author:guy keyword:nice - cp's OEIS frontend

A051280 Numbers n such that n = k/d(k) has exactly 3 solutions, where d(k) = number of divisors of k.

3, 25, 40, 49, 54, 121, 125, 135, 140, 169, 189, 216, 220, 250, 260, 289, 297, 340, 351, 361, 375, 380, 400, 459, 460, 500, 513, 529, 580, 620, 621, 675, 729, 740, 770, 783, 820, 837, 841, 860, 875, 882, 910, 940, 961, 999, 1060, 1107, 1152, 1161, 1180, 1188, 1190

Offset: 1

N. J. A. Sloane, R. K. Guy, David W. Wilson

Many terms are of the form a(k) * p^m/(m+1), where p is coprime to the three solutions for k. The sequence of "primitive" terms (i.e. not expressible this way) begins 3, 40, 54, 125, 135, 216, 250.... Are there any such numbers that admit a fourth solution? - Charlie Neder, Feb 13 2019

			There are exactly 3 numbers k, 9, 18 and 24, with k/d(k) = 3.

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

Cf. A033950, A036763, A051278, A051279, A051346.

(Select[Table[k / Length @ Divisors[k], {k, 1, 200000}], IntegerQ] // Sort // Split // Select[#, Length[#] == 3 &] &)[[All, 1]][[1 ;; 53]] (* Jean-François Alcover, Apr 22 2011 *)

A003320 a(n) = max_{k=0..n} k^(n-k).

Original entry on oeis.org

1, 1, 1, 2, 4, 9, 27, 81, 256, 1024, 4096, 16384, 78125, 390625, 1953125, 10077696, 60466176, 362797056, 2176782336, 13841287201, 96889010407, 678223072849, 4747561509943, 35184372088832, 281474976710656, 2251799813685248

Offset: 0

N. J. A. Sloane, R. K. Guy

For n > 0, a(n+1) = largest term of row n in triangles A051129 and A247358. - Reinhard Zumkeller, Sep 14 2014

			a(5) = max(5^0, 4^1, 3^2, 2^3, 1^4, 0^5) = max(1,4,9,8,1,0) = 9.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
I. Tomescu, Introducere in Combinatorica. Editura Tehnica, Bucharest, 1972, p. 231.

Seiichi Manyama, Table of n, a(n) for n = 0..599 (terms 0..100 from T. D. Noe).
D. Easdown, Minimal faithful permutation and transformation representations of groups and semigroups, Contemporary Math. (1992), Vol. 131 (Part 3), 75-84.
R. Gray and J. D. Mitchell, Largest subsemigroups of the full transformation monoid, Discrete Math., 308 (2008), 4801-4810.
W. S. Gray and M. Thitsa, System Interconnections and Combinatorial Integer Sequences, in: System Theory (SSST), 2013 45th Southeastern Symposium on, Date of Conference: 11-11 March 2013, Digital Object Identifier: 10.1109/SSST.2013.6524939.
R. K. Guy, Letter to N. J. A. Sloane, Mar 1974
I. Tomescu, Excerpts from "Introducese in Combinatorica" (1972), pp. 230-1, 44-5, 128-9. (Annotated scanned copy)

Cf. A003992, A031435, A056155.

a003320 n = maximum $ zipWith (^) [0 .. n] [n, n-1 ..]
-- Reinhard Zumkeller, Jun 24 2013

Join[{1},Max[#]&/@Table[k^(n-k),{n,25},{k,n}]] (* Harvey P. Dale, Jun 20 2011 *)

a(n) = vecmax(vector(n+1, k, (k-1)^(n-k+1))); \\ Michel Marcus, Jun 13 2017

a(n) = A056155(n-1)^(n - A056155(n-1)), for n >= 2. - Ridouane Oudra, Dec 09 2020

Easdown reference from Michail Kats (KatsMM(AT)info.sgu.ru)

More terms from James Sellers, Aug 21 2000

A007586 11-gonal (or hendecagonal) pyramidal numbers: a(n) = n(n+1)(3*n-2)/2.

