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Consider the number m formed from the first n digits of the repeating string 102010201020... Then a(n) = number of prime factors (with repetition) of m.
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%I #18 Jul 09 2023 09:47:38

%S 0,2,3,5,2,4,5,7,2,4,8,10,1,3,7,9,1,3,8,10,3,5,11,13,3,5,9,11,7,9,12,

%T 14,2,4,13,15,5,7,12,14,5,7,12,14,1,3,14,16,4,6,10,12,2,4,13,15,4,6,

%U 21,23,6,8,16,18,4,6,11,13,2,4,19,21,4,6,7,9,2,4,16,18,7,9,23,25,4,6,16,18,4

%N Consider the number m formed from the first n digits of the repeating string 102010201020... Then a(n) = number of prime factors (with repetition) of m.

%C a(2n) = a(2n-1)+2.

%C Consider the number m formed from the first n digits of the repeating string 102010201020... as S(n). Obviously S(2n) = 10*S(2n-1). Also define Repunits (A002275) as R(n) = (10^n-1)/9 = {1, 11, 111, 1111, 11111, ...} and consider a generalized plateau number sequence as X(n) = (34*10^(n+2)-67)/3 = { 1111, 11311, 113311, 1133311, 11333311, ... }. Then S(4n-1) = 34*(10000^n-1)/3333 = 102*R(4n)/R(4) and S(4n+1) = (3400*10000^n-67)/3333 = X(4n)/R(4). - Remarks communicated to _Robert G. Wilson v_ by KAMADA Makoto (m_kamada(AT)nifty.com), Feb 18 2005

%H Dario Alejandro Alpern, <a href="https://www.alpertron.com.ar/ECM.HTM">Factorization using the Elliptic Curve Method</a>.

%H Makoto Kamada, <a href="https://stdkmd.net/nrr/#DRwD">Factorizations of near-repdigits numbers</a>.

%H Makoto Kamada, <a href="https://stdkmd.net/nrr/repunit/">Factorizations of 11...11 (Repunit)</a>.

%e a(1) = 0 because 1 is neither prime nor semiprime.

%e a(2) = 2 because 10 = 2 * 5 is semiprime.

%e a(3) = 3 because 102 = 2 * 3 * 17, 3 prime factors.

%e a(4) = 5 because 1020 = 2^2 * 3 * 5 * 17, 5 prime factors with repetition (2 is counted twice because of 2^2 in factorization).

%e a(5) = 2 because 10201 = 101^2 is a semiprime (and a square) (and a palindrome)

%e a(6) = 4 because 102010 = 2 * 5 * 101^2

%t f[n_] := Plus @@ Transpose[ FactorInteger[ IntegerPart[10^(n - 1)*3400 / 3333]]][[2]]; Table[ f[n], {n, 2, 67}] (* _Robert G. Wilson v_, Feb 14 2005 *)

%Y Cf. A004023, A103833.

%K nonn,base

%O 1,2

%A _Jonathan Vos Post_, Feb 12 2005

%E Edited, corrected and extended, a(70)-a(92), by _Robert G. Wilson v_, Feb 14 2005

%E Edited by _Charles R Greathouse IV_, Aug 03 2010