Search: seq:2,6,8,12,16
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A008407
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Minimal difference s(n) between beginning and end of n consecutive large primes (n-tuplet) permitted by divisibility considerations.
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+30
29
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0, 2, 6, 8, 12, 16, 20, 26, 30, 32, 36, 42, 48, 50, 56, 60, 66, 70, 76, 80, 84, 90, 94, 100, 110, 114, 120, 126, 130, 136, 140, 146, 152, 156, 158, 162, 168, 176, 182, 186, 188, 196, 200, 210, 212, 216, 226, 236, 240, 246, 252, 254, 264, 270, 272, 278
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OFFSET
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1,2
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COMMENTS
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Tony Forbes defines a prime k-tuplet (distinguished from a prime k-tuple) to be a maximally possible dense cluster of primes (a prime constellation) which will necessarily involve consecutive primes whereas a prime k-tuple is a prime cluster which may not necessarily be of maximum possible density (in which case the primes are not necessarily consecutive.)
a(n) >> n log log n; in particular, for any eps > 0, there is an N such that a(n) > (e^gamma - eps) n log log n for all n > N. Probably N can be chosen as 1; the actual rate of growth is larger. Can a larger growth rate be established? Perhaps a(n) ~ n log n. - Charles R Greathouse IV, Apr 19 2012
Conjecture: (i) The sequence a(n)^(1/n) (n=3,4,...) is strictly decreasing (to the limit 1). (ii) We have 0 < a(n)/n - H_n < (gamma + 2)/(log n) for all n > 4, where H_n denotes the harmonic number 1+1/2+1/3+...+1/n, and gamma refers to the Euler constant 0.5772... [The second inequality has been verified for n = 5, 6, ..., 5000.] - Zhi-Wei Sun, Jun 28 2013.
Conjecture: For any integer n > 2, there is 1 < k < n such that 2*n - a(k)- 1 and 2*n - a(k) + 1 are twin primes. Also, every n = 3, 4, ... can be written as p + a(k)/2 with p a prime and k an integer greater than one. - Zhi-Wei Sun, Jun 29-30 2013.
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REFERENCES
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R. K. Guy, "Unsolved Problems in Number Theory", lists a number of relevant papers in Section A8.
John Leech, "Groups of primes having maximum density", Math. Tables Aids to Comput., 12 (1958) 144-145.
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LINKS
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FORMULA
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s(k), k >= 2, is smallest s such that there exist B = {b_1, b_2, ..., b_k} with s = b_k - b_1 and such that for all primes p <= k, not all residues modulo p are represented by B.
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CROSSREFS
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KEYWORD
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nonn,nice,changed
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AUTHOR
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T. Forbes (anthony.d.forbes(AT)googlemail.com)
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EXTENSIONS
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Correction from Pat Weidhaas (weidhaas(AT)wotan.llnl.gov), Jun 15 1997
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STATUS
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approved
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A111051
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Numbers m such that 3*m^2 + 1 is prime.
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+30
7
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2, 6, 8, 12, 16, 20, 22, 26, 34, 36, 40, 58, 64, 68, 78, 82, 84, 86, 98, 112, 120, 126, 142, 146, 148, 152, 156, 160, 168, 188, 190, 194, 196, 208, 216, 218, 222, 238, 240, 244, 246, 254, 264, 272, 282, 286, 294, 300, 302, 306, 308, 316, 320, 330, 338, 344, 348
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OFFSET
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1,1
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COMMENTS
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The resulting primes are the generalized cuban primes of the form (x^3-y^3)/(x-y), x=y+2 (see A002648). - Jani Melik, Jul 18 2007
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LINKS
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FORMULA
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EXAMPLE
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1 + 3*2^2 = 13 = A002648(1) is the 1st prime of this form, so a(1) = 2.
1 + 3*6^2 = 109 = A002648(2) is the 2nd prime of this form, so a(2) = 6.
1 + 3*8^2 = 193 = A002648(3) is the 3rd prime of this form, so a(3) = 8.
