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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.
5
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
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).
Sequence in context: A359821 A047396 A276936 * A243373 A085304 A015843
KEYWORD
nonn,easy
AUTHOR
Antti Karttunen, Sep 24 2016
STATUS
approved