A360883 Smallest powerful (1) number which is at the end of an arithmetic progression of n terms.

1, 4, 49, 144, 4500, 5400, 308700, 352800, 396900, 441000, 58697100, 64033200, 11723411700, 12625212600, 13527013500, 14428814400

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

Table of n, a(n) for n=1..16.

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

Subsequence of A001694.

Smallest number at the end of an arithmetic progression of n terms of various types of numbers: A005115 (primes).

Sequence in context: A147803 A112533 A016874 * A092866 A340662 A041065

Adjacent sequences: A360880 A360881 A360882 * A360884 A360885 A360886

Keyword

nonn,hard

Author

Charles R Greathouse IV, Feb 24 2023

Extensions

a(13)-a(16) from Bert Dobbelaere, Feb 25 2023

Status

approved