A005180 -id:A005180 - cp's OEIS frontend

A001228 Orders of sporadic simple groups.

7920, 95040, 175560, 443520, 604800, 10200960, 44352000, 50232960, 244823040, 898128000, 4030387200, 145926144000, 448345497600, 460815505920, 495766656000, 42305421312000, 64561751654400, 273030912000000, 51765179004000000, 90745943887872000, 4089470473293004800, 4157776806543360000, 86775571046077562880, 1255205709190661721292800, 4154781481226426191177580544000000, 808017424794512875886459904961710757005754368000000000

Offset: 1

N. J. A. Sloane

Numbers of divisors: A174601(n) = A000005(a(n)); squarefree kernels: A174848(n) = A007947(a(n)). - Reinhard Zumkeller, Apr 02 2010

By historical convention, the Tits group is often excluded from the list of sporadic simple groups. It could be inserted as a(7) = 17971200 giving this sequence 27 rather than 26 elements. - Charles R Greathouse IV, Jul 09 2020

			The first term is 7920 because the order of the sporadic group M_{11} is 7920, the smallest order of any sporadic group.

J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, ATLAS of Finite Groups. Oxford Univ. Press, 1985 [for best online version see https://oeis.org/wiki/Welcome#Links_to_Other_Sites].
J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 296.
Martin Gardner, "The Last Recreations", 1997, chap 9, p. 153.

David Madore, Table of sporadic simple groups.
Grant Sanderson, Group theory and 808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000, 3Blue1Brown video (2020)
Eric Weisstein's World of Mathematics, Sporadic Group
Index entries for sequences related to groups

Cf. A001034, A005180.

Entries checked by Pab Ter (pabrlos(AT)yahoo.com), May 29 2004

A001149 A self-generating sequence: a(1)=1, a(2)=2, a(n+1) chosen so that a(n+1)-a(n-1) is the first number not obtainable as a(j)-a(i) for 1<=i

Original entry on oeis.org

1, 2, 3, 5, 8, 13, 17, 26, 34, 45, 54, 67, 81, 97, 115, 132, 153, 171, 198, 228, 256, 288, 323, 357, 400, 439, 488, 530, 581, 627, 681, 732, 790, 843, 908, 963, 1029, 1085, 1152, 1213, 1284, 1346, 1418, 1484, 1561, 1630, 1710, 1785, 1867, 1945, 2034, 2116

Offset: 1

N. J. A. Sloane

nonn

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Manfred Scheucher, Table of n, a(n) for n = 1..2000
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.
N. J. A. Sloane, Handwritten notes on Self-Generating Sequences, 1970 (note that A1148 has now become A005282)
Manfred Scheucher, Python Script

Cf. A005282, A054540.

Description corrected and moved to name line by Franklin T. Adams-Watters, Nov 01 2009

More terms from Manfred Scheucher, Jul 01 2015

A001524 Number of stacks, or arrangements of n pennies in contiguous rows, each touching 2 in row below.

Original entry on oeis.org

1, 1, 1, 2, 3, 5, 8, 12, 18, 26, 38, 53, 75, 103, 142, 192, 260, 346, 461, 607, 797, 1038, 1348, 1738, 2234, 2856, 3638, 4614, 5832, 7342, 9214, 11525, 14369, 17863, 22142, 27371, 33744, 41498, 50903, 62299, 76066, 92676, 112666, 136696, 165507, 200018

Offset: 0

N. J. A. Sloane

Also n-stacks with strictly receding left wall.
Weakly unimodal compositions such that each up-step is by at most 1 (and first part 1). By dropping the requirement for weak unimodality one obtains A005169. - Joerg Arndt, Dec 09 2012
The values of a(19) and a(20) in Auluck's table on page 686 are wrong (they have been corrected here). - David W. Wilson, Mar 07 2015
Also the number of overpartitions of n having more overlined parts than non-overlined parts.   For example, a(5) = 5 counts the overpartitions [5'], [4',1'], [3',2'], [3',1',1] and [2',2,1']. - Jeremy Lovejoy, Jan 15 2021

			For a(6)=8 we have the following stacks:
..x
.xx .xx. ..xx .x... ..x.. ...x. ....x
xxx xxxx xxxx xxxxx xxxxx xxxxx xxxxx xxxxxx
From _Franklin T. Adams-Watters_, Jan 18 2007: (Start)
For a(7) = 12 we have the following stacks:
..x. ...x
.xx. ..xx .xxx .xx.. ..xx. ...xx
xxxx xxxx xxxx xxxxx xxxxx xxxxx
and
.x.... ..x... ...x.. ....x. .....x
xxxxxx xxxxxx xxxxxx xxxxxx xxxxxx xxxxxxx
(End)