If m=98 then 3*m^2 + 1 = 28813 = A002648(19) is prime (the 19th of this form), so 98 is a term (the 19th).
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MAPLE
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ts_kubpra_ind:=proc(n) local i, tren, ans; ans:=[ ]: for i from 0 to n do tren:=1+3*i^2: if (isprime(tren)='true') then ans:=[ op(ans), i ] fi od: RETURN(ans); end: ts_kubpra_ind(2000); # Jani Melik, Jul 18 2007
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MATHEMATICA
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PROG
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A083769
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a(1)=2; for n >= 2, a(n) = smallest even number such that a(1)*a(2)*...*a(n) + 1 is prime.
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+30
5
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2, 6, 8, 12, 16, 10, 4, 30, 26, 22, 24, 14, 50, 42, 18, 64, 46, 60, 32, 36, 20, 34, 28, 108, 48, 44, 68, 282, 90, 54, 76, 62, 180, 66, 132, 86, 74, 38, 58, 106, 120, 52, 244, 94, 100, 82, 138, 156, 98, 72, 172, 150, 248, 154, 166, 114, 162, 126, 124, 208, 222, 324, 212
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OFFSET
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1,1
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COMMENTS
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Is this a permutation of the even numbers?
For any even positive integers a_1, a_2, ..., a_n, there are infinitely many even positive integers t such that a_1 a_2 ... a_n t + 1 is prime: this follows from Dirichlet's theorem on primes in arithmetic progressions. As far as I know there is no guarantee that the sequence defined here leads to a permutation of the even numbers, i.e. there might be some even integer that never appears in the sequence. However, if the partial products a_1 ... a_n grow like 2^n n!, heuristically the probability of a_1 ... a_n t + 1 being prime is on the order of 1/log(a_1 ... a_n) ~ 1/(n log n), and since sum_n 1/(n log n) diverges we might expect that there should be infinitely many n for which some a_1 ... a_n t + 1 is prime, and thus every even integer should occur. - Robert Israel, Dec 20 2012
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LINKS
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EXAMPLE
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2+1=3, 2*6+1=13, 2*6*8+1=97, 2*6*8*12+1=1153, etc. are primes.
After 200 terms the prime is
224198929826405912196464851358435330956778558123234657623126\
069546460095464785674042966210907411841359152393200850271694\
899718487202330385432243578646330245831108247815285116235792\
875886417750289946171599027675234787802312202111702704952223\
563058999855839876391430601719636148884060097930252529666254\
756431522481046758186320659298713737639441014068272279177710\
551232067814381240340990584869121776471244800000000000000000\
00000000000000000000000000000 (449 digits). - Robert Israel, Dec 21 2012
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MAPLE
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N := 200: # number of terms desired
P := 2:
a[1] := 2:
C := {seq(2*j, j = 2 .. 10)}:
Cmax := 20:
for n from 2 to N do
for t in C do
if isprime(t*P+1) then
a[n]:= t;
P:= t*P;
C:= C minus {t};
break;
end if;
end do;
while not assigned(a[n]) do
t0:= Cmax+2;
Cmax:= 2*Cmax;
C:= C union {seq(j, j=t0 .. Cmax, 2)};
for t from t0 to Cmax by 2 do
if isprime(t*P+1) then
a[n]:= t;
P:= t*P;
C:= C minus {t};
break;
end if
end do;
end do;
end do;
[seq(a[n], n=1..N)];
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MATHEMATICA
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f[s_List] := Block[{k = 2, p = Times @@ s}, While[ MemberQ[s, k] || !PrimeQ[k*p + 1], k += 2]; Append[s, k]]; Nest[f, {2}, 62] (* Robert G. Wilson v, Dec 24 2012 *)
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CROSSREFS
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Cf. A036013, A046966, A046972, A051957, A073673, A073674, A083769, A083770, A083771, A084401, A084402, A084724, A087338.
