Michael Taktikos - cp's OEIS frontend

A102084 a(1) = 0; for n>0, write 2n=p+q (p, q prime), pq maximal; then a(n)=pq (see A073046).

0, 4, 9, 15, 25, 35, 49, 55, 77, 91, 121, 143, 169, 187, 221, 247, 289, 323, 361, 391, 437, 403, 529, 551, 589, 667, 713, 703, 841, 899, 961, 943, 1073, 1147, 1189, 1271, 1369, 1363, 1517, 1591, 1681, 1763, 1849, 1927, 2021, 1891, 2209, 2279, 2257, 2491

Offset: 1

Michael Taktikos, Feb 16 2005

nonn

For n>1, largest semiprime whose sum of prime factors = 2n. Assumes the Goldbach conjecture is true. Also the largest semiprime <= n^2.

Also the greatest integer x such that x' = 2*n, or 0 if there is no such x, where x' is the arithmetic derivative (A003415). Bisection of A099303. The only even number without an anti-derivative is 2. All terms are <= n^2, with equality only when n is prime. In fact a(n) = n^2 - k^2, where k is the least number such that both n-k and n+k are prime; k = A047160(n). It appears that the anti-derivatives of even numbers are overwhelmingly semiprimes of the form n^2 - k^2. For example, 1000 has 28 anti-derivatives, all of this form. Sequence A189763 lists the even numbers that have anti-derivatives not of this form. - T. D. Noe, Apr 27 2011

			n=13: 2n = 26; 26 = 23 + 3 = 19 + 7 = 13 + 13; 13*13 = maximal => p*q = 13*13 = 169.

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A073046, A003415, A047160, A099303, A189762.

f[n_] := Block[{pf = FactorInteger[n]}, If[Plus @@ Last /@ pf == 2, If[ Length[pf] == 2, Plus @@ First /@ pf, 2pf[[1, 1]]], 0]]; t = Table[0, {51}]; Do[a = f[n]; If[ EvenQ[a] && 0 < a < 104, t[[a/2]] = n], {n, 2540}]; t (* Robert G. Wilson v, Jun 14 2005 *)
Table[k = 0; While[k < n && (! PrimeQ[n - k] || ! PrimeQ[n + k]), k++]; If[k == n, 0, (n - k)*(n + k)], {n, 100}] (* T. D. Noe, Apr 27 2011 *)

a(n) = n^2 - A047160(n)^2. - Jason Kimberley, Jun 26 2012

Edited by N. J. A. Sloane, Aug 29 2008 at the suggestion of R. J. Mathar

A096233 Number of digits of the 10^n-th tribonacci number (A000073).

Original entry on oeis.org

2, 26, 264, 2646, 26464, 264649, 2646494, 26464944, 264649443, 2646494434, 26464944348, 264649443484, 2646494434842, 26464944348425, 264649443484250, 2646494434842508, 26464944348425087

Offset: 0

Michael Taktikos, Aug 09 2004

a(n)/10^n tends towards log[10](rho) = .26464944348425087191..., where rho is the real root of x^3-x^2-x-1 = 0. - Vladeta Jovovic, Sep 01 2004

Cf. A068070.

rho=1/3*((19+3*Sqrt[33])^(1/3)+(19-3*Sqrt[33])^(1/3)+1); triboappr[n_]:=N[(rho-1)/(4rho-6)*rho^(n-3), 3000]; Table[MantissaExponent[triboappr[10^i]][[2]], {i, 1, 7}]

A097353 Number of digits of the (10^n)-th tetranacci number (A000078(10^n)).

Original entry on oeis.org

1, 2, 28, 284, 2849, 28500, 285008, 2850083, 28500834, 285008350, 2850083504, 28500835049, 285008350498, 2850083504986, 28500835049863, 285008350498633, 2850083504986335, 28500835049863359, 285008350498633597, 2850083504986335973

Offset: 0

Michael Taktikos, Sep 17 2004

a(n)/10^n converges to 0.28500835...

			Let t(n) = A000078(n). Then we have t(1) = 0, t(10) = 56, t(100) = 2505471397838180985096739296, with respectively 1, 2, 28 and 284 digits.

Herman Jamke (hermanjamke(AT)fastmail.fm), May 01 2007, Table of n, a(n) for n = 0..25
Tito Piezas III, On Fibonacci Numbers and Their Friends.

# This Maple code will at least get the first few terms correctly!
f:=proc(n) option remember; if n <= 2 then RETURN(0); fi; if n = 3 then RETURN(1); fi; f(n-1) + f(n-2) + f(n-3) +f(n-4); end; for n from 0 to 4 do lprint(f(10^n), length(f(10^n))); od;

a = b = c = 0; d = i = 1; Do[e = a + b + c + d; a = b; b = c; c = d; d = e; If[n == 10^i, Print[Length[IntegerDigits[e]]]; i++ ], {n, 4, 10^6}] (* Ryan Propper, Jul 22 2005 *)

