A084967 -id:A084967 - cp's OEIS frontend

A249746 Permutation of natural numbers: a(n) = A126760(A249735(n)) = A249824(A064216(n)).

1, 2, 3, 4, 9, 5, 6, 12, 7, 8, 19, 10, 17, 42, 11, 13, 22, 26, 14, 29, 15, 16, 59, 18, 41, 32, 20, 31, 39, 21, 23, 92, 40, 24, 49, 25, 27, 82, 48, 28, 209, 30, 45, 52, 33, 63, 62, 54, 34, 109, 35, 36, 129, 37, 38, 69, 43, 68, 142, 70, 57, 72, 115, 44, 79, 46, 85, 292, 47, 50, 89, 74, 73, 202, 51, 53, 159, 87, 55, 99, 107, 56, 152, 58, 97, 192, 60

Offset: 1

Antti Karttunen, Nov 23 2014

nonn

Permutation obtained from the odd bisection of A003961 (or from the odd bisection of A048673).

			a(5) = 9 because of the following. 2*A064216(5) = 2*4 = 8 = 2^3. We replace the prime factor 2 of 8 with the next prime 3 to get 3^3, then replace 3 with 5 to get 5^3 = 125. The smallest prime factor of 125 is 5. 125 is the 9th term of A084967: 5, 25, 35, 55, 65, 85, 95, 115, 125, ..., thus a(5) = 9.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences that are permutations of the natural numbers

Inverse: A249745.
Row 2 of A251722.
Cf. A003961, A007310, A048673, A084967, A126760, A249735, A064216, A249824.
Cf. also A273664, A273669.

t = PositionIndex[FactorInteger[#][[1, 1]] & /@ Range[10^6]]; f[n_] := Times @@ Power[If[# == 1, 1, NextPrime@ #] & /@ First@ #, Last@ #] &@ Transpose@ FactorInteger@ n; Flatten@ Map[Position[Lookup[t, FactorInteger[#][[1, 1]] ], #] &[f@ f[2 #]] &, Table[Times @@ Power[If[# == 1, 1, NextPrime[#, -1]] & /@ First@ #, Last@ #] &@ Transpose@ FactorInteger[2 n - 1], {n, 87}]] (* Michael De Vlieger, Jul 25 2016, Version 10 *)

(define (A249746 n) (define (Ainv_of_A007310off0 n) (+ (* 2 (floor->exact (/ n 6))) (/ (- (modulo n 6) 1) 4))) (+ 1 (Ainv_of_A007310off0 (A003961 (+ n n -1)))))

a(n) = 1 + f(A003961(2n - 1)), where f(n) = 2*floor[n/6] + ((n mod 6)-1)/4. [Here 1 + f(A007310(n)) = n.]
a(n) = A126760(A249735(n)). - Antti Karttunen, Jul 25 2016
As a composition of related permutations:
a(n) = A249824(A064216(n)).
Other identities. For all n >= 1:
A249735(n) = A007310(a(n)).
a(3n-1) = A273669(a(n)) and a(A254049(n)) = A273664(a(n)). - Antti Karttunen, Aug 07 2016

A084969 Numbers whose smallest prime factor is 11.

Original entry on oeis.org

11, 121, 143, 187, 209, 253, 319, 341, 407, 451, 473, 517, 583, 649, 671, 737, 781, 803, 869, 913, 979, 1067, 1111, 1133, 1177, 1199, 1243, 1331, 1397, 1441, 1507, 1529, 1573, 1639, 1661, 1727, 1793, 1837, 1859, 1903, 1969, 1991, 2057, 2101, 2123, 2167, 2189, 2299, 2321

Offset: 1

Robert G. Wilson v, Jun 15 2003

Fifth row of A083140.

Integers k such that gcd(11*k, 210) = 1.

			a(2) = 11*11, a(3) = 11*13.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1).

Cf. A084967 (5), A084968 (7), A084970 (13), A332799 (17), A332798 (19), A332797 (23), A008364 (11-rough numbers).

