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

1,1

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

Links

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

Example

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.

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 *)

Prog

(Scheme, with Antti Karttunen's IntSeq-library)
(define A276937 (MATCHING-POS 1 1 (lambda (n) (and (not (zero? (A276935 n))) (zero? (A276077 n))))))
(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

Crossrefs

Intersection of A276078 and A276936.

Topmost row of A276941 (leftmost column in A276942).

Cf. A276935, A276077.

Sequence in context: A359821 A047396 A276936 * A243373 A085304 A015843

Adjacent sequences: A276934 A276935 A276936 * A276938 A276939 A276940

Keyword

nonn,easy

Author

Antti Karttunen, Sep 24 2016

Status

approved