\p 100 x=solve(x=1.9274,1.9276,x^4-x^3-x^2-x-1); r=solve(x=0.2937,0.2939,563*x^4-20*x^2-5*x-1); for(k=1,25,n=10^k;print1(floor( (log(r)+(n-2)*log(x))/log(10) )+1",")) \\ Herman Jamke (hermanjamke(AT)fastmail.fm), May 01 2007

a(n) = floor(log_10(r) + (10^n-2)*log_10(x)) + 1 for n >= 1, where x is the positive real root of the tetranacci limit equation x^4 - x^3 - x^2 - x - 1 = 0, x = 1.92756... and r is the positive real root of the tetranacci auxiliary equation 563r^4 - 20r^2 - 5r - 1 = 0, r = 0.293813... - Herman Jamke (hermanjamke(AT)fastmail.fm), May 01 2007

2 more terms from Ryan Propper, Jul 22 2005

More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), May 01 2007

A094776 a(n) = largest k such that the decimal representation of 2^k does not contain the digit n.

Original entry on oeis.org

86, 91, 168, 153, 107, 71, 93, 71, 78, 108

Offset: 0

Michael Taktikos, Jun 09 2004

These values are only conjectural.

The sequence could be extended to any nonnegative integer index n defining a(n) to be the largest k such that n does not appear as substring in the decimal expansion of 2^k. I conjecture that for n = 10, 11, 12, ... it continues (2000, 3020, 1942, 1465, 1859, 2507, 1950, 1849, 1850, ...). For example, curiously enough, the largest power of 2 in which the string "10" does not appear seems to be 2^2000. - M. F. Hasler, Feb 10 2023

			a(0) = 86 because 2^86 = 77371252455336267181195264 is conjectured to be the highest power of 2 that doesn't contain the digit 0.

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 71, p. 25, Ellipses, Paris 2008.

Tanya Khovanova, 86 Conjecture, T. K.'s Math Blog, Feb. 15, 2011.
Popular Computing (Calabasas, CA), Two Tables, Vol. 1, (No. 9, Dec 1973), page PC9-16.

Cf. A259081 - A259083.
Cf. A027870 and A065712 - A065744 (number of '0's, ..., '9's in 2^n).
Cf. A034293 (numbers k such that 2^k has no '2').

f[n_] := Block[{a = {}, k = 1}, While[k < 10000, If[ Position[ Union[ IntegerDigits[ 2^k, 10]], n] == {}, AppendTo[a, k]]; k++ ]; a]; Table[ f[n][[ -1]], {n, 0, 9}] (* Robert G. Wilson v, Jun 12 2004 *)

A094776(n,L=10*20^#Str(n))={forstep(k=L, 0, -1, foreach(digits(1<M. F. Hasler, Feb 13 2023

def A094776(n, L=0):
   n = str(n)
   for k in range(L if L else 10*20**len(n), 0, -1):
      if n not in str(2**k): return k # M. F. Hasler, Feb 13 2023

A095807 Number of integers from 0 to 10^n - 1 whose decimal digits include at least one 0.

Original entry on oeis.org

1, 10, 181, 2620, 33571, 402130, 4619161, 51572440, 564151951, 6077367550, 64696307941, 682266771460, 7140400943131, 74263608488170, 768372476393521, 7915352287541680, 81238170587875111

Offset: 1

Michael Taktikos, Aug 25 2004

			a(3)=181 because among the integers from 0 to 999 there are 181 numbers which contain at least 1 zero.

Vincenzo Librandi, Table of n, a(n) for n = 1..200
Index entries for linear recurrences with constant coefficients, signature (20,-109,90).

Cf. A016189.

[10^n + 9/8 - 9^(1+n)/8: n in [1..20]]; // Vincenzo Librandi, Aug 14 2013

LinearRecurrence[{20,-109,90},{1,10,181},20] (* or *) Rest[ CoefficientList[ Series[(1-19x+99x^2)/((1-x)(1-10x)(1-9x)),{x,0,20}], x]] (* Harvey P. Dale, Jun 20 2015 *)

a(n) = 10^n + 9/8 - 9^(1+n)/8; \\ Michel Marcus, Aug 13 2013

a(n) = 10^n + 9/8 - 9^(1+n)/8.
G.f.: (1-19*x+99*x^2)/((1-x)*(1-10*x)*(1-9*x)). - Vincenzo Librandi, Aug 14 2013
a(n) = 20*a(n-1) - 109*a(n-2) + 90*a(n-3); a(0)=1, a(1)=10, a(2)=181. - Harvey P. Dale, Jun 20 2015
Limit_{n->oo} a(n+1)/a(n) = 10. - Bernard Schott, Feb 28 2023

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User: Michael Taktikos

A102084 a(1) = 0; for n>0, write 2n=p+q (p, q prime), p*q maximal; then a(n)=p*q (see A073046).

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A096233 Number of digits of the 10^n-th tribonacci number (A000073).

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A097353 Number of digits of the (10^n)-th tetranacci number (A000078(10^n)).

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A094776 a(n) = largest k such that the decimal representation of 2^k does not contain the digit n.

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A095807 Number of integers from 0 to 10^n - 1 whose decimal digits include at least one 0.

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Magma

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A102084 a(1) = 0; for n>0, write 2n=p+q (p, q prime), pq maximal; then a(n)=pq (see A073046).