Original entry on oeis.org

0, 1, 12, 42, 100, 195, 336, 532, 792, 1125, 1540, 2046, 2652, 3367, 4200, 5160, 6256, 7497, 8892, 10450, 12180, 14091, 16192, 18492, 21000, 23725, 26676, 29862, 33292, 36975, 40920, 45136, 49632, 54417, 59500, 64890, 70596, 76627, 82992, 89700, 96760, 104181

Offset: 0

N. J. A. Sloane, R. K. Guy

Starting with 1 equals binomial transform of [1, 11, 19, 9, 0, 0, 0, ...]. - Gary W. Adamson, Nov 02 2007

			From _Vincenzo Librandi_, Feb 12 2014: (Start)
After 0, the sequence is provided by the row sums of the triangle (see above, third formula):
  1;
  2, 10;
  3, 20, 19;
  4, 30, 38, 28;
  5, 40, 57, 56, 37;
  6, 50, 76, 84, 74, 46; etc. (End)

A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 194.
E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 93.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A051682.
Cf. A093644 ((9,1) Pascal, column m=3).
Cf. similar sequences listed in A237616.

List([0..45], n-> n*(n+1)*(3*n-2)/2); # G. C. Greubel, Aug 30 2019

I:=[0,1,12,42]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4) : n in [1..50]]; // Vincenzo Librandi, Feb 12 2014

seq(n*(n+1)*(3*n-2)/2, n=0..45); # G. C. Greubel, Aug 30 2019

Table[n(n+1)(3n-2)/2,{n,0,45}] (* or *) LinearRecurrence[{4,-6,4,-1},{0,1,12,42}, 45] (* Harvey P. Dale, Apr 09 2012 *)
CoefficientList[Series[x(1+8x)/(1-x)^4, {x, 0, 45}], x] (* Vincenzo Librandi, Feb 12 2014 *)

a(n)=n*(n+1)*(3*n-2)/2 \\ Charles R Greathouse IV, Oct 07 2015

[n*(n+1)*(3*n-2)/2 for n in (0..45)] # G. C. Greubel, Aug 30 2019

G.f.: x*(1+8*x)/(1-x)^4.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n>3, a(0)=0, a(1)=1, a(2)=12, a(3)=42. - Harvey P. Dale, Apr 09 2012
a(n) = Sum_{i=0..n-1} (n-i)*(9*i+1), with a(0)=0. - Bruno Berselli, Feb 10 2014
From Amiram Eldar, Jun 28 2020: (Start)
Sum_{n>=1} 1/a(n) = (9*log(3) + sqrt(3)*Pi - 4)/10.
Sum_{n>=1} (-1)^(n+1)/a(n) = (sqrt(3)*Pi + 2 - 4*log(2))/5. (End)
E.g.f.: exp(x)*x*(2 + 10*x + 3*x^2)/2. - Elmo R. Oliveira, Aug 03 2025

More terms from Vincenzo Librandi, Feb 12 2014

A006036 Primitive pseudoperfect numbers.

Original entry on oeis.org

6, 20, 28, 88, 104, 272, 304, 350, 368, 464, 490, 496, 550, 572, 650, 748, 770, 910, 945, 1184, 1190, 1312, 1330, 1376, 1430, 1504, 1575, 1610, 1696, 1870, 1888, 1952, 2002, 2030, 2090, 2170, 2205, 2210, 2470, 2530, 2584, 2590, 2870, 2990, 3010, 3128, 3190, 3230, 3290, 3410, 3465, 3496, 3710, 3770, 3944, 4070, 4095, 4130, 4216, 4270, 4288, 4408, 4510, 4544, 4672, 4690, 4712, 4730, 4970

Offset: 1

N. J. A. Sloane, R. K. Guy

A primitive pseudoperfect number is a pseudoperfect number that is not a multiple of any other pseudoperfect number.
The odd entries so far are identical to the odd primitive abundant A006038. - Walter Kehowski, Aug 12 2005
Zachariou and Zachariou (1972) called these numbers "irreducible semiperfect numbers". - Amiram Eldar, Dec 04 2020

Richard K. Guy, Unsolved Problems in Number Theory, 3rd edition, Springer, 2004, Section B2, pp. 74-75.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Donovan Johnson, Table of n, a(n) for n = 1..10000
Richard K. Guy, Letter to N. J. A. Sloane with attachment, Jun. 1991
Eric Weisstein's World of Mathematics, Primitive Pseudoperfect Number.
Andreas Zachariou and Eleni Zachariou, Perfect, Semi-Perfect and Ore Numbers, Bull. Soc. Math. Grèce (New Ser.), Vol. 13, No. 13A (1972), pp. 12-22; alternative link.