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 1001 terms from T. D. Noe)
F. C. Auluck, On some new types of partitions associated with generalized Ferrers graphs, Proc. Cambridge Philos. Soc. 47, (1951), 679-686.
F. C. Auluck, On some new types of partitions associated with generalized Ferrers graphs (annotated scanned copy)
J. S. Birman, Letter to N. J. A. Sloane, Apr 09 1994
Sergi Elizalde, Symmetric peaks and symmetric valleys in Dyck paths, arXiv:2008.05669 [math.CO], 2020.
Erich Friedman, Illustration of initial terms
H. W. Gould, R. K. Guy, and N. J. A. Sloane, Correspondence, 1987.
D. Gouyou-Beauchamps and P. Leroux, Enumeration of symmetry classes of convex polyominoes on the honeycomb lattice, arXiv:math/0403168 [math.CO], 2004.
R. K. Guy, Letter to N. J. A. Sloane, Apr 08 1988 (annotated scanned copy, included with permission)
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.
B. Kim, E. Kim, and J. Lovejoy, Parity bias in partitions, European J. Combin., 89 (2020), 103159, 19 pp.
E. M. Wright, Stacks, III, Quart. J. Math. Oxford, 23 (1972), 153-158.

Cf. A001522, A001523, A171604, A007293, A015128.

Row sums of triangle A259095.

s := 1+sum(z^(n*(n+1)/2)/((1-z^(n))*product((1-z^i), i=1..n-1)^2), n=1..50): s2 := series(s, z, 300): for j from 1 to 100 do printf(`%d,`,coeff(s2, z, j)) od: # James Sellers, Feb 27 2001
# second Maple program:
b:= proc(n, i) option remember; `if`(i>n, 0, `if`(
      irem(n, i)=0, 1, 0)+add(j*b(n-i*j, i+1), j=1..n/i))
    end:
a:= n-> `if`(n=0, 1, b(n, 1)):
seq(a(n), n=0..100);  # Alois P. Heinz, Oct 03 2018

m = 45; CoefficientList[ Series[Sum[ z^(n*(n+1)/2)/((1-z^(n))*Product[(1-z^i), {i, 1, n-1}]^2), {n, 1, m}], {z, 0, m}], z] // Prepend[Rest[#], 1] &
(* Jean-François Alcover, May 19 2011, after Maple prog. *)

{a(n) = if( n<0, 0, polcoeff( sum( k=0,(sqrt(8*n + 1) - 1) / 2, x^((k^2 + k) / 2) / prod( i=1, k, (1 - x^i + x * O(x^n))^((iMichael Somos, Apr 27 2003 */

G.f.: sum(n>=1, q^(n*(n+1)/2) / prod(k=1..n-1, 1-q^k)^2 / (1-q^n) ). [Joerg Arndt, Jun 28 2013]
a(n) = sum_{m>0,k>0,2*k^2+k+2*m<=n-1} A008289(m,k)*A000041(n-k*(1+2k)-2*m-1). - [Auluck eq 29]
From Vaclav Kotesovec, Mar 03 2020: (Start)
Pi * sqrt(2/3) <= n^(-1/2)*log(a(n)) <= Pi * sqrt(5/6). [Auluck, 1951]
log(a(n)) ~ 2*Pi*sqrt(n/5). [Wright, 1971]
a(n) ~ exp(2*Pi*sqrt(n/5)) / (sqrt(2) * 5^(3/4) * (1 + sqrt(5)) * n). (End)
a(n) = A143184(n) - A340659(n). - Vaclav Kotesovec, Jun 06 2021

Corrected by R. K. Guy, Apr 08 1988

More terms from James Sellers, Feb 27 2001

A005579 a(n) = smallest number k such that Product_{i=1..k} prime(i)/(prime(i)-1) > n.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 9, 14, 22, 35, 55, 89, 142, 230, 373, 609, 996, 1637, 2698, 4461, 7398, 12301, 20503, 34253, 57348, 96198, 161659, 272124, 458789, 774616, 1309627, 2216968, 3757384, 6375166, 10828012, 18409028, 31326514, 53354259, 90945529, 155142139

Offset: 0

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

nonn

Laatsch (1986) proved that for n >= 2, a(n) gives the smallest number of distinct prime factors in even numbers having an abundancy index > n.
The abundancy index of a number k is sigma(k)/k. - T. D. Noe, May 08 2006
The first differences of this sequence, A005347, begin the same as the Fibonacci sequence A000045. - T. D. Noe, May 08 2006
Equal to A256968 except for n = 2 and n = 3.  See comment in A256968. - Chai Wah Wu, Apr 17 2015

			The products Product_{i=1..k} prime(i)/(prime(i)-1) for k >= 0 start with 1, 2, 3, 15/4, 35/8, 77/16, 1001/192, 17017/3072, 323323/55296, 676039/110592, 2800733/442368, 86822723/13271040, 3212440751/477757440, 131710070791/19110297600, 5663533044013/802632499200, ... = A060753/A038110. So a(3) = 3.

