A276079 Numbers n such that prime(k)^(k+1) divides n for some k.

4, 8, 12, 16, 20, 24, 27, 28, 32, 36, 40, 44, 48, 52, 54, 56, 60, 64, 68, 72, 76, 80, 81, 84, 88, 92, 96, 100, 104, 108, 112, 116, 120, 124, 128, 132, 135, 136, 140, 144, 148, 152, 156, 160, 162, 164, 168, 172, 176, 180, 184, 188, 189, 192, 196, 200, 204, 208, 212, 216, 220, 224, 228, 232, 236, 240, 243, 244, 248, 252, 256, 260, 264, 268, 270, 272

Offset: 1

Author

Antti Karttunen, Aug 18 2016

Keywords

• nonn

Comments

The asymptotic density of this sequence is 1 - Product_{i>=1} 1-prime(i)^(-1-i) = 0.2789766... - Amiram Eldar, Oct 21 2020

Examples

			625 = 5*5*5*5 = prime(3)^4 so it is divisible by prime(3)^(3+1), and thus 625 is included in the sequence.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..5000

Crossrefs

Positions of nonzeros in A276077.

Complement: A276078.

Cf. A000040, A000720, A008586 (a subsequence).

Differs from its subsequence A100716 for the first time at n=175, where a(175) = 625, while that value is missing from A100716.

Programs

Python

from sympy import primepi, isprime, primefactors, factorint
def a028234(n):
    f=factorint(n)
    minf = min(f)
    return 1 if n==1 else n//(minf**f[minf])
def a067029(n):
    f=factorint(n)
    return 0 if n==1 else f[min(f)]
def a049084(n): return primepi(n) if isprime(n) else 0
def a055396(n): return 0 if n==1 else a049084(min(primefactors(n)))
def a(n): return 0 if n==1 else a(a028234(n)) + (1 if a067029(n) > a055396(n) else 0)
print([n for n in range(1, 301) if a(n)!=0]) # Indranil Ghosh, Jun 21 2017