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

A007587 12-gonal (or dodecagonal) pyramidal numbers: a(n) = n(n+1)(10*n-7)/6.

0, 1, 13, 46, 110, 215, 371, 588, 876, 1245, 1705, 2266, 2938, 3731, 4655, 5720, 6936, 8313, 9861, 11590, 13510, 15631, 17963, 20516, 23300, 26325, 29601, 33138, 36946, 41035, 45415, 50096, 55088, 60401, 66045, 72030, 78366, 85063, 92131, 99580, 107420, 115661

Offset: 0

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

Binomial transform of [1, 12, 21, 10, 0, 0, 0, ...] = (1, 13, 46, 110, ...). - Gary W. Adamson, Nov 28 2007

This sequence is related to A000566 by a(n) = n*A000566(n) - Sum_{i=0..n-1} A000566(i) and this is the case d=5 in the identity n*(n*(d*n-d+2)/2) - Sum_{k=0..n-1} k*(d*k-d+2)/2 = n*(n+1)*(2*d*n - 2*d + 3)/6. - Bruno Berselli, Oct 18 2010

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
B. Berselli, A description of the recursive method in Comments lines: website Matem@ticamente (in Italian), 2008.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A093645 ((10, 1) Pascal, column m=3). Partial sums of A051624.
Cf. A000566.
See similar sequences listed in A237616.

List([0..45], n-> n*(n+1)*(10*n-7)/6); # G. C. Greubel, Aug 30 2019

[ n eq 1 select 0 else Self(n-1)+(n-1)*(5*n-9): n in [1..45] ]; // Klaus Brockhaus, Nov 20 2008

A007587:=n->n*(n+1)*(10*n-7)/6: seq(A007587(n), n=0..50); # Wesley Ivan Hurt, Oct 23 2014

CoefficientList[Series[x(1+9x)/(1-x)^4, {x,0,45}], x] (* Vincenzo Librandi, Jun 20 2013 *)
Table[n(n+1)(10n-7)/6,{n,0,50}] (* Harvey P. Dale, Nov 13 2013 *)

a(n)=if(n,([0,1,0,0; 0,0,1,0; 0,0,0,1; -1,4,-6,4]^n*[0;1;13;46])[1,1],0) \\ Charles R Greathouse IV, Oct 07 2015

vector(45, n, n*(n-1)*(10*n-17)/6) \\ G. C. Greubel, Aug 30 2019

[n*(n+1)*(10*n-7)/6 for n in (0..45)] # G. C. Greubel, Aug 30 2019

a(n) = (10*n-7)*binomial(n+1, 2)/3.
G.f.: x*(1+9*x)/(1-x)^4.
a(n) = Sum_{k=0..n} k*(5*k-4). - Klaus Brockhaus, Nov 20 2008
a(n) = Sum_{i=0..n-1} (n-i)*(10*i+1), with a(0)=0. - Bruno Berselli, Feb 10 2014
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Wesley Ivan Hurt, Oct 23 2014
E.g.f.: exp(x)*x*(6 + 33*x + 10*x^2)/6. - Elmo R. Oliveira, Aug 04 2025

A011774 Nonprimes k that divide sigma(k) + phi(k).

Original entry on oeis.org

1, 312, 560, 588, 1400, 23760, 59400, 85632, 147492, 153720, 556160, 569328, 1590816, 2013216, 3343776, 4563000, 4695456, 9745728, 12558912, 22013952, 23336172, 30002960, 45326160, 52021242, 75007400, 113315400, 137617728, 153587720, 402831360, 699117024

Offset: 1

R. K. Guy

2*k = sigma(k) + phi(k) if and only if k is 1 or a prime.
If 7*2^j - 1 is prime then m = 2^(j+2)*3*(7*2^j - 1) is in the sequence. Because phi(m) = 2^(j+2)*(7*2^j - 2); sigma(m) = 7*2^(j+2)*(2^(j+3) - 1) so phi(m) + sigma(m) = 2^(j+2)*((7*2^j - 2) + (7*2^(j+3) - 7)) = 2^(j+2)* (63*2^(j+2) - 9) = 3*(2^(j+2)*3*(7*2^j - 1)) = 3*m, hence m is a term of A011251 and consequently m is a term of this sequence. A112729 gives such m's. - Farideh Firoozbakht, Dec 01 2005
Conjecture: For n > 1, a(n) is a Zumkeller number (A083207). Verified for all n in [2,63]. - Ivan N. Ianakiev, Jan 25 2023

			a(26) = 113315400: sigma = 426535200, phi = 26726400, quotient = 4.

Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004. See Section B42, p. 151.
Zhang Ming-Zhi, typescript submitted to Unsolved Problems section of Monthly, 96-01-10.

Giovanni Resta, Table of n, a(n) for n = 1..63 (terms < 10^13; first 53 terms from Donovan Johnson)
Richard K. Guy, Divisors and desires, Amer. Math. Monthly, 104 (1997), 359-360.
Q.-X. Jin and M. Tang, The 4-Nicol Numbers Having Five Different Prime Divisors, J. Int. Seq. 14 (2011) # 11.7.2.
Eric Weisstein's World of Mathematics, Prime Number.

Cf. A065387, A011251, A011254, A055681, A001771, A112729.

Do[If[Mod[DivisorSigma[1, n]+EulerPhi[n], n]==0, Print[n]], {n, 1, 2*10^7}]
Do[ If[ ! PrimeQ[n] && Mod[ DivisorSigma[1, n] + EulerPhi[n], n] == 0, Print[n] ], {n, 1, 10^8} ]

sp(n)=my(f=factor(n));n*prod(i=1, #f[,1], 1-1/f[i,1]) + prod(i=1, #f[,1], (f[i,1]^(f[i,2]+1)-1)/(f[i,1]-1))
p=2;forprime(q=3, 1e6, for(n=p+1, q-1, if(sp(n)%n==0, print1(n", ")));p=q) \\ Charles R Greathouse IV, Mar 19 2012

More terms from David W. Wilson

Corrected by Labos Elemer, Feb 12 2004

A051487 Numbers k such that phi(k) = phi(k - phi(k)).

Original entry on oeis.org

2, 6, 12, 24, 48, 96, 150, 192, 300, 384, 600, 726, 750, 768, 1200, 1452, 1500, 1536, 2310, 2400, 2904, 3000, 3072, 3174, 3750, 4620, 4800, 5046, 5808, 5874, 6000, 6090, 6144, 6348, 6930, 7500, 7986, 9240, 9600, 10086, 10092, 10374, 11550, 11616, 11748, 12000

Offset: 1

R. K. Guy

This sequence is infinite, in fact 3*2^n is a subsequence because if m = 3*2^n then phi(m-phi(m)) = phi(3*2^n-2^n) = 2^n = phi(m). Also, if p is a Sophie Germain prime greater than 3 then for each natural number n, 2^n*3*p^2 is in the sequence. Note that there exist terms of this sequence like 750 or 2310 that they aren't of either of these forms. - Farideh Firoozbakht, Jun 19 2005
If n is an even term greater than 2 in this sequence then 2n is also in the sequence. Because for even numbers m we have phi(2m) = 2*phi(m) so phi(2n) = 2*phi(n) = 2*phi(n-phi(n)) and since n is an even number greater than 2, n-phi(n) is even so 2*phi(n-phi(n)) = phi(2n-2*phi(n)) = phi(2n-phi(2n)) hence phi(2n) = phi(2n-phi(2n)) and 2n is in the sequence. Conjecture: All terms of this sequence are even. - Farideh Firoozbakht, Jul 04 2005
If n is in the sequence and the natural number m divides gcd(n,phi(n)) then m*n is in the sequence. The facts that I have found about this sequence earlier (Jun 19 2005 and Jul 04 2005) are consequences of this. If p is a Sophie Germain prime greater than 3, k>1 and k & n are natural numbers then 2^n*3*p^k are in the sequence. - Farideh Firoozbakht, Dec 10 2005
Numbers n such that phi(n) = phi(n + phi(n)) includes all n = 2^k. - Jonathan Vos Post, Oct 25 2007

Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B42, p. 150.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
Aleksander Grytczuk, Florian Luca and Marek Wojtowicz, A conjecture of Erdős concerning inequalities for the Euler totient function, Publ. Math. Debrecen, Vol. 59, No. 1-2, (2001), pp. 9-16.

Cf. A005384, A051488.

Cf. A000010, A051953.

a051487 n = a051487_list !! (n-1)
a051487_list = [x | x <- [2..], let t = a000010 x, t == a000010 (x - t)]
-- Reinhard Zumkeller, Jun 03 2013

Select[Range[11700], EulerPhi[ # ] == EulerPhi[ # - EulerPhi[ # ]] &] (* Farideh Firoozbakht, Jun 19 2005 *)

isA051487(n) = eulerphi(n) == eulerphi(n - eulerphi(n)) \\ Michael B. Porter, Dec 07 2009

More terms from James Sellers

A004401 Least number of edges in graph containing all trees on n nodes.