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

Amiram Eldar, Table of n, a(n) for n = 0..44
Richard K. Guy, Letter to N. J. A. Sloane, 1988-04-12 (annotated scanned copy).
Richard K. Guy and N. J. A. Sloane, Correspondence, 1988.
Richard Laatsch, Measuring the abundancy of integers, Mathematics Magazine 59 (2) (1986) 84-92.
Popular Computing (Calabasas, CA), Problem 182 (Suggested by Victor Meally), Annotated and scanned copy of page 10 of Vol. 5 (No. 53, Aug 1977).

A001611 is similar but strictly different.

Cf. A000720, A001611, A005580, A060753, A023199, A038110, A072986, A091440, A256968.

(* For speed and accuracy, the second Mathematica program uses 30-digit real numbers and interval arithmetic. *)
prod=1; k=0; Table[While[prod<=n, k++; prod=prod*Prime[k]/(Prime[k]-1)]; k, {n,0,25}] (* T. D. Noe, May 08 2006 *)
prod=Interval[1]; k=0; Table[While[Max[prod]<=n, k++; p=Prime[k]; prod=N[prod*p/(p-1),30]]; If[Min[prod]>n, k, "too few digits"], {n,0,38}]

a(n)=my(s=1,k); forprime(p=2,, s*=p/(p-1); k++; if(s>n, return(k))) \\ Charles R Greathouse IV, Aug 20 2015

from sympy import nextprime
def a_list(upto: int) -> list[int]:
    L: list[int] = [0]
    count = 1; bn = 1; bd = 1; p = 2
    for k in range(1, upto + 1):
        bn *= p
        bd *= p - 1
        while bn > count * bd:
            L.append(k)
            count += 1
        p = nextprime(p)
    return L
print(a_list(1000))  # Chai Wah Wu, Apr 17 2015, adapted by Peter Luschny, Jan 25 2025

a(n) = smallest k such that A002110(k)/A005867(k) > n. - Artur Jasinski, Nov 06 2008

a(n) = PrimePi(A091440(n)) = A000720(A091440(n)) for n >= 4. - Amiram Eldar, Apr 18 2025

Edited by T. D. Noe, May 08 2006
a(26) added by T. D. Noe, Sep 18 2008
Typo corrected by Vincent E. Yu (yu.vincent.e(AT)gmail.com), Aug 14 2009
a(27)-a(36) from Vincent E. Yu (yu.vincent.e(AT)gmail.com), Aug 14 2009
Comment corrected by T. D. Noe, Apr 04 2010
a(37)-a(39) from T. D. Noe, Nov 16 2010
Edited and terms a(0)-a(1) prepended by Max Alekseyev, Jan 25 2025

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

A109379 Orders of non-cyclic simple groups (with repetition).

Original entry on oeis.org

60, 168, 360, 504, 660, 1092, 2448, 2520, 3420, 4080, 5616, 6048, 6072, 7800, 7920, 9828, 12180, 14880, 20160, 20160, 25308, 25920, 29120, 32736, 34440, 39732, 51888, 58800, 62400, 74412, 95040, 102660, 113460, 126000, 150348

Offset: 1

N. J. A. Sloane, Jul 29 2006

The first repetition is at 20160 (= 8!/2) and the first proof that there exist two nonisomorphic simple groups of this order was given by the American mathematician Ida May Schottenfels (1869-1942). - David Callan, Nov 21 2006

By the Feit-Thompson theorem, all terms in this sequence are even. - Robin Jones, Dec 25 2023

See A001034 for references and other links.

David A. Madore, Table of n, a(n) for n = 1..493 [taken from link below]
David A. Madore, More terms
John McKay, The non-abelian simple groups g, |g|<10^6 - character tables, Commun. Algebra 7 (1979) no. 13, 1407-1445.
Ida May Schottenfels, Two Non-Isomorphic Simple Groups of the Same Order 20,160, Annals of Math., 2nd Ser., Vol. 1, No. 1/4 (1899), pp. 147-152.
Wikipedia, Feit-Thompson theorem.
Index entries for sequences related to groups

Cf. A000001, A000679, A005180, A001228, A060793, A056866, A056868, A119630.