Cf. A005835.

Cf. A210455, A027751.

a006036 n = a006036_list !! (n-1)
a006036_list = filter (all (== 0) . map a210455 . a027751_row) a005835_list
-- Reinhard Zumkeller, Jan 21 2013

with(numtheory): with(combinat): issemiperfect := proc(n) local b, S;
b:=false; S:=subsets(divisors(n) minus {n}); while not S[finished] do if
convert(S[nextvalue](),`+`)=n then b:=true; break fi od; return b end:
L:=remove(proc(z) isprime(z) end,[$1..5000]): PP:=[]: for zz from 1 to 1 do
for n in L do if issemiperfect(n) then PP:=[op(PP),n] fi od od;
sr := proc(l::list) local x, R, S, P, L; S:=sort(l); R:=[]; P:=S;
for x in S do
if not(x in R) then
L:=selectremove(proc(z) z>x and z mod x = 0 end, P);
R:=[op(R),op(L[1])]; P:=L[2];
fi; od; return P; end:
PPP:=sr(PP); # primitive pseudoperfect numbers less than 5000 # Walter Kehowski, Aug 12 2005

(* First run one of the programs for A005835 *) A006036 = A005835; curr = 1; max = A005835[[-1]]; While[curr < Length[A006036], currMult = A006036[[curr]]; A006036 = Complement[A006036, Range[2currMult, Ceiling[max/currMult] currMult, currMult]]; curr++]; A006036 (* Alonso del Arte, Sep 08 2012 *)

More terms from Walter Kehowski, Aug 12 2005

A014090 Numbers that are not the sum of a square and a prime.

Original entry on oeis.org

1, 10, 25, 34, 58, 64, 85, 91, 121, 130, 169, 196, 214, 226, 289, 324, 370, 400, 526, 529, 625, 676, 706, 730, 771, 784, 841, 1024, 1089, 1225, 1255, 1351, 1414, 1444, 1521, 1681, 1849, 1906, 1936, 2116, 2209, 2304, 2500, 2809, 2986, 3136, 3364, 3481, 3600

Offset: 1

N. J. A. Sloane, R. K. Guy

Sequence is infinite: if 2n-1 is composite then n^2 is in the sequence. (Proof: If n^2 = x^2 + p with p prime, then p = (n-x)(n+x), so n-x=1 and n+x=p. Hence 2n-1=p is prime, not composite.) - Dean Hickerson, Nov 27 2002
21679 is the last known nonsquare in this sequence. See A020495. - T. D. Noe, Aug 05 2006
A002471(a(n))=0; complement of A014089. - Reinhard Zumkeller, Sep 07 2008
There are no prime numbers in this sequence because at the very least they can be represented as p + 0^2. - Alonso del Arte, May 26 2012
Number of terms <10^k,k=0..8: 1, 8, 27, 75, 223, 719, 2361, 7759, ..., . - Robert G. Wilson v, May 26 2012
So far there are only 21 terms which are not squares and they are the terms of A020495. Those that are squares, their square roots are members of A104275. - Robert G. Wilson v, May 26 2012

			From _Alonso del Arte_, May 26 2012: (Start)
10 is in the sequence because none of 10 - p_i are square (8, 7, 5, 3) and none of 10 - b^2 are prime (10, 9, 6, 1); i goes from 1 to pi(10) or b goes from 0 to floor(sqrt(10)).
11 is not in the sequence because it can be represented as 3^2 + 2 or 0^2 + 11. (End)

Robert G. Wilson v, Table of n, a(n) for n = 1..10000 (first 115 terms from T. D. Noe)

Cf. A020495, A104275.