Original entry on oeis.org

0, 1, 2, 4, 6, 8, 11, 13, 16, 18

Offset: 1

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

Paul Tabatabai's program only tests graphs on n nodes, but the term a(9) = 16 is now confirmed. It seems to be sufficient to test only graphs on n vertices (cf. Eqn. 3 and Related Question 1 in Chung and Graham). Moreover, if this assumption is true, then the next terms are a(11) = 22 and a(12) = 24. Also note that my program can be used to enumerate all possible solutions. - Manfred Scheucher, Jan 25 2018
Chung and Graham use n and T_n to refer to trees on n *edges* (i.e., n-1 nodes). - Eric W. Weisstein, Jan 30 2025
Graphs containing all trees on n nodes could be referred to as fully n-forested graphs. - Eric W. Weisstein, Jan 31 2025

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

F. R. K. Chung and R. L. Graham, On graphs which contain all small trees, J. Combinatorial Theory Ser. B 24 (1978), no. 1, 14--23. MR0505812 (58 #21808a)
F. R. K. Chung, R. L. Graham and N. Pippenger, On graphs which contain all small trees. II. Combinatorics (Proc. Fifth Hungarian Colloq., Keszthely, 1976), Vol. I, pp. 213-223. Colloq. Math. Soc. Janos Bolyai, 18. North-Holland, Amsterdam, 1978.
Manfred Scheucher, Sage Program
Manfred Scheucher, Graph on 10 vertices and 18 edges containing all trees on 10 vertices.
Paul Tabatabai, Exhaustive search proving a(9) = 16. (Sage script)
Paul Tabatabai, Graph on 9 vertices and 16 edges containing all trees on 9 vertices.
Eric Weisstein's World of Mathematics, Fully Forested Graph.
Index entries for sequences related to trees

Cf. A212555, A097911, A348638.

Cf. A380740 (numbers of smallest fully forested graphs).

a(9) by Paul Tabatabai, Jul 17 2016

a(10) by Manfred Scheucher, Jan 25 2018

A046936 Same rule as Aitken triangle (A011971) except a(0,0)=0, a(1,0)=1.

Original entry on oeis.org

0, 1, 1, 1, 2, 3, 3, 4, 6, 9, 9, 12, 16, 22, 31, 31, 40, 52, 68, 90, 121, 121, 152, 192, 244, 312, 402, 523, 523, 644, 796, 988, 1232, 1544, 1946, 2469, 2469, 2992, 3636, 4432, 5420, 6652, 8196, 10142, 12611, 12611, 15080, 18072, 21708, 26140

Offset: 0

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

			Triangle starts:
0,
1, 1,
1, 2, 3,
3, 4, 6, 9,
9, 12, 16, 22, 31,
31, 40, 52, 68, 90, 121,
121, 152, 192, 244, 312, 402, 523,
523, 644, 796, 988, 1232, 1544, 1946, 2469,
2469, 2992, 3636, 4432, 5420, 6652, 8196, 10142, 12611,
12611, 15080, ...

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened
R. K. Guy, Letters to N. J. A. Sloane, June-August 1968
Don Knuth, Email to N. J. A. Sloane, Jan 29 2018

Borders give A040027. Reading across rows gives A007604.

Cf. A121207, A298804.

a046936 n k = a046936_tabl !! n !! k
a046936_row n = a046936_tabl !! n
a046936_tabl = [0] : iterate (\row -> scanl (+) (last row) row) [1,1]
-- Reinhard Zumkeller, Jan 01 2014

a[0, 0] = 0; a[1, 0] = 1; 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, Nov 15 2013 *)

from itertools import accumulate
def A046936(): # Compare function Gould_diag in A121207.
    yield [0]
    accu = [1, 1]
    while True:
        yield accu
        accu = list(accumulate([accu[-1]] + accu))
g = A046936()
[next(g) for  in range(9)] # _Peter Luschny, Apr 25 2016

A060984 a(1) = 1; a(n+1) = a(n) + (largest square <= a(n)).

