A398316 - OEIS

A398316

Numbers of the form p^2*s, where p is prime, s is squarefree, gcd(p,s) = 1, and A001221(s) >= 2.

60, 84, 90, 126, 132, 140, 150, 156, 198, 204, 220, 228, 234, 260, 276, 294, 306, 308, 315, 340, 342, 348, 350, 364, 372, 380, 414, 420, 444, 460, 476, 490, 492, 495, 516, 522, 525, 532, 550, 558, 564, 572, 580, 585, 620, 630, 636, 644, 650, 660, 666, 693, 708, 726, 732, 735, 738, 740, 748

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

The prime signature of every term is [1,1,...,1,2], with at least two 1's. Equivalently, exactly one prime occurs with exponent 2, all other prime exponents are 1, and A001221(a(n)) >= 3.

The subsequence of terms with exactly three distinct prime factors is A085987. The first term with more than three distinct prime factors is 420 = 2^2*3*5*7.

A072357 is the disjoint union of A054753 and this sequence.

From their prime signatures, A060687 is the disjoint union of A001248, A054753, and this sequence.

For each k >= 3, A088860(k) is the smallest term having exactly k distinct prime factors. Thus the tail A088860(3), A088860(4), ... = 60, 420, 4620, 60060, ... is a subsequence of this sequence.

The asymptotic density of this sequence is (6/Pi^2) * Sum_{p prime} 1/(p(p+1)) = 0.200755... (A271971), the same as the density of A060687. - Amiram Eldar, Jul 26 2026

Terms k in this sequence are of the form p*m for p | m, m in A350352, where A350352 = {m : m = A007947(m), A001221(m) > 2}. - Michael De Vlieger, Jul 26 2026

LINKS

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Michael De Vlieger, Plot prime(i)^m | a(n) at (x,y) = (n,i), n = 1..256, 4X horizontal and 8X vertical exaggeration, with a color function showing m = 1 in black and m = 2 in red.

Michael De Vlieger, Plot prime(i)^m | a(n) at (x,y) = (n,i), n = 1..2048, 3X vertical exaggeration, with a color function showing m = 1 in black and m = 2 in red.

FORMULA

Integers m such that A001222(m) - A001221(m) = 1 and A001221(m) >= 3.

Intersection of A060687 and A000977.

Equals A072357 \ A054753.

For k >= 3, min{a(n) : A001221(a(n)) = k} = A088860(k) = 2*A002110(k).

EXAMPLE

The smallest terms having respectively 3, 4, 5, and 6 distinct prime factors are

60 = A088860(3) = 2^2*3*5,

420 = A088860(4) = 2^2*3*5*7,

4620 = A088860(5) = 2^2*3*5*7*11,

60060 = A088860(6) = 2^2*3*5*7*11*13,

with prime signatures [1,1,2], [1,1,1,2], [1,1,1,1,2], and [1,1,1,1,1,2], respectively.

MATHEMATICA

q[k_] := Module[{e = FactorInteger[k][[;; , 2]]}, Length[e] >= 3 && Select[e, # > 1 &] == {2}]; Select[Range[750], q] (* Amiram Eldar, Jul 26 2026 *)

PROG

(PARI) isok(k) = my(f=factor(k)[, 2]~); (#select(x->(x==2), f) == 1) && (#select(x->(x==1), f) >= 2); \\ Michel Marcus, Jul 26 2026

(Python)

from math import isqrt

from sympy import primepi, primerange, integer_nthroot

from oeis_sequences.OEISsequences import bisection, squarefreepi

def A398316(n):

def f(x): return int(n+x+sum(primepi(x//p**2)+sum(squarefreepi(x//p**j) if j&1 else -squarefreepi(x//p**j) for j in range(2, x.bit_length())) for p in primerange(isqrt(x)+1))+primepi(isqrt(x))-primepi(integer_nthroot(x, 3)[0]))

return bisection(f, n, n) # Chai Wah Wu, Jul 30 2026

CROSSREFS

Cf. A000977, A001221, A001222, A001248, A002110, A054753, A060687, A072357, A085987, A088860, A126706, A271971, A332785, A350352, A378878, A392926, A396269.

Subsequence of A000977, A060687, A072357, A126706, A332785, A392926, and A396269.

A085987 is the subsequence of terms with exactly three distinct prime factors.

The terms A088860(k), k >= 3, form the subsequence of least terms for each fixed value k = A001221(a(n)).

A378878 = A126706 \ A072357. Hence A126706 is the disjoint union of A054753, this sequence, and A378878.

Sequence in context: A009129 A174292 A396269 * A085987 A356413 A398787

Adjacent sequences: A398310 A398314 A398315 * A398318 A398322 A398323

KEYWORD

nonn

AUTHOR

Charles Kusniec, Jul 25 2026

STATUS

approved