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KEYWORD
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nonn
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AUTHOR
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Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), May 06 2003
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EXTENSIONS
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Comment edited, Maple code and additional terms by Robert Israel, Dec 20 2012
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STATUS
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approved
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A084724
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Beginning with 2, the smallest even number greater than the previous term such that every partial product + 1 is a prime.
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+30
5
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2, 6, 8, 12, 16, 18, 20, 22, 26, 36, 42, 44, 100, 120, 124, 162, 168, 174, 192, 218, 272, 278, 338, 364, 380, 392, 502, 512, 532, 560, 594, 614, 698, 790, 814, 838, 922, 938, 1072, 1082, 1092, 1102, 1146, 1182, 1256, 1354, 1360, 1484, 1508, 1566, 1662, 1690
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OFFSET
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1,1
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LINKS
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MATHEMATICA
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ppp[{p_, a_}]:=Module[{n=a+2}, While[!PrimeQ[p*n+1], n=n+2]; {p*n, n}]; NestList[ ppp, {2, 2}, 60][[All, 2]] (* Harvey P. Dale, Aug 12 2017 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), Jun 13 2003
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EXTENSIONS
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STATUS
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approved
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A136513
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Number of unit square lattice cells inside half-plane (two adjacent quadrants) of origin centered circle of diameter n.
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+30
5
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0, 0, 2, 2, 6, 8, 12, 16, 26, 30, 38, 44, 56, 60, 74, 82, 96, 108, 128, 138, 154, 166, 188, 196, 220, 238, 262, 278, 304, 324, 344, 366, 398, 416, 452, 468, 506, 526, 562, 588, 616, 644, 686, 714, 754, 780, 824, 848, 894, 930, 976, 1008, 1056, 1090, 1134, 1170
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OFFSET
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1,3
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LINKS
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FORMULA
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Lim_{n -> oo} a(n)/(n^2) -> Pi/8.
a(n) = 2 * Sum_{k=1..floor(n/2)} floor(sqrt((n/2)^2 - k^2)).
a(n) = [x^(n^2)] (theta_3(x^4) - 1)^2 / (2 * (1 - x)). - Ilya Gutkovskiy, Nov 24 2021
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EXAMPLE
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a(3) = 2 because a circle centered at the origin and of radius 3/2 encloses (-1,1) and (1,1) in the upper half plane.
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MATHEMATICA
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Table[2*Sum[Floor[Sqrt[(n/2)^2 -k^2]], {k, Floor[n/2]}], {n, 100}]
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PROG
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(Magma)
A136513:= func< n | n eq 1 select 0 else 2*(&+[Floor(Sqrt((n/2)^2-j^2)): j in [1..Floor(n/2)]]) >;
(SageMath)
def A136513(n): return 2*sum(isqrt((n/2)^2-k^2) for k in range(1, (n//2)+1))
(PARI) a(n) = 2*sum(k=1, n\2, sqrtint((n/2)^2-k^2)); \\ Michel Marcus, Jul 27 2023
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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Glenn C. Foster (gfoster(AT)uiuc.edu), Jan 02 2008
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STATUS
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approved
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2, 6, 8, 12, 16, 18, 22, 24, 28, 32, 34, 38, 42, 44, 48, 50, 54, 58, 60, 64, 66, 70, 74, 76, 80, 84, 86, 90, 92, 96, 100, 102, 106, 110, 112, 116, 118, 122, 126, 128, 132, 134, 138, 142, 144, 148, 152, 154, 158, 160, 164, 168, 170, 174, 176, 180, 184, 186
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OFFSET
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1,1
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COMMENTS
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This is one of three sequences that partition the positive integers. In general, suppose that r, s, t are positive real numbers for which the sets {i/r: i>=1}, {j/s: j>=1}, {k/t: k>=1} are pairwise disjoint. Let a(n) be the rank of n/r when all the numbers in the three sets are jointly ranked. Define b(n) and c(n) as the ranks of n/s and n/t. It is easy to prove that
a(n)=n+[ns/r]+[nt/r],
b(n)=n+[nr/s]+[nt/s],
c(n)=n+[nr/t]+[ns/t], where [ ]=floor.