Original entry on oeis.org

1, 2, 3, 4, 8, 12, 21, 37, 73, 137, 258, 514, 998, 1959, 3895, 7739, 15308, 30437, 60713, 121229, 242333, 484397, 967422, 1933711, 3865811, 7730967, 15459367, 30912128, 61814609, 123625653, 247235577, 494448306, 988888002, 1977738918, 3955408759, 7910812423

Offset: 1

R. K. Guy, May 11 2001

Arises in analyzing "put-or-take" games (see Winning Ways, 484-486, 501-503), the prototype being Epstein's Put-or-Take-a-Square game.

E. R. Berlekamp, J. H. Conway and R. K. Guy, Winning Ways, Academic Press, NY, 2 vols., 1982.
R. K. Guy, Unsolved Problems in Number Theory, Springer, 1st edition, 1981. See section E26.

Henry J. Smith and Harvey P. Dale, Table of n, a(n) for n = 1..1000 (first 500 terms from Henry J. Smith)

Cf. A060985, A237888.

a060984 n = a060984_list !! (n-1)
a060984_list = iterate (\x -> x + a048760 x) 1
-- Reinhard Zumkeller, Dec 24 2013

a[1] = 1; a[n_] := a[n] = a[n - 1] + Floor[ Sqrt[ a[n - 1] ] ]^2; Table[ a[n], {n, 1, 40} ]
RecurrenceTable[{a[1]==1,a[n]==a[n-1]+Floor[Sqrt[a[n-1]]]^2},a,{n,40}] (* Harvey P. Dale, Nov 19 2011 *)
NestList[#+Floor[Sqrt[#]]^2&,1,40] (* Harvey P. Dale, Jan 22 2013 *)

{ default(realprecision, 100); for (n=1, 500, if (n==1, a=1, a+=floor(sqrt(a))^2); write("b060984.txt", n, " ", a); ) } \\ Harry J. Smith, Jul 15 2009

from sympy import integer_nthroot
A060984_list = [1]
for i in range(10**3): A060984_list.append(A060984_list[-1]+integer_nthroot(A060984_list[-1],2)[0]**2) # Chai Wah Wu, Apr 02 2021

from math import isqrt
from itertools import accumulate
def f(an, _): return an + isqrt(an)**2
print(list(accumulate([1]*36, f))) # Michael S. Branicky, Apr 02 2021

a(n+1) = a(n)+[sqrt(a(n))]^2 = a(n)+A061886(n) = a(n)+A048760(a(n)) = A061887(a(n)). - Henry Bottomley, May 12 2001

a(n) ~ c * 2^n, where c = 0.11511532187216693... (see A237888). - Vaclav Kotesovec, Feb 15 2014

More terms from David W. Wilson, Henry Bottomley and Robert G. Wilson v, May 12 2001

A060985 a(1) = 1; a(n+1) = a(n) + (largest triangular number <= a(n)).

Original entry on oeis.org

1, 2, 3, 6, 12, 22, 43, 79, 157, 310, 610, 1205, 2381, 4727, 9383, 18699, 37227, 74355, 148660, 296900, 593735, 1187240, 2373810, 4746741, 9491481, 18981027, 37956907, 75910735, 151820416, 303627016, 607253419, 1214497244, 2428978214, 4857918665

Offset: 1

R. K. Guy, May 11 2001

Arises in analyzing 'put-or-take' games (see Winning Ways, 484-486, 501-503), the prototype being Epstein's Put-or-Take-a-Square game.

E. R. Berlekamp, J. H. Conway and R. K. Guy, Winning Ways, Academic Press, NY, 2 vols., 1982.

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A060984, A237889.

a060985 n = a060985_list !! (n-1)
a060985_list = iterate a061885 1  -- Reinhard Zumkeller, Feb 03 2012

a[1] = 1; a[n_] := a[n] = Block[ {k = 1}, While[ k*(k + 1)/2 <= a[n - 1], k++ ]; a[n - 1] + k*(k - 1)/2]; Table[ a[n], {n, 1, 40} ]
f[n_]:=Module[{c=Floor[(Sqrt[1+8n]-1)/2]},(c(c+1))/2]; NestList[#+f[#]&, 1, 40] (* Harvey P. Dale, Jun 19 2011 *)

{ default(realprecision, 1000); for (n=1, 1000, if (n<2, a=1, k=(sqrt(1 + 8*a) - 1)\2; a+=k*(k + 1)/2 ); write("b060985.txt", n, " ", a) ) } \\ Harry J. Smith, Jul 16 2009

a(n+1) = a(n) + A061883(n) = a(n) + A057944(a(n)) = A061885(a(n)). - Henry Bottomley, May 12 2001

a(n) ~ 0.28276... * 2^n. - Charles R Greathouse IV, Jun 19 2011

More terms from David W. Wilson, Henry Bottomley and Robert G. Wilson v, May 12 2001

A063980 Pillai primes: primes p such that there exists an integer m such that m! + 1 == 0 (mod p) and p != 1 (mod m).

