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Smallest powerful (1) number which is at the end of an arithmetic progression of n terms.
0

%I #11 Feb 26 2023 02:20:36

%S 1,4,49,144,4500,5400,308700,352800,396900,441000,58697100,64033200,

%T 11723411700,12625212600,13527013500,14428814400

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

%C Makowski proves that this sequence is infinite.

%D A. Makowski, Remarks on some problems in the elementary theory of numbers, Acta Math. Univ. Comenian. 50/51 (1987), 277-281.

%H Prajeet Bajpai, Michael A. Bennett, and Tsz Ho Chan, <a href="https://arxiv.org/abs/2302.03113">Arithmetic Progressions in Squarefull Numbers</a>, arXiv preprint (2023). arXiv:2302.03113 [math.NT]

%H Tsz Ho Chan, <a href="https://arxiv.org/abs/2210.00281">Arithmetic progressions among powerful numbers</a>, Journal of Integer Sequences, Vol. 26 (2023), Article 23.1.1. arXiv:2210.00281 [math.NT]

%e Corresponding arithmetic progressions:

%e 1

%e 1, 4

%e 1, 25, 49

%e 36, 72, 108, 144

%e 900, 1800, 2700, 3600, 4500

%e 900, 1800, 2700, 3600, 4500, 5400

%e 44100, 88200, 132300, 176400, 220500, 264600, 308700

%e 44100, 88200, 132300, 176400, 220500, 264600, 308700, 352800

%e 44100, 88200, 132300, 176400, 220500, 264600, 308700, 352800, 396900

%e 44100, 88200, 132300, 176400, 220500, 264600, 308700, 352800, 396900, 441000

%e 5336100, 10672200, 16008300, 21344400, 26680500, 32016600, 37352700, 42688800, 48024900, 53361000, 58697100

%e 5336100, 10672200, 16008300, 21344400, 26680500, 32016600, 37352700, 42688800, 48024900, 53361000, 58697100, 64033200

%e From _Bert Dobbelaere_, Feb 25 2023: (Start)

%e a(13): k*901800900 (k=1..13)

%e a(14): k*901800900 (k=1..14)

%e a(15): k*901800900 (k=1..15)

%e a(16): k*901800900 (k=1..16)

%e (End)

%o (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)

%o v=pow(1e11); \\ Set as needed

%o 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")

%Y Subsequence of A001694.

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

%K nonn,hard

%O 1,2

%A _Charles R Greathouse IV_, Feb 24 2023

%E a(13)-a(16) from _Bert Dobbelaere_, Feb 25 2023