Cf. A001034 (orders without repetition), A119648 (orders that are repeated).

A002230 Primes with record values of the least positive primitive root.

Original entry on oeis.org

2, 3, 7, 23, 41, 71, 191, 409, 2161, 5881, 36721, 55441, 71761, 110881, 760321, 5109721, 17551561, 29418841, 33358081, 45024841, 90441961, 184254841, 324013369, 831143041, 1685283601, 6064561441, 7111268641, 9470788801, 28725635761, 108709927561, 386681163961, 1990614824641, 44384069747161, 89637484042681

Offset: 1

N. J. A. Sloane

R. Osborn, Tables of All Primitive Roots of Odd Primes Less Than 1000, Univ. Texas Press, 1961.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
A. E. Western and J. C. P. Miller, Tables of Indices and Primitive Roots. Royal Society Mathematical Tables, Vol. 9, Cambridge Univ. Press, 1968, p. XLIV.

Michel Marcus, Table of n, a(n) for n = 1..38 (using McGown and Sorenson).
Stephen D. Cohen, Tomás Oliveira e Silva, and Tim Trudgian, On Grosswald's conjecture on primitive roots, arXiv:1503.04519 [math.NT], 2015.
R. K. Guy and N. J. A. Sloane, Correspondence, 1988.
Kevin J. McGown and Jonathan P. Sorenson, Computation of the least primitive root, arXiv:2206.14193 [math.NT], 2022.
Tomás Oliveira e Silva, Least prime primitive root of prime numbers
A. E. Western and J. C. P. Miller, Tables of Indices and Primitive Roots, Royal Society Mathematical Tables, Vol. 9, Cambridge Univ. Press, 1968 [Annotated scans of selected pages]
Index entries for primes by primitive root

Cf. A002229 (for the primitive roots in question).

Records in A023048, indices in A114885.

s = {2}; rm = 1; Do[p = Prime[k]; r = PrimitiveRoot[p]; If[r > rm, Print[p]; AppendTo[s, p]; rm = r], {k, 10^6}]; s (* Jean-François Alcover, Apr 05 2011 *)
DeleteDuplicates[Table[{p,PrimitiveRoot[p,1]},{p,Prime[Range[61100]]}],GreaterEqual[ #1[[2]],#2[[2]]]&][[All,1]] (* The program generates the first 15 terms of the sequence. *) (* Harvey P. Dale, Aug 22 2022 *)

from sympy import isprime, primitive_root
from itertools import count, islice
def f(n): return 0 if not isprime(n) or (r:=primitive_root(n))==None else r
def agen(r=0): yield from ((m, r:=f(m))[0] for m in count(1) if f(m) > r)
print(list(islice(agen(), 15))) # Michael S. Branicky, Feb 13 2023

More terms from Scott Lindhurst (ScottL(AT)alumni.princeton.edu)

A005181 a(n) = ceiling(exp((n-1)/2)).

Original entry on oeis.org

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 91, 149, 245, 404, 666, 1097, 1809, 2981, 4915, 8104, 13360, 22027, 36316, 59875, 98716, 162755, 268338, 442414, 729417, 1202605, 1982760, 3269018, 5389699, 8886111, 14650720, 24154953, 39824785, 65659970, 108254988, 178482301

Offset: 0

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

nonn

This sequence illustrates the second law of small numbers because it is a coincidence that its first ten terms are the same as the first ten Fibonacci numbers (A000045). - Alonso del Arte, Mar 18 2013

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
I. Stewart, L'univers des nombres, pp. 27 Belin-Pour La Science, Paris 2000.

Vladimir Pletser, Table of n, a(n) for n = 0..1000
R. K. Guy, The Second Strong Law of Small Numbers, Math. Mag, 63 (1990), no. 1, 3-20. [Annotated scanned copy]
R. K. Guy, The Second Strong Law of Small Numbers, Math. Mag, 63 (1990), no. 1, 3-20.
R. K. Guy and N. J. A. Sloane, Correspondence, 1988.
I. Stewart, Fibonacci Forgeries
Eric Weisstein's World of Mathematics, Fibonacci Number.
Eric Weisstein's World of Mathematics, Strong Law of Small Numbers.