Cf. A064233 (does not allow 0^2).

t={}; Do[k=0; While[k^2=n, AppendTo[t,n]], {n,25000}]; t (* T. D. Noe, Aug 05 2006 *)
max = 5000; Complement[Range[max], Flatten[Table[Prime[p] + b^2, {p, PrimePi[max]}, {b, 0, Ceiling[Sqrt[max]]}]]] (* Alonso del Arte, May 26 2012 *)
fQ[n_] := Block[{j = Sqrt[n], k}, If[ IntegerQ[j] && !PrimeQ[2j - 1], True, k = Floor[j]; While[k > -1 && !PrimeQ[n - k^2], k--]; If[k == -1, True, False]]]; Select[ Range[3600], fQ] (* Robert G. Wilson v, May 26 2012 *)

A045947 Triangles in open triangular matchstick arrangement (triangle minus one side) of side n.

Original entry on oeis.org

0, 0, 2, 7, 17, 33, 57, 90, 134, 190, 260, 345, 447, 567, 707, 868, 1052, 1260, 1494, 1755, 2045, 2365, 2717, 3102, 3522, 3978, 4472, 5005, 5579, 6195, 6855, 7560, 8312, 9112, 9962, 10863, 11817, 12825, 13889, 15010, 16190, 17430, 18732, 20097, 21527

Offset: 0

R. K. Guy

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (3,-2,-2,3,-1).

First differences of A082289.

Cf. A000447, A002412, A051895.

[Floor((4*n^3+2*n^2-4*n)/16): n in [0..50]]; // Vincenzo Librandi, Aug 29 2011

LinearRecurrence[{3, -2, -2, 3, -1}, {0, 0, 2, 7, 17}, 45] (* Jean-François Alcover, Dec 12 2016 *)
CoefficientList[Series[(2x^2+x^3)/((1-x)^3(1-x^2)),{x,0,50}],x] (* Harvey P. Dale, Jun 26 2021 *)

PARI
```
a(n)=(4*n^3+2*n^2-4*n)\16
```

G.f.: (2*x^2+x^3)/((1-x)^3*(1-x^2)). - Michael Somos
a(n) = (1/16)*(2*n*(2*n^2+n-2)+(-1)^n-1). - Bruno Berselli, Aug 29 2011
a(2*n) = A000447(n)+A002412(n); a(2*n+1) = A051895(n). - J. M. Bergot, Apr 12 2018
E.g.f.: (x*(1 + 7*x + 2*x^2)*cosh(x) - (1 - x - 7*x^2 - 2*x^3)*sinh(x))/8. - Stefano Spezia, Aug 22 2023

A045946 Star of David matchstick numbers: a(n) = 6n(3*n+1).

Original entry on oeis.org

0, 24, 84, 180, 312, 480, 684, 924, 1200, 1512, 1860, 2244, 2664, 3120, 3612, 4140, 4704, 5304, 5940, 6612, 7320, 8064, 8844, 9660, 10512, 11400, 12324, 13284, 14280, 15312, 16380, 17484, 18624, 19800, 21012, 22260, 23544, 24864, 26220, 27612, 29040, 30504, 32004

Offset: 0

R. K. Guy

Vertical spoke of triangular spiral in A051682. - Paul Barry, Mar 15 2003

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

Cf. A005449, A033580, A049451, A045945, A051682, A081266.

Table[6n(3n+1),{n,0,40}] (* or *) LinearRecurrence[{3,-3,1},{0,24,84},40] (* Harvey P. Dale, Nov 23 2012 *)

a(n)=18*n^2+6*n \\ Charles R Greathouse IV, Feb 19 2017

a(n) = 24*C(n,1) + 36*C(n,2); binomial transform of (0, 24, 36, 0, 0, 0, ...). - Paul Barry, Mar 15 2003
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3); a(0)=0, a(1)=24, a(2)=84. - Harvey P. Dale, Nov 23 2012
G.f.: 12*x*(2+x)/(1-x)^3. - Ivan Panchenko, Nov 13 2013
a(n) = 2*A045945(n). - Michel Marcus, Nov 13 2013
a(n) = 12*A005449(n). - R. J. Mathar, Feb 08 2016
From Amiram Eldar, Jan 14 2021: (Start)
Sum_{n>=1} 1/a(n) = 1/2 - Pi/(12*sqrt(3)) - log(3)/4.
Sum_{n>=1} (-1)^(n+1)/a(n) = -1/2 + Pi/(6*sqrt(3)) + log(2)/3. (End)
From Elmo R. Oliveira, Dec 12 2024: (Start)
E.g.f.: 6*exp(x)*x*(4 + 3*x).
a(n) = 6*A049451(n) = 4*A081266(n) = 3*A033580(n). (End)

A039955 Squarefree numbers congruent to 1 (mod 4).

