OFFSET
1,2
COMMENTS
Makowski proves that this sequence is infinite.
REFERENCES
A. Makowski, Remarks on some problems in the elementary theory of numbers, Acta Math. Univ. Comenian. 50/51 (1987), 277-281.
LINKS
Prajeet Bajpai, Michael A. Bennett, and Tsz Ho Chan, Arithmetic Progressions in Squarefull Numbers, arXiv preprint (2023). arXiv:2302.03113 [math.NT]
Tsz Ho Chan, Arithmetic progressions among powerful numbers, Journal of Integer Sequences, Vol. 26 (2023), Article 23.1.1. arXiv:2210.00281 [math.NT]
EXAMPLE
Corresponding arithmetic progressions:
1
1, 4
1, 25, 49
36, 72, 108, 144
900, 1800, 2700, 3600, 4500
900, 1800, 2700, 3600, 4500, 5400
44100, 88200, 132300, 176400, 220500, 264600, 308700
44100, 88200, 132300, 176400, 220500, 264600, 308700, 352800
44100, 88200, 132300, 176400, 220500, 264600, 308700, 352800, 396900
44100, 88200, 132300, 176400, 220500, 264600, 308700, 352800, 396900, 441000
5336100, 10672200, 16008300, 21344400, 26680500, 32016600, 37352700, 42688800, 48024900, 53361000, 58697100
5336100, 10672200, 16008300, 21344400, 26680500, 32016600, 37352700, 42688800, 48024900, 53361000, 58697100, 64033200
From Bert Dobbelaere, Feb 25 2023: (Start)
a(13): k*901800900 (k=1..13)
a(14): k*901800900 (k=1..14)
a(15): k*901800900 (k=1..15)
a(16): k*901800900 (k=1..16)
(End)
PROG
(PARI) pow(lim, mn=2)=my(v=List(), t); for(m=1, sqrtnint(lim\1, 3), t=m^3; for(n=1, sqrtint(lim\t), listput(v, t*n^2))); Set(v)
v=pow(1e11); \\ Set as needed
a(n)=my(k=n-1); for(i=n, #v, for(j=1, i-k, my(g=v[i]-v[j], m); if(g%k, next); m=g/k; forstep(a=v[j]+m, v[i]-m, m, if(!setsearch(v, a), next(2))); return(v[i]))); warning("More powerful numbers needed in v")
CROSSREFS
KEYWORD
nonn,hard
AUTHOR
Charles R Greathouse IV, Feb 24 2023
EXTENSIONS
a(13)-a(16) from Bert Dobbelaere, Feb 25 2023
STATUS
approved