A276937 Numbers m with at least one prime factor prime(k) such that prime(k)^k is a divisor of m, but with no factor prime(j) such that prime(j)^(j+1) divides m.

2, 6, 9, 10, 14, 18, 22, 26, 30, 34, 38, 42, 45, 46, 50, 58, 62, 63, 66, 70, 74, 78, 82, 86, 90, 94, 98, 99, 102, 106, 110, 114, 117, 118, 122, 125, 126, 130, 134, 138, 142, 146, 150, 153, 154, 158, 166, 170, 171, 174, 178, 182, 186, 190, 194, 198, 202, 206, 207, 210, 214, 218, 222, 225, 226, 230, 234, 238, 242, 246, 250

Offset: 1

Author

Antti Karttunen, Sep 24 2016

Keywords

• nonn
• easy

Comments

Numbers m for which A276077(m) = 0 and A276935(m) > 0.

The asymptotic density of this sequence is Product_{k>=1} (1 - 1/prime(k)^k) - Product_{k>=1} (1 - 1/prime(k)^(k-1)) = 0.2803209124521781114031... . - Amiram Eldar, Sep 30 2023

Examples

			14 = 2*7 = prime(1)^1 * prime(4)^1 is a member as the first prime factor (2) satisfies the first condition, and neither prime factor violates the second condition.
36 = 4*9 = prime(1)^2 * prime(2)^2 is NOT a member because prime(1)^2 does not satisfy the second condition.
45 = 5*9 = prime(3)^1 * prime(2)^2 is a member as the latter prime factor satisfies the first condition, and neither prime factor violates the second condition.
		

Links

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

Crossrefs

Intersection of A276078 and A276936.

Topmost row of A276941 (leftmost column in A276942).

Cf. A276935, A276077.

Programs

Mathematica

p[n_]:=FactorInteger[n][[All,1]];f[n_]:=PrimePi/@p[n];
yQ[n_]:=Select[n/(Prime[f[n]]^f[n]),IntegerQ]!={};
nQ[n_]:=Select[n/(Prime[f[n]]^(f[n]+1)),IntegerQ]=={};
Select[Range[2,250],yQ[#]&&nQ[#]&] (* Ivan N. Ianakiev, Sep 28 2016 *)

PARI

is(n) = {my(f = factor(n), c = 0, k); for (i=1, #f~, k = primepi(f[i, 1]); if(f[i, 2] > k, return(0), if( f[i, 2] == k, c++))); c > 0;} \\ Amiram Eldar, Sep 30 2023