Cf. A008364, A038110, A038111, A083140.

11Select[ Range[210], GCD[ #, 2*3*5*7] == 1 & ]
Select[11*Range[0,200],GCD[#,210]==1&] (* Harvey P. Dale, Dec 23 2013 *)

is(n)=gcd(n,2310)==11 \\ Charles R Greathouse IV, Nov 19 2014

G.f.: 11*x*(x^48 +10*x^47 +2*x^46 +4*x^45 +2*x^44 +4*x^43 +6*x^42 +2*x^41 +6*x^40 +4*x^39 +2*x^38 +4*x^37 +6*x^36 +6*x^35 +2*x^34 +6*x^33 +4*x^32 +2*x^31 +6*x^30 +4*x^29 +6*x^28 +8*x^27 +4*x^26 +2*x^25 +4*x^24 +2*x^23 +4*x^22 +8*x^21 +6*x^20 +4*x^19 +6*x^18 +2*x^17 +4*x^16 +6*x^15 +2*x^14 +6*x^13 +6*x^12 +4*x^11 +2*x^10 +4*x^9 +6*x^8 +2*x^7 +6*x^6 +4*x^5 +2*x^4 +4*x^3 +2*x^2 +10*x +1) / (x^49 -x^48 -x +1). - Colin Barker, Feb 22 2013
a(n) = a(n-48) + 2310 = a(n-1) + a(n-48) - a(n-49). - Charles R Greathouse IV, Nov 19 2014
Lim_{n->infinity} a(n)/n = A038111(5)/A038110(5) = 385/8 = 48.125. - Vladimir Shevelev, Jan 20 2015
a(n) = 11*A008364(n).

a(47)-a(49) from Georg Fischer, Nov 07 2019

New name from Frank Ellermann, Feb 25 2020

A249824 Permutation of natural numbers: a(n) = A078898(A003961(A003961(2*n))).

Original entry on oeis.org

1, 2, 3, 9, 4, 12, 5, 42, 17, 19, 6, 59, 7, 22, 26, 209, 8, 82, 10, 92, 31, 29, 11, 292, 41, 32, 115, 109, 13, 129, 14, 1042, 40, 39, 48, 409, 15, 49, 45, 459, 16, 152, 18, 142, 180, 52, 20, 1459, 57, 202, 54, 159, 21, 572, 63, 542, 68, 62, 23, 642, 24, 69, 213

Offset: 1

Antti Karttunen, Nov 06 2014

nonn

			a(4) = 9 because of the following. 2n = 2*4 = 8 = 2^3. We replace the prime factor 2 of 8 with the next prime 3 to get 3^3, then replace 3 with 5 to get 5^3 = 125. The smallest prime factor of 125 is 5. 125 is the 9th term of A084967: 5, 25, 35, 55, 65, 85, 95, 115, 125, ..., thus a(4) = 9.

Antti Karttunen, Table of n, a(n) for n = 1..255
Index entries for sequences that are permutations of the natural numbers

Inverse: A249823.
Row 3 of A249822.
Cf. A003961, A048673, A078898, A084967, A243071, A246278, A249734, A249746, A249826, A250475, A275716.
Cf. also A273664, A273669.

t = PositionIndex[FactorInteger[#][[1, 1]] & /@ Range[10^4]]; f[n_] := Times @@ Power[If[# == 1, 1, NextPrime@ #] & /@ First@ #, Last@ #] &@ Transpose@ FactorInteger@ n; Flatten@ Table[Position[Lookup[t, FactorInteger[#][[1, 1]] ], #] &[f@ f[2 n]], {n, 120}] (* Michael De Vlieger, Jul 25 2016, Version 10 *)

(define (A249824 n) (A078898 (A003961 (A003961 (* 2 n)))))

a(n) = A078898(A003961(A003961(2*n))) = A078898(A003961(A249734(n))).
a(n) = A078898(A246278(3,n)).
As a composition of other permutations:
a(n) = A249746(A048673(n)).
a(n) = A250475(A249826(n)).
a(n) = A275716(A243071(n)).
Other identities. For all n >= 1:
a(2n) = A273669(a(n)) and a(A003961(n)) = A273664(a(n)). -- Antti Karttunen, Aug 07 2016

A255413 a(n) = 15*n - 11 + (n mod 2). Row 3 of Ludic array A255127.

