A276937 - OEIS

%I #18 Sep 30 2023 21:56:37

%S 2,6,9,10,14,18,22,26,30,34,38,42,45,46,50,58,62,63,66,70,74,78,82,86,

%T 90,94,98,99,102,106,110,114,117,118,122,125,126,130,134,138,142,146,

%U 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

%N 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.

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

%C 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

%H Antti Karttunen, <a href="/A276937/b276937.txt">Table of n, a(n) for n = 1..5000</a>

%e 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.

%e 36 = 4*9 = prime(1)^2 * prime(2)^2 is NOT a member because prime(1)^2 does not satisfy the second condition.

%e 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.

%t p[n_]:=FactorInteger[n][[All,1]];f[n_]:=PrimePi/@p[n];

%t yQ[n_]:=Select[n/(Prime[f[n]]^f[n]),IntegerQ]!={};

%t nQ[n_]:=Select[n/(Prime[f[n]]^(f[n]+1)),IntegerQ]=={};

%t Select[Range[2,250],yQ[#]&&nQ[#]&] (* _Ivan N. Ianakiev_, Sep 28 2016 *)

%o (Scheme, with _Antti Karttunen_'s IntSeq-library)

%o (define A276937 (MATCHING-POS 1 1 (lambda (n) (and (not (zero? (A276935 n))) (zero? (A276077 n))))))

%o (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

%Y Intersection of A276078 and A276936.

%Y Topmost row of A276941 (leftmost column in A276942).

%Y Cf. A276935, A276077.

%K nonn,easy

%O 1,1

%A _Antti Karttunen_, Sep 24 2016