2*A004525(n+1) = A104564(n) + a(n+1).
Superseeker results:
a(2*n+1) = A001844(n) = 2*n*(n+1) + 1 (Centered square numbers);
a(n+1) - a(n) = A098180(n) (Odd numbers with two times the odd numbers repeated in order between them);
a(n) + a(n+2) = A059100(n+1) = A010000(n+1);
a(n+2) - a(n) = A047599(n+1) (Numbers that are congruent to {0, 3, 4, 5} mod 8);
a(n+2) - 2*a(n+1) + a(n) = A007877(n+3) (Period 4 sequence with initial period (0, 1, 2, 1));
Coefficients of g.f.*(1-x)/(1+x) = convolution of this with A280560 gives A004525;
Coefficients of g.f./(1+x) = convolution of this with A033999 gives A054925.
a(n) = (1/2)*(n^2 + 1 - cos(n*Pi/2)). - Ralf Stephan, May 20 2007
From Colin Barker, Apr 29 2019: (Start)
a(n) = (2 - (-i)^n - i^n + 2*n^2) / 4 where i=sqrt(-1).
a(n) = 3*a(n-1) - 4*a(n-2) + 4*a(n-3) - 3*a(n-4) + a(n-5) for n>4. (End)
a(n) = A011848(n-1)+A011848(n+2). - R. J. Mathar, Sep 11 2019
Sum_{n>=1} 1/a(n) = Pi^2/48 + (Pi/2) * tanh(Pi/2) + (Pi/(4*sqrt(2)) * tanh(Pi/(2*sqrt(2)))). - Amiram Eldar, Dec 14 2024

Stephan's formula corrected by Bruno Berselli, Apr 29 2019

cp's OEIS Frontend

A104659 Number of distinct prime divisors of 44...441 (with n 4s).

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A104889 Number of distinct prime divisors of 44...447 (with n 4s).

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A104890 Number of distinct prime divisors of 66...661 (with n 6s).

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A105972 Number of distinct prime divisors of 88...881 (with n 8's).

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A104563 A floretion-generated sequence relating to centered square numbers.

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