A348119 -id:A348119 - cp's OEIS frontend

A348120 Starts of runs of 3 consecutive numbers whose powerful part is larger than their powerfree part (A328014).

48, 350, 9800, 11374, 31211, 32798, 48373, 59534, 63000, 103246, 118579, 373827, 488187, 625974, 629693, 830464, 1193983, 1294298, 2989439, 3815174, 4231248, 5132699, 5331248, 6674166, 7616950, 7970157, 8388223, 8670375, 9235520, 9516680, 9841094, 11121382, 12708359

Offset: 1

Amiram Eldar, Oct 01 2021

nonn

There are no runs of 4 consecutive numbers below 10^10.

It is conjectured that there are no runs of 3 consecutive numbers that are powerful (A001694), but if they do exist, their starts are contained in this sequence.

			48 is a term since 48, 49 and 50 are all in A328014.

Amiram Eldar, Table of n, a(n) for n = 1..280

Subsequence of A328014 and A348119.

Cf. A001694.

f[p_, e_] := If[e==1, p, 1]; s[n_] := Times @@ (f @@@ FactorInteger[n]); q[n_] := s[n]^2 < n; v = q /@ Range[3]; seq = {}; Do[v = Append[Drop[v, 1], q[k]]; If[And @@ v, AppendTo[seq, k - 2]], {k, 4, 10^6}]; seq

A348122 Numbers k such that k and k+1 both have more nonunitary than unitary prime divisors (A348121).

Original entry on oeis.org

8, 288, 360, 675, 1224, 1331, 1368, 2196, 2400, 2600, 2808, 3024, 5328, 6075, 6859, 9408, 9800, 10647, 11448, 12167, 16128, 17199, 19844, 20448, 21024, 23275, 25920, 26568, 26900, 28899, 29791, 33524, 38024, 38808, 39600, 40400, 41624, 42875, 45324, 46224, 46475

Offset: 1

Amiram Eldar, Oct 01 2021

nonn

			8 is a term since 8 = 2^3 has one nonunitary prime divisor, 2, and no unitary prime divisors, and 8 + 1 = 9 = 3^2 has one nonunitary prime divisor, 3, and no unitary prime divisors.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Subsequence of A348121.
A060355 is a subsequence.
Similar sequence: A348119.

q[n_] := 2*Count[(e = FactorInteger[n][[;; , 2]]), 1] < Length[e]; Select[Range[5*10^5], q[#] && q[# + 1] &]

A355433 Numbers k such that k is sqrt(k)-smooth and k+1 is sqrt(k+1)-smooth.

Original entry on oeis.org

8, 24, 48, 49, 63, 80, 120, 125, 168, 175, 195, 224, 242, 288, 324, 350, 351, 360, 363, 374, 384, 399, 440, 441, 455, 475, 494, 512, 528, 539, 560, 575, 594, 624, 675, 714, 728, 735, 759, 832, 840, 874, 896, 935, 960, 968, 1000, 1014, 1023, 1044, 1053, 1088, 1104

Offset: 1

Amiram Eldar, Jul 02 2022

nonn

Numbers k such that k and k+1 are both in A048098.

This sequence is infinite: if p is an odd prime then p^2-1 is a term.

			8 is a term since 8 is sqrt(8)-smooth (2^2 <= 8) and 9 is sqrt(9)-smooth (3^2 <= 9).

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A048098, A355434.

Subsequences: A084920 \ {3}, A060355, A348119.

smQ[n_] := FactorInteger[n][[-1, 1]]^2 <= n; Select[Range[1000], smQ[#] && smQ[# + 1] &]

cp's OEIS Frontend

A348120 Starts of runs of 3 consecutive numbers whose powerful part is larger than their powerfree part (A328014).

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A348122 Numbers k such that k and k+1 both have more nonunitary than unitary prime divisors (A348121).

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A355433 Numbers k such that k is sqrt(k)-smooth and k+1 is sqrt(k+1)-smooth.

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