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LINKS
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FORMULA
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a(n) = 2*floor(n*r), where r = (1+sqrt(5))/2.
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MATHEMATICA
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r = 1; s = (-1 + 5^(1/2))/2; t = (1 + 5^(1/2))/2;
a[n_] := n + Floor[n*s/r] + Floor[n*t/r];
b[n_] := n + Floor[n*r/s] + Floor[n*t/s];
c[n_] := n + Floor[n*r/t] + Floor[n*s/t]
Table[a[n], {n, 1, 120}] (* A283233 *)
Table[b[n], {n, 1, 120}] (* A283234 *)
Table[c[n], {n, 1, 120}] (* A005408 *)
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PROG
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(Python)
from math import isqrt
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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2, 6, 8, 12, 16, 20, 22, 26, 28, 32, 36, 40, 42, 46, 50, 54, 56, 60, 64, 68, 70, 74, 76, 80, 84, 88, 90, 94, 96, 100, 104, 108, 110, 114, 118, 122, 124, 128, 132, 136, 138, 142, 144, 148, 152, 156, 158, 162, 166, 170, 172, 176, 180, 184, 186, 190, 192, 196
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OFFSET
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1,1
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COMMENTS
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Conjecture: 0 < n*r - a(n) < 4 for n>=1, where r = 2 + sqrt(2).
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LINKS
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EXAMPLE
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As a word, A285341 = 1011101..., in which 0 is in positions 2,6,8,12,16,...
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MATHEMATICA
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s = Nest[Flatten[# /. {0 -> {1, 0}, 1 -> {1, 0, 1, 1}}] &, {0}, 10]; (* A285341 *)
u = Flatten[Position[s, 0]]; (* A285342 *)
Flatten[Position[s, 1]]; (* A285343 *)
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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A189400
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n+[nr/s]+[nt/s]; r=1, s=sqrt(e), t=e.
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+30
3
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2, 6, 8, 12, 16, 18, 22, 25, 28, 32, 35, 38, 41, 45, 48, 51, 55, 57, 61, 64, 67, 71, 73, 77, 81, 83, 87, 90, 93, 97, 100, 103, 107, 110, 113, 116, 120, 123, 126, 129, 132, 136, 139, 142, 146, 148, 152, 156, 158, 162, 165, 168, 172, 175, 178, 181, 184, 188, 191, 194, 197, 201, 204, 207, 211, 214, 217, 221, 223, 227, 231, 233, 237, 240, 243, 247, 249, 253, 256, 259, 263, 266, 269
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OFFSET
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1,1
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COMMENTS
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LINKS
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MATHEMATICA
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Table[n+Floor[n/Sqrt[E]]+Floor[n*Sqrt[E]], {n, 90}] (* Harvey P. Dale, Aug 20 2014 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A057656
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Number of points (x,y) in square lattice with (x-1/2)^2+y^2 <= n.
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+30
2
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2, 6, 8, 12, 16, 16, 22, 26, 26, 30, 34, 38, 40, 44, 44, 48, 56, 56, 60, 60, 62, 70, 74, 74, 78, 82, 82, 86, 90, 94, 96, 104, 104, 104, 108, 108, 116, 120, 124, 128, 128, 128, 134, 138, 138, 142, 150, 150, 154, 158, 158, 166, 166, 166, 166, 174
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OFFSET
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0,1
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REFERENCES
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J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 106.
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LINKS
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A077561
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Indices of terms of A025487 which divide the terms. Numbers k such that A025487(k) is a multiple of k.
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+30
2
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1, 2, 6, 8, 12, 16, 20, 24, 32, 45, 48, 66, 84, 90, 96, 108, 120, 140, 144, 150, 154, 162, 168, 175, 180, 192, 198, 200, 216, 220, 224, 240, 252, 264, 280, 288, 300, 315, 324, 336, 360, 375, 390, 396, 432, 486, 504, 525, 570, 576, 594
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OFFSET
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1,2
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LINKS
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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