Original entry on oeis.org

23, 29, 59, 61, 67, 71, 79, 83, 109, 137, 139, 149, 193, 227, 233, 239, 251, 257, 269, 271, 277, 293, 307, 311, 317, 359, 379, 383, 389, 397, 401, 419, 431, 449, 461, 463, 467, 479, 499, 503, 521, 557, 563, 569, 571, 577, 593, 599, 601, 607

Offset: 1

R. K. Guy, Sep 08 2001

Hardy & Subbarao prove that this sequence is infinite. An upper bound can be extracted from their proof: a(n) < e^e^...^e^O(n log n) with e appearing n times. This tetrational bound could be improved with results on the disjointness of the factorizations of numbers of the form k! + 1. - Charles R Greathouse IV, Sep 15 2015

Named after the Indian mathematician Subbayya Sivasankaranarayana Pillai (1901-1950). - Amiram Eldar, Jun 16 2021

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
G. E. Hardy and M. V. Subbarao, A modified problem of Pillai and some related questions, Amer. Math. Monthly, Vol. 109, No. 6 (2002), pp. 554-559; alternative link.
Wikipedia, Pillai prime.

Smallest m is given in A063828, largest in A211411.

ok[p_] := (r = False; Do[If[Mod[m! + 1, p] == 0 && Mod[p, m] != 1, r = True; Break[]], {m, 2, p}]; r); Select[Prime /@ Range[111], ok] (* Jean-François Alcover, Apr 22 2011 *)
nn=1000; fact=1+Rest[FoldList[Times,1,Range[nn]]]; t={}; Do[p=Prime[i]; m=2; While[mT. D. Noe, Apr 22 2011 *)

is(p)=my(t=Mod(5040,p)); for(m=8, p-2, t*=m; if(t==-1 && p%m!=1, return(isprime(p)))); 0 \\ Charles R Greathouse IV, Feb 10 2013

More terms from David W. Wilson, Sep 08 2001

A122036 Odd abundant numbers (A005231) which are not in A136446, i.e., not sum of some of their proper divisors > 1.

Original entry on oeis.org

351351

Offset: 1

N. J. A. Sloane, Apr 11 2008, following correspondence from R. K. Guy, M. F. Hasler and others

It is conjectured that there are no odd weird numbers (A006037), i.e., that all odd abundant numbers (A005231) are pseudoperfect (A005835); this sequence lists those which are not equal to the sum of a subset of proper divisors > 1.
No second term in the range <= 53850001. - R. J. Mathar, Mar 21 2011
No other terms congruent to 21 (mod 30) below 10^9. - M. F. Hasler, Jul 16 2016
a(2) > 10^16. - Wenjie Fang, Jul 17 2017

			a(1) = 351351 = 3^3 * 7 * 11 * 13^2 is the sum of all its 47 proper divisors (including 1) except 7 and 11, but it is not possible to get the same sum without using the trivial divisor 1: The sum of all proper divisors *larger than 1* yields 351351 + 7 + 11 - 1 = 351351 + 17, and it is not possible to get 17 as sum of a subset of {3, 7, 9, 11, 13, 21, ...}. Thus, 351351 is not in A136446, and therefore in this sequence. - _M. F. Hasler_, Jul 16 2016, edited Mar 15 2021

Index entries for one-term sequences

Cf. A005231, A005835, A006037, A136446.

is_A122036(n)={n>351350 && !is_A005835(n,n=divisors(n)[2..-2]) && n && vecsum(n)>=n[1]*n[#n] && n[1]>2} \\ (Checking for abundant & odd after is_A005835() rather than before, to make it faster when operating on candidates known to satisfy these conditions.) Updated for current PARI syntax by M. F. Hasler, Jul 16 2016, further edits Jan 31 2020
forstep(n=1,10^7,2, is_A122036(n) && print1(n","))

Comments and PARI code from M. F. Hasler, Apr 12 2008

Edited by M. F. Hasler, Jul 16 2016, Mar 15 2021

A169677 The first of a pair of sequences A and B with property that all the differences |a_i - b_j| are distinct - for precise definition see Comments lines.