Original entry on oeis.org

1, 5, 13, 17, 21, 29, 33, 37, 41, 53, 57, 61, 65, 69, 73, 77, 85, 89, 93, 97, 101, 105, 109, 113, 129, 133, 137, 141, 145, 149, 157, 161, 165, 173, 177, 181, 185, 193, 197, 201, 205, 209, 213, 217, 221, 229, 233, 237, 241, 249, 253, 257, 265, 269

Offset: 1

R. K. Guy

The subsequence of primes is A002144.
The subsequence of semiprimes (intersection with A001358) begins: 21, 33, 57, 65, 69, 77, 85, 93, 129, 133, 141, 145, 161, 177, 185, 201, 205, 209, 213, 217, 221, 237, 249, 253, 265.
The subsequence with more than two prime factors (intersection with A033942) begins: 105 = 3 * 5 * 7, 165 = 3 * 5 * 11, 273, 285, 345, 357, 385, 429, 465. - Jonathan Vos Post, Feb 19 2011
Except for a(1) = 1 these are the squarefree members of A079896 (i.e., squarefree determinants D of indefinite binary quadratic forms). - Wolfdieter Lang, Jun 01 2013
The asymptotic density of this sequence is 2/Pi^2 = 0.202642... (A185197). - Amiram Eldar, Feb 10 2021

Richard A. Mollin, Quadratics, CRC Press, 1996, Tables B1-B3.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
A. M. Legendre, Diviseurs de la formule t^2+a*u^2, a étant de la forme 4 n + 1, Essai sur la Théorie des Nombres An VI, Table IV. See first column. [_Paul Curtz_, Aug 14 2019]

Cf. A002144, A039956, A039957, A005117, A185197.

a039955 n = a039955_list !! (n-1)
a039955_list = filter ((== 1) . (`mod` 4)) a005117_list
-- Reinhard Zumkeller, Aug 15 2011

[4*n+1: n in [0..67] | IsSquarefree(4*n+1)];  // Bruno Berselli, Mar 03 2011

fQ[n_] := Max[Last /@ FactorInteger@ n] == 1 && Mod[n, 4] == 1; Select[ Range@ 272, fQ] (* Robert G. Wilson v *)
Select[Range[1,300,4],SquareFreeQ[#]&] (* Harvey P. Dale, Mar 27 2020 *)

list(lim)=my(v=List([1])); forfactored(n=5,lim\1, if(vecmax(n[2][,2])==1 && n[1]%4==1, listput(v,n[1]))); Vec(v) \\ Charles R Greathouse IV, Nov 05 2017

is(n)=n%4==1 && issquarefree(n) \\ Charles R Greathouse IV, Nov 05 2017

A046934 Same rule as Aitken triangle (A011971) except a(0,0)=1, a(1,0)=0.

Original entry on oeis.org

1, 0, 1, 1, 1, 2, 2, 3, 4, 6, 6, 8, 11, 15, 21, 21, 27, 35, 46, 61, 82, 82, 103, 130, 165, 211, 272, 354, 354, 436, 539, 669, 834, 1045, 1317, 1671, 1671, 2025, 2461, 3000, 3669, 4503, 5548, 6865, 8536, 8536, 10207, 12232, 14693, 17693, 21362, 25865, 31413, 38278, 46814

Offset: 0

N. J. A. Sloane, R. K. Guy

First differences of A046935 = this triangle seen as flattened list without the initial term. - Reinhard Zumkeller, Nov 10 2013

			[0] [   1]
[1] [   0,     1]
[2] [   1,     1,     2]
[3] [   2,     3,     4,     6]
[4] [   6,     8,    11,    15,    21]
[5] [  21,    27,    35,    46,    61,    82]
[6] [  82,   103,   130,   165,   211,   272,   354]
[7] [ 354,   436,   539,   669,   834,  1045,  1317,  1671]
[8] [1671,  2025,  2461,  3000,  3669,  4503,  5548,  6865,  8536]
[9] [8536, 10207, 12232, 14693, 17693, 21362, 25865, 31413, 38278, 46814]