Original entry on oeis.org

5, 19, 35, 49, 65, 79, 95, 109, 125, 139, 155, 169, 185, 199, 215, 229, 245, 259, 275, 289, 305, 319, 335, 349, 365, 379, 395, 409, 425, 439, 455, 469, 485, 499, 515, 529, 545, 559, 575, 589, 605, 619, 635, 649, 665, 679, 695, 709, 725, 739, 755, 769, 785, 799, 815, 829, 845, 859, 875, 889, 905, 919, 935, 949, 965, 979, 995, 1009

Offset: 1

Antti Karttunen, Feb 22 2015

Antti Karttunen, Table of n, a(n) for n = 1..10001
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Row 3 of A255127.

Cf. A007310, A016921, A084967, A255407.

a[n_] := 15 n + Mod[n, 2] - 11;
Array[a, 100] (* Jean-François Alcover, Mar 14 2016 *)

apply( {A255413(n)=15*n-11+n%2}, [1..50]) \\ M. F. Hasler, Nov 09 2024

;; two alternatives:
(define (A255413 n) (A255127bi 3 n)) ;; Code for A255127bi given in A255127.
(define (A255413 n) (A007310 (- (* 5 n) 3)))

a(n) = A007310((5*n)-3).
a(n) = A255407(A084967(n)) = A255407(5*A007310(n)).
a(2n+1) = 5*A016921(n) for all n >= 0.
From M. F. Hasler, Nov 09 2024: (Start)
a(n) = a(n-1) + a(n-2) + a(n-3) for n > 3, a(1..3) = (5, 19, 35).
a(n) = a(n-2) + 30 for n > 2, with a(1..2) = (5, 19).
a(2n-1) = 30n - 25, a(2n) = 30n - 11.
G.f.: x*(5 + 14*x + 11*x^2)/((1 - x)^2*(1 + x)). (End)
E.g.f.: 11 + (15*x - 11)*cosh(x) + 5*(3*x - 2)*sinh(x). - Stefano Spezia, Nov 12 2024

New definition from M. F. Hasler, Nov 09 2024

A332797 Numbers whose smallest prime factor is 23.

Original entry on oeis.org

23, 529, 667, 713, 851, 943, 989, 1081, 1219, 1357, 1403, 1541, 1633, 1679, 1817, 1909, 2047, 2231, 2323, 2369, 2461, 2507, 2599, 2921, 3013, 3151, 3197, 3427, 3473, 3611, 3749, 3841, 3979, 4117, 4163, 4393, 4439, 4531, 4577, 4853, 5129, 5221, 5267, 5359, 5497

Offset: 1

Frank Ellermann, Feb 24 2020

The asymptotic density of this sequence is 55296/7436429. - Amiram Eldar, Dec 06 2020

			a(2) = 23*23, a(3) = 23*29.

Emmanuel Desurvire, Classical and Quantum Information Theory: An Introduction for the Telecom Scientist, Cambridge University Press, 2009, table 20.5 p. 421.

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

Cf. A084967 (5), A084968 (7), A084969 (11), A084970 (13), A332799 (17), A332798 (19), A166063 (23-rough numbers).

23 * Select[Range[240], CoprimeQ[#, 9699690] &] (* Amiram Eldar, Feb 24 2020 *)
Select[Range[6000],FactorInteger[#][[1,1]]==23&] (* Harvey P. Dale, Aug 29 2025 *)

P = 23         ;  S = P
do N = P by 2 while length( S ) < 255
   do I = 1 until P = X
      X = PRIME( I )
      if P = X       then  leave I
      if N // X = 0  then  iterate N
   end I
   S = S || ',' P*N
end N
say S          ;  return S

a(n) = 23*A166063(n).

