A222311 -id:A222311 - cp's OEIS frontend

A222313 A222311 sorted and duplicates removed (conjectured).

1, 2, 3, 5, 6, 15, 17, 33, 41, 55, 57, 65, 70, 105, 129, 257, 273, 385, 561, 897, 969, 1001, 1105, 1353, 1430, 1785, 2049, 2145, 2337, 2665, 3553, 4097, 4305, 4745, 4845, 5633, 6105, 6545, 8193, 8385

Offset: 1

N. J. A. Sloane, Feb 16 2013

Obtained by sorting and removing duplicates from the first 500 terms of A222311. There is no proof as yet that this list is complete up to 105. Only the first three terms shown are certain. Is there a proof that 4 cannot appear?

Cristian Cobeli, Mihai Prunescu, Alexandru Zaharescu, A growth model based on the arithmetic Z-game, arXiv:1511.04315 [math.NT], 2015.
Cristian Cobeli, Alexandru Zaharescu, A game with divisors and absolute differences of exponents, arXiv:1411.1334 [math.NT], 2014 (see page 12).

Cf. A222310, A222311.

terms = 40; nmax0 = 5000;
seq[nmax_] := seq[nmax] = Union[Print[nmax]; Join[r = {1}, Table[Reverse[r = FoldList[#1*(#2/GCD[#1, #2]^2) & , n, r]], {n, 2, nmax}][[All, 1]]]][[1 ;; terms]];
seq[nmax = nmax0]; seq[nmax = 2 nmax]; While[seq[nmax] == seq[nmax/2], nmax = 2 nmax]; seq[nmax] (* Jean-François Alcover, Sep 04 2018, after Ivan Neretin in A222310 *)

Corrected and extended using data from Cobeli et al., 2015. - N. J. A. Sloane, Aug 27 2016

More terms (computed from a list of 10000) from Jean-François Alcover, Sep 04 2018

A275914 A close cousin of A222311 (see Cobeli et al. 2015 for precise definition).

Original entry on oeis.org

1, 2, 3, 3, 5, 15, 105, 35, 3, 15, 11, 165, 195, 91, 3003, 2145, 17, 51, 969, 1615, 1785, 19635, 37145, 245157, 255, 221, 53295, 4849845, 44863, 16269, 14325749295, 6678671, 33, 561, 385

Offset: 1

N. J. A. Sloane, Aug 27 2016

nonn

Cristian Cobeli, Mihai Prunescu, Alexandru Zaharescu, A growth model based on the arithmetic Z-game, arXiv:1511.04315 [math.NT], 2015.

Cf. A222311.

A222310 Array read by antidiagonals: first row is 1, 2, 3, 4, ...; for subsequent rows, write i*j/gcd(i,j)^2 under ...i.j... in previous row.

Original entry on oeis.org

1, 2, 2, 3, 6, 3, 6, 2, 12, 4, 5, 30, 15, 20, 5, 15, 3, 10, 6, 30, 6, 105, 7, 21, 210, 35, 42, 7, 70, 6, 42, 2, 420, 12, 56, 8, 1, 70, 105, 10, 5, 84, 63, 72, 9, 5, 5, 14, 30, 3, 15, 1260, 20, 90, 10, 33, 165, 33, 462, 385, 1155, 77, 1980, 99, 110, 11, 55, 15, 11, 3, 154, 10, 462, 6, 330, 30, 132, 12, 65, 143, 2145, 195, 65, 10010, 1001, 78

Offset: 1

N. J. A. Sloane, Feb 16 2013

			Array begins:
1...2...3.....4......5......6.....7.....8.....9.....10
..2...6....12....20.....30....42.....56...72.....90
....3...2....15......6.....35....12....63....20
......6....30....10....210...420.....84..1260
........5.....3.....21......2.....5....15
...........15.....7.....42....10......3
.............105.....6.....105...30
........

Ivan Neretin, Table of n, a(n) for n = 1..8001
Cristian Cobeli, Mihai Prunescu, Alexandru Zaharescu, A growth model based on the arithmetic Z-game, arXiv:1511.04315 [math.NT], 2015.
C. Cobeli and A. Zaharescu, Promenade around Pascal Triangle-Number Motives, Bull. Math. Soc. Sci. Math. Roumanie, Tome 56(104) No. 1, 2013, 73-98.

Cf. A036262. Leading diagonal is A222311 (cf. A222313).

Similar array with primes in the starting row is A255483.

# To get first M rows of the array (s0 is A222311):
g:=(i,j)->i*j/gcd(i,j)^2;
M:=50;
s0:=[1]:
s1:=[seq(n,n=1..M)]:
for i1 from 1 to M-1 do
lprint(s1);
s2:=[seq(g(s1[i],s1[i+1]),i=1..nops(s1)-1)];
s0:=[op(s0),s2[1]];
s1:=[seq(s2[i],i=1..nops(s2))];
od:
# To produce A222310 (i.e., to read the array by antidiagonals):
g:=(i,j)->i*j/gcd(i,j)^2;
M:=15;
b1:=Array(1..M);
s0:=[1]:
s1:=[seq(n,n=1..M)]:
b1[1]:=s1;
for i1 from 1 to M-1 do
#lprint(s1);
s2:=[seq(g(s1[i],s1[i+1]),i=1..nops(s1)-1)];
b1[i1+1]:=s2;
s0:=[op(s0),s2[1]];
s1:=[seq(s2[i],i=1..nops(s2))];
od:
#[seq(s0[i],i=1..nops(s0))]; (that gives A222311)
lis:=[]:
for i from 1 to M do for j from 1 to i do
lis:=[op(lis),b1[i-j+1][j]];
od: od:
[seq(lis[k],k=1..nops(lis))];

a = r = {1}; Do[a = Join[a, Reverse[r = FoldList[#1*#2/GCD[#1, #2]^2 &, n, r]]], {n, 2, 13}]; a (* Ivan Neretin, May 14 2015 *)

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A222313 A222311 sorted and duplicates removed (conjectured).

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A275914 A close cousin of A222311 (see Cobeli et al. 2015 for precise definition).

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A222310 Array read by antidiagonals: first row is 1, 2, 3, 4, ...; for subsequent rows, write i*j/gcd(i,j)^2 under ...i.j... in previous row.

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