A272981 -id:A272981 - cp's OEIS frontend

A272980 Least k>1 such that all powers k^e, 1 <= e <= n, are divisible by the number of their divisors, d(k^e).

2, 60, 1056, 1260, 1441440, 551350800, 42226984800, 111924212400, 11251629148359600, 284440457440339200, 582249616380374342400, 621260340677859423340800, 621260340677859423340800, 921088919608373507667359523840000000

Offset: 1

Paolo P. Lava, May 12 2016

nonn

Cf. A000005, A272981.

with(numtheory): P:= proc(q) local a,j,k,ok,n,p; a:=2;
for k from 1 to q do for n from a to q do ok:=1;
for j from 1 to k do if not type(n^j/tau(n^j),integer) then ok:=0; break; fi; od;
if ok=1 then a:=n; print(n); break; fi; od; od; end: P(10^9);

Table[SelectFirst[Range[2, 2*10^6], AllTrue[#^Range@ n, Divisible[#, DivisorSigma[0, #]] &] &], {n, 5}] (* Michael De Vlieger, May 12 2016, Version 10 *)

2 / d(2) = 2 / 2 = 1 but 2^2 / d(2^2) = 4 / 3;

60 / d(60) = 60 / 12 = 5, 60^2 / d (60^2) = 3600 / 45 = 80 but 60^3 / d(60^3) = 216000 / 112 = 13500 / 7.

a(6)-a(14) from Giovanni Resta, May 12 2016

A272857 Least k>1 such that the Euler totient function of powers k^e, 1 <= e <= n, are divisible by the number their divisors, d(k^e).

Original entry on oeis.org

3, 3, 13, 61, 61, 421, 2521, 2521, 2521, 55441, 55441, 4324321, 4324321, 4324321, 4324321, 85765681, 85765681, 232792561, 232792561, 232792561, 232792561

Offset: 1

Paolo P. Lava, May 12 2016

a(22) <= 10708457761. - Amiram Eldar, May 27 2024

			phi(3) / d(3) = 2 / 2 = 1, phi(3^2) / d(3^2) = 6 / 3 = 2 but phi(3^3) / d(3^3) = 18 / 4 = 9 / 2;
phi(13) / d(13) = 12 / 2 = 6, phi(13^2) / d(13^2) = 156 / 3 = 52, phi(13^3) / d(13^3) = 2028 / 4 = 507 but phi(13^4) / d(13^4) = 26364 / 5.

Cf. A000005, A000010, A070858, A272980, A272981.

with(numtheory): P:= proc(q) local a, j, k, ok, p; global n; a:=2;
for k from 1 to q do for n from a to q do ok:=1;
for j from 1 to k do if not type(phi(n^j)/tau(n^j), integer) then ok:=0; break; fi; od;
if ok=1 then a:=n; print(n); break; fi; od; od; end: P(10^9);

cp's OEIS Frontend

A272980 Least k>1 such that all powers k^e, 1 <= e <= n, are divisible by the number of their divisors, d(k^e).

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