A332798 Numbers whose smallest prime factor is 19.

Original entry on oeis.org

19, 361, 437, 551, 589, 703, 779, 817, 893, 1007, 1121, 1159, 1273, 1349, 1387, 1501, 1577, 1691, 1843, 1919, 1957, 2033, 2071, 2147, 2413, 2489, 2603, 2641, 2831, 2869, 2983, 3097, 3173, 3287, 3401, 3439, 3629, 3667, 3743, 3781, 4009, 4237, 4313, 4351, 4427

Offset: 1

Frank Ellermann, Feb 24 2020

The asymptotic density of this sequence is 3072/323323. - Amiram Eldar, Dec 06 2020

			a(2) = 19*19, a(3) = 19*23.

Emmanuel Desurvire, Classical and Quantum Information Theory: An Introduction for the Telecom Scientist, Cambridge University Press, 2009, table 20.5 p. 421.

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

Cf. A084967 (5), A084968 (7), A084969 (11), A084970 (13), A332799 (17), A332797 (23), A166061 (19-rough numbers).

19 * Select[Range[235], CoprimeQ[#, 510510] &] (* Amiram Eldar, Feb 24 2020 *)

P = 19         ;  S = P
do N = P by 2 while length( S ) < 255
   do I = 1 until P = X
      X = PRIME( I )
      if P = X       then  leave I
      if N // X = 0  then  iterate N
   end I
   S = S || ',' P*N
end N
say S          ;  return S

a(n) = 19*A166061(n).

A247225 a(n) = n if n <= 3, a(4)=5, otherwise the smallest number not occurring earlier having at least one common factor with a(n-3), but none with a(n-1)*a(n-2).

Original entry on oeis.org

1, 2, 3, 5, 4, 9, 25, 8, 21, 55, 16, 7, 11, 6, 35, 121, 12, 49, 143, 10, 63, 13, 20, 27, 91, 22, 15, 119, 26, 33, 17, 14, 39, 85, 28, 57, 65, 32, 19, 45, 34, 133, 69, 40, 77, 23, 18, 175, 253, 24, 95, 161, 36, 125, 203, 38, 75, 29, 44, 51, 145, 46, 81, 155, 52

Offset: 1

Vladimir Shevelev, Jan 11 2015

nonn

Conjecturally the sequence is a permutation of the positive integers. However, to prove this we need more subtle arguments than were used to prove the corresponding property for A098550. - Vladimir Shevelev, Jan 14 2015
For n <= 2000, a(3n-1) is even and both a(3n) and a(3n-2) are odd numbers. I conjecture that this is true for all positive integers n. This conjecture is true iff for all positive integers n, a(3n-1) is even. - Farideh Firoozbakht, Jan 14 2015
From Vladimir Shevelev, Jan 19 2015:  (Start)
A generalization of A098550 and A247225.
Let p_n=prime(n). Define the following sequence
a(1)=1, a(2)=p_1,...,a(k+2)=p_(k+1), otherwise the smallest number not occurring earlier having at least one common factor with a(n-(k+1)), but none with a(n-1)*a(n-2)*...*a(n-k).
The sequence begins
1, p_1, p_2, ..., p_(k+1), p_1^2, p_2^2, ..., p_(k+1)^2, p_1^3, ... (*)
[ p_1^3 is followed by p_2*p_(k+2), k<=2,
p_2^3, k>=3, etc.]
In particular, if k=1, it is A098550, if k=2, it is A247225.
Conjecturally for every k>=2, as in the case k=1, the sequence (*) is a permutation of the positive integers. For k>=3, at first glance, already the appearance of the number 6 seems problematic. However, at the author's request, Peter J. C. Moses found that the positions of 6 are 83, 157, 1190, 206, ... in cases k=3,4,5,6,... respectively (A254003).
Note also that for every k>=2, every even term is followed by k odd terms. This is explained by the minimal growth of even numbers (2n) relatively with one of the numbers with the smallest prime divisor p>=3 (asymptotically 6n, 15n, 105n/4, 385n/8, ... for p = 3,5,7,11,... respectively (cf. A084967 - A084970)).
(End)

Peter J. C. Moses, Table of n, a(n) for n = 1..2000
David L. Applegate, Hans Havermann, Bob Selcoe, Vladimir Shevelev, N. J. A. Sloane, and Reinhard Zumkeller, The Yellowstone Permutation, arXiv preprint arXiv:1501.01669, 2015.