Original entry on oeis.org

0, 1, 7, 18, 35, 59, 88, 125, 178, 233, 285, 344, 352, 442, 557, 675, 796, 797, 957, 1011, 1220, 1411, 1564, 1579, 1888, 2120, 2152, 2503, 2829, 2953, 3393, 3464, 3593, 3724, 4237, 4956, 5310, 5388, 5968, 6478, 6756, 7344, 7698, 8004, 8182

Offset: 1

R. K. Guy and N. J. A. Sloane, Mar 27 2010

Consider pairs of sequences A = a_1 a_2 a_3 a_4 ... and B = b_1 b_2 b_3 ... such that
All the terms are nonnegative integers
The terms of A are strictly increasing
The terms of B are strictly increasing
All the numbers |a_i - b_j| are distinct
The terms are computed in the following order: a(1), b(1), a(2), b(2), ..., b(n-1), a(n), b(n), a(n+1), ... and always the smallest value is chosen that satisfies constraints 1-4.
Computed by Alois P. Heinz and Wouter Meeussen, Mar 27 2010

Alois P. Heinz, Table of n, a(n) for n = 1..200

Cf. A169678, A169679, A169680, A169690-A169693.

# Maple program from Alois P. Heinz:
ab:=proc() false end: ab(0):=true:
a:= proc(n) option remember;
local ok,i,k,s;
if n=1 then 0
else b(n-1);
for k from a(n-1)+1 do
ok:=true;
for i from 1 to n-1 do
if ab(abs(k-b(i))) then ok:= false; break fi
od;
if ok then s:={};
for i from 1 to n-1 do
s:= s union {abs(k-b(i))};
od
fi;
if ok and nops(s)=n-1 then break fi
od;
for i from 1 to n-1 do
ab(abs(k-b(i))):=true
od;
k
fi
end;
b:= proc(n) option remember;
local ok,i,k,s;
if n=1 then 0
else a(n);
for k from b(n-1)+1 do
ok:=true;
for i from 1 to n do
if ab(abs(k-a(i))) then ok:= false; break fi
od;
if ok then s:={};
for i from 1 to n do
s:= s union {abs(k-a(i))};
od
fi;
if ok and nops(s)=n then break fi
od;
for i from 1 to n do
ab(abs(k-a(i))):=true
od;
k
fi
end;
seq(a(n), n=1..80);
seq(b(n), n=1..80);

ClearAll[ab, a, b]; ab[] = False; ab[0] = True; a[n] := a[n] = Module[{ ok, i, k, s}, If[ n == 1 , 0, b[n-1]; For[ k = a[n-1] + 1 , True, k++, ok = True; For[ i = 1 , i <= n-1, i++, If[ ab[Abs[k - b[i]]] , ok = False; Break[] ]]; If[ ok , s = {}; For[ i=1 , i <= n-1 , i++, s = s ~Union~ {Abs[k - b[i]]};]]; If[ ok && (Length[s] == n-1) , Break[] ]]; For[ i=1 , i <= n-1 , i++, ab[Abs[k - b[i]]] = True]; k]]; b[n_] := b[n] = Module[{ ok, i, k, s}, If[ n == 1 , 0, a[n]; For[ k = b[n-1] + 1 , True, k++, ok = True; For[ i=1 , i <= n, i++, If[ ab[Abs[k - a[i]]] , ok = False; Break[] ]]; If[ ok , s = {}; For[ i=1 , i <= n , i++, s = s ~Union~ {Abs[k - a[i]]};]]; If[ ok && Length[s] == n , Break[] ]]; For[ i=1 , i <= n, i++, ab[Abs[k - a[i]]] := True]; k]]; Table[a[n], {n, 1, 45}] (* Jean-François Alcover, Aug 13 2012, translated from Alois P. Heinz's Maple program *)

Comments clarified by Zak Seidov and Alois P. Heinz, Apr 13 2010.

cp's OEIS Frontend

A007587 12-gonal (or dodecagonal) pyramidal numbers: a(n) = n*(n+1)*(10*n-7)/6.

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A011774 Nonprimes k that divide sigma(k) + phi(k).

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A051487 Numbers k such that phi(k) = phi(k - phi(k)).

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A004401 Least number of edges in graph containing all trees on n nodes.

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A046936 Same rule as Aitken triangle (A011971) except a(0,0)=0, a(1,0)=1.

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A060984 a(1) = 1; a(n+1) = a(n) + (largest square <= a(n)).

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A007587 12-gonal (or dodecagonal) pyramidal numbers: a(n) = n(n+1)(10*n-7)/6.