Cf. A000045, A019774.

seq(round(ceil(exp((n-1)/2))), n=0..50); # Vladimir Pletser, Sep 15 2013

Table[Ceiling[E^((n - 1)/2)], {n, 0, 39}] (* Alonso del Arte, Mar 18 2013 *)

import math
for n in range(99):
    print(str(int(math.ceil(math.e**((n-1)*0.5)))), end=', ')
# Alex Ratushnyak, Mar 18 2013

Limit_{n->oo} a(n+1)/a(n) = sqrt(e) = 1.64872127... = A019774. - Alois P. Heinz, Feb 19 2019

A few more terms from Alonso del Arte, Mar 18 2013

A119630 Orders of non-Abelian simple groups of rank at least two.

Original entry on oeis.org

60, 168, 360, 2520, 5616, 6048, 7920, 20160, 20160, 25920, 29120, 62400, 95040, 126000, 175560, 181440, 372000, 443520, 604800, 979200, 1451520, 1814400, 1876896, 3265920, 4245696, 4680000, 5515776, 5663616, 6065280, 9999360

Offset: 1

N. J. A. Sloane, Jun 10 2006

nonn

This omits (from A001034) only the non-Abelian simple groups of Lie type of rank 1, i.e. the linear groups L(2,q).

David A. Madore, Table of n, a(n) for n = 1..243 [taken from link below]
David A. Madore, More terms
Index entries for sequences related to groups

Subsequence of A005180.

Cf. A001034.

A001856 A self-generating sequence: every positive integer occurs as a(i)-a(j) for a unique pair i,j.

Original entry on oeis.org

1, 2, 4, 8, 16, 21, 42, 51, 102, 112, 224, 235, 470, 486, 972, 990, 1980, 2002, 4004, 4027, 8054, 8078, 16156, 16181, 32362, 32389, 64778, 64806, 129612, 129641, 259282, 259313, 518626, 518658, 1037316, 1037349, 2074698, 2074734, 4149468

Offset: 1

N. J. A. Sloane

This is a B_2 sequence. More economical recursion: a(1)=1, a(2n)=2a(2n-1), a(2n+1)=a(2n)+r(n), where r(n) is the smallest positive integer not of the form a(j)-a(i) with 1<=iA247556. - Thomas Ordowski, Sep 28 2014

R. K. Guy, Unsolved Problems in Number Theory, E25.
W. Sierpiński, Elementary Theory of Numbers. Państ. Wydaw. Nauk., Warsaw, 1964, p. 444.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
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. L. Graham, Problem E1910, Amer. Math. Monthly, 73 (1966), 775.
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.
M. Hall, Cyclic projective planes, Duke Math. J., 4 (1947), 1079-1090.
C. B. A. Peck, Remark on Problem E1910, Amer. Math. Monthly, 75 (1968), 80-81.
W. Sierpiński, Elementary Theory of Numbers, Warszawa 1964.
N. J. A. Sloane, Handwritten notes on Self-Generating Sequences, 1970 (note that A1148 has now become A005282)

Cf. A054540, A004978, A247556.

a[1] = 1; a[2] = 2; a[n_?OddQ] := a[n] = 2*a[n-1]; a[n_?EvenQ] := a[n] = a[n-1] + r[(n-2)/2]; r[n_] := ( diff = Table[a[j] - a[i], {i, 1, 2*n+1}, {j, i+1, 2*n+1}] // Flatten // Union; max = diff // Last; notDiff = Complement[Range[max], diff]; If[notDiff == {}, max+1, notDiff // First]); Table[a[n], {n, 1, 39}] (* Jean-François Alcover, Dec 31 2012 *)

a(1)=1, a(2)=2, a(2n+1) = 2a(2n), a(2n+2) = a(2n+1) + r(n), where r(n) = smallest positive number not of form a(j) - a(i) with 1 <= i < j <= 2n+1.

More terms from Larry Reeves (larryr(AT)acm.org), Sep 14 2000

cp's OEIS Frontend

A001228 Orders of sporadic simple groups.

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A001149 A self-generating sequence: a(1)=1, a(2)=2, a(n+1) chosen so that a(n+1)-a(n-1) is the first number not obtainable as a(j)-a(i) for 1<=i

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A001524 Number of stacks, or arrangements of n pennies in contiguous rows, each touching 2 in row below.

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A005579 a(n) = smallest number k such that Product_{i=1..k} prime(i)/(prime(i)-1) > n.

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

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Haskell

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A109379 Orders of non-cyclic simple groups (with repetition).

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A002230 Primes with record values of the least positive primitive root.

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