A293556 -id:A293556 - cp's OEIS frontend

1, 6, 10, 18, 30, 42, 60, 78, 84, 90, 126, 150, 210, 330, 390, 420, 630, 840, 990, 1050, 1470, 1890, 2100, 2310, 2730, 3570, 3990, 4620, 5460, 6930, 8190, 9240, 10920, 11550, 13650, 13860, 16170, 19110, 20790, 23100, 25410, 30030, 39270, 43890, 46410, 51870

Offset: 1

Michael De Vlieger, Oct 22 2017

nonn

From Michael De Vlieger, Nov 17 2017: (Start)
Terms in a(n) appear in A244052 except {78, 126, 990, 19110, 6276270, ...}.
Primorials A002110(t) seem to divide this sequence into "tiers" thus: all terms A002110(t) <= m < A002110(t + 1), wherein A001221(m) = t as seen in A244052.
Terms in A244052 appear in a(n) except {2, 4, 12, 24, 120, 180, 1260, 1680, 18480, 27720, 360360, ...}. These numbers seem to have significantly more divisors than terms that are slightly greater or lesser in a(n).
Conjecture: all terms of a(n) with n > 92 also appear in A244052, and all terms in A244052 greater than a(92) = 6276270 appear in a(n).
(End)

			From _Michael De Vlieger_, Nov 17 2017: (Start)
Consider A243822(n), a function that counts numbers k < n such that k | n^e with e >= 2. The numbers k themselves appear in A272618(n).
a(1) = 1 since the number 1 has 0 such k. Primes p also have 0 such k, since 1 | p and all other numbers k < p are coprime to p. Prime powers p^e have 0 such k since any number k | n^e divides n^1.
a(2) = 6 since it is the smallest number to have 1 such k (i.e., 4 | 6^2). The numbers 7, 8, and 9 are prime powers having 0 such k.
a(3) = 10 since it has 2 such k (i.e., 4 | 10^2, 8 | 10^3), etc.
(End)

Michael De Vlieger, Table of n, a(n) for n = 1..110
Michael De Vlieger, Records and indices of records in A243822

Cf. A000005, A010846, A243822, A244052, A272618, A293556.

With[{s = Table[Count[Range@ n, ?(PowerMod[n, #, #] == 0 &)] - DivisorSigma[0, n], {n, 10^4}]}, Position[s, #][[1, 1]] & /@ Union@ FoldList[Max, s]] (* _Michael De Vlieger, Oct 22 2017 *)

cp's OEIS Frontend

A293555 Indices of records in A243822.

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