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened

Borders give A032347 and A032346. Cf. A046935.

a046934 n k = a046934_tabl !! n !! k
a046934_row n = a046934_tabl !! n
a046934_tabl = [1] : iterate (\row -> scanl (+) (last row) row) [0,1]
a046934_list = concat a046934_tabl
-- Reinhard Zumkeller, Nov 10 2013

alias(PS = ListTools:-PartialSums):
A046934Triangle := proc(len) local a, k, P, T; a := 0; P := [1]; T := [];
for k from 1 to len  do T := [op(T), P]; P := PS([a, op(P)]); a := P[-1] od;
ListTools:-Flatten(T) end: A046934Triangle(10);  # Peter Luschny, Mar 29 2022

a[0, 0] = 1; a[1, 0] = 0; a[n_, 0] := a[n, 0] = a[n-1, n-1]; a[n_, k_] := a[n, k] = a[n, k-1] + a[n-1, k-1]; Table[a[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Oct 14 2013 *)

A005350 a(1) = a(2) = a(3) = 1, a(n) = a(a(n-1)) + a(n-a(n-1)) for n >= 4.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 3, 3, 4, 5, 5, 5, 5, 6, 7, 7, 8, 8, 8, 8, 8, 9, 10, 11, 11, 12, 12, 12, 13, 13, 13, 13, 13, 13, 14, 15, 16, 16, 17, 18, 18, 19, 19, 19, 20, 20, 20, 20, 21, 21, 21, 21, 21, 21, 21, 22, 23, 24, 25, 25, 26, 27, 27, 28

Offset: 1

N. J. A. Sloane, R. K. Guy

a(n) - a(n-1) = 0 or 1 (see the 1991 Monthly reference). - Emeric Deutsch, Jun 06 2005

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..1000
R. K. Guy, The Second Strong Law of Small Numbers, Math. Mag, 63 (1990), no. 1, 3-20.
R. K. Guy, The Second Strong Law of Small Numbers, Math. Mag, 63 (1990), no. 1, 3-20. [Annotated scanned copy]
R. K. Guy and N. J. A. Sloane, Correspondence, 1988.
D. Kleitman, Solution to Problem E3274, Amer. Math. Monthly, 98 (1991), 958-959.

Cf. A004001, A005185, A005707.

a005350 n = a005350_list !! (n-1)
a005350_list = 1 : 1 : 1 : h 4 1 where
   h x y = z : h (x + 1) z where z = a005350 y + a005350 (x - y)
-- Reinhard Zumkeller, Jul 20 2012

A005350 := proc(n) option remember; if n<=3 then 1 else procname(procname(n-1)) + procname(n-procname(n-1)); end if; end proc:
seq(A005350(n),n=1..64) ;

a[1] = a[2] = a[3] = 1; a[n_] := a[n] = a[a[n-1]] + a[n-a[n-1]]; Table[a[n], {n, 1, 64}] (* Jean-François Alcover, Feb 11 2014 *)

@CachedFunction
def a(n): return 1 if (n<4) else a(a(n-1)) + a(n-a(n-1))
[a(n) for n in range(1,100)]  # G. C. Greubel, Nov 14 2022

cp's OEIS Frontend

A051280 Numbers n such that n = k/d(k) has exactly 3 solutions, where d(k) = number of divisors of k.

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A003320 a(n) = max_{k=0..n} k^(n-k).

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A007586 11-gonal (or hendecagonal) pyramidal numbers: a(n) = n*(n+1)*(3*n-2)/2.

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A006036 Primitive pseudoperfect numbers.

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A014090 Numbers that are not the sum of a square and a prime.

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A045947 Triangles in open triangular matchstick arrangement (triangle minus one side) of side n.

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Magma

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PARI

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A045946 Star of David matchstick numbers: a(n) = 6*n*(3*n+1).

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A007586 11-gonal (or hendecagonal) pyramidal numbers: a(n) = n(n+1)(3*n-2)/2.

A045946 Star of David matchstick numbers: a(n) = 6n(3*n+1).