Cf. A098550, A247942, A249167, A251604, A254003.

a[n_ /; n <= 3] := n; a[4]=5; a[n_] := a[n] = For[aa = Table[a[j], {j, 1, n-1}]; k=4, True, k++, If[FreeQ[aa, k] && !CoprimeQ[k, a[n-3]] && CoprimeQ[k, a[n-1]*a[n-2]], Return[k]]]; Table[ a[n], {n, 1, 65}] (* Jean-François Alcover, Jan 12 2015 *)

More terms from Peter J. C. Moses, Jan 12 2015

A328589 Numbers n that are multiples of 6 and for which A257993(A276086(A276086(n))) is larger than A257993(n), where A276086 converts the primorial base expansion of n into its prime product form, and A257993 returns the index of the least prime not present in its argument.

Original entry on oeis.org

240, 270, 300, 330, 360, 390, 630, 660, 690, 720, 750, 780, 810, 1050, 1080, 1110, 1140, 1170, 1200, 1230, 1500, 1530, 1560, 1590, 1620, 1650, 1890, 1920, 1950, 1980, 2010, 2040, 2070, 2550, 2580, 2610, 2640, 2670, 2700, 2940, 2970, 3000, 3030, 3060, 3090, 3120, 3360, 3390, 3420, 3450, 3480, 3510, 3540, 3810, 3840, 3870, 3900, 3930, 3960, 4200

Offset: 1

Antti Karttunen, Oct 21 2019

nonn

Multiples of six such that the least nondivisor prime of the original n is less than the least nondivisor prime of the number obtained after two iterations of A276086 is.

All terms are multiples of 5 (and thus of 30), because when applied to any number which is a multiple of 6, but not of 5 (and thus not a multiple of 30, so the primorial expansion ends with "x00", where x <> 0, and A257993(n) = 3), A276086 will yield a number of the form 30k+5 or 30k+25 (A084967) whose primorial expansion ends either as "...021" or as "...401" (with the least significant zero either in position 2 or 3), thus then A328578(n) = A257993(A276086(A276086(n))) is definitely not larger than 3, while A257993(6k) >= 3 for all k >= 1.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences related to primorial base

Subsequence of A328588.
Other multiples of 6 are either in A328586 or in A328587.
Cf. A049345, A084967, A257993, A276086, A328578, A328585, A328588, A328590, A328762.

A257993(n) = { for(i=1,oo,if(n%prime(i),return(i))); }
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A328578(n) = A257993(A276086(A276086(n)));
isA328589(n) = ((!(n%6))&&(A328578(n) > A257993(n)));

A332799 Numbers whose smallest prime factor is 17.

Original entry on oeis.org

17, 289, 323, 391, 493, 527, 629, 697, 731, 799, 901, 1003, 1037, 1139, 1207, 1241, 1343, 1411, 1513, 1649, 1717, 1751, 1819, 1853, 1921, 2159, 2227, 2329, 2363, 2533, 2567, 2669, 2771, 2839, 2941, 3043, 3077, 3247, 3281, 3349, 3383, 3587, 3791, 3859, 3893

Offset: 1

Frank Ellermann, Feb 24 2020

The asymptotic density of this sequence is 192/17017. - Amiram Eldar, Dec 06 2020

			a(2) = 17*17, a(3) = 17*19.

Emmanuel Desurvire, Classical and Quantum Information Theory: An Introduction for the Telecom Scientist, Cambridge University Press, 2009, table 20.5 p. 421.

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

Cf. A084967 (5), A084968 (7), A084969 (11), A084970 (13), A332798 (19), A332797 (23), A008366 (17-rough numbers).

17 * Select[Range[230], CoprimeQ[#, 30030] &] (* Amiram Eldar, Feb 24 2020 *)

P = 17         ;  S = P
do N = P by 2 while length( S ) < 255
   do I = 1 until P = X
      X = PRIME( I )
      if P = X       then  leave I
      if N // X = 0  then  iterate N
   end I
   S = S || ',' P*N
end N
say S          ;  return S

a(n) = 17*A008366(n).

A328633 Numbers n for which A328578(n) = A257993(A276086(A276086(n))) = 3, where A276086 converts the primorial base expansion of n into its prime product form, and A257993 returns the index of the least prime not present in its argument.

Original entry on oeis.org

2, 6, 18, 34, 36, 48, 66, 78, 96, 108, 122, 126, 138, 154, 156, 168, 186, 198, 212, 222, 234, 244, 252, 264, 282, 294, 312, 324, 332, 342, 354, 364, 372, 384, 402, 414, 422, 426, 438, 454, 456, 468, 486, 498, 516, 528, 542, 546, 558, 574, 576, 588, 606, 618, 632, 642, 654, 664, 672, 684, 702, 714, 732, 744, 752, 762, 774, 784, 792, 804

Offset: 1

Antti Karttunen, Oct 27 2019

nonn

Numbers n for which A276087(n) is a multiple of 6, but not of 5.
Question: Is the even bisection of A328316, starting from A328316(4) as: 6, 18, 43218, ..., a subsequence of this sequence? See also A328317.
Subsequence such that both k and A276087(k) are in this sequence starts as: 2, 6, 18, 34, 36, 48, 66, 154, 156, 186, 234, 244, 294, 312, 324, 354, 364, 384, 426, 438, 454, 456, 542, 546, 558, 588, 606, ...
When A276086 is applied to any number which is a multiple of 6, but not of 5 (and thus not a multiple of 30, implying that the number's primorial expansion ends with "x00", where x <> 0, and A257993(n) = 3), the original number will be converted to a number of the form 30k+5 or 30k+25 (A084967) whose primorial expansion ends either as "...021" or as "...401", with the least significant zero in position A328578(n), which is seen to be always either 3 or 2.

			294 = 7^2 * 3 * 2 has primorial base expansion (A049345) "12400", which, when converted to a prime product form (A276086) yields 11^1 * 7^2 * 5^4 * 3^0 * 2^0 = 336875. This in turn has primorial base representation [11,2,9,1,0,2,1], which when converted to prime product form gives 17^11 * 13^2 * 11^9 * 7^1 * 5^0 * 3^2 * 2^1 = 1720796647657111567992931482, which has the required property of being a multiple of 6 but not of 5, thus 294 is included in this sequence.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences related to primorial base

Row 3 of A328631.
Cf. A049345, A257993, A276086, A276087, A328578.
Cf. also A328316, A328317, A328589, A328762.

A257993(n) = { for(i=1,oo,if(n%prime(i),return(i))); }
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A328578(n) = A257993(A276086(A276086(n)));
isA328633(n) = (3==A328578(n));

isA328633(n) = { my(u=A276086(A276086(n))); ((u%5)&&!(u%6)); };

cp's OEIS Frontend

A249746 Permutation of natural numbers: a(n) = A126760(A249735(n)) = A249824(A064216(n)).

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A084969 Numbers whose smallest prime factor is 11.

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A249824 Permutation of natural numbers: a(n) = A078898(A003961(A003961(2*n))).

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A255413 a(n) = 15*n - 11 + (n mod 2). Row 3 of Ludic array A255127.

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A332797 Numbers whose smallest prime factor is 23.

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A332798 Numbers whose smallest prime factor is 19.

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A247225 a(n) = n if n <= 3, a(4)=5, otherwise the smallest number not occurring earlier having at least one common factor with a(n-3), but none with a(n-1)*a(n-2).

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