cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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.

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%I A276937 #18 Sep 30 2023 21:56:37
%S A276937 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 A276937 90,94,98,99,102,106,110,114,117,118,122,125,126,130,134,138,142,146,
%U A276937 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 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.
%C A276937 Numbers m for which A276077(m) = 0 and A276935(m) > 0.
%C A276937 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 A276937 Antti Karttunen, <a href="/A276937/b276937.txt">Table of n, a(n) for n = 1..5000</a>
%e A276937 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 A276937 36 = 4*9 = prime(1)^2 * prime(2)^2 is NOT a member because prime(1)^2 does not satisfy the second condition.
%e A276937 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 A276937 p[n_]:=FactorInteger[n][[All,1]];f[n_]:=PrimePi/@p[n];
%t A276937 yQ[n_]:=Select[n/(Prime[f[n]]^f[n]),IntegerQ]!={};
%t A276937 nQ[n_]:=Select[n/(Prime[f[n]]^(f[n]+1)),IntegerQ]=={};
%t A276937 Select[Range[2,250],yQ[#]&&nQ[#]&] (* _Ivan N. Ianakiev_, Sep 28 2016 *)
%o A276937 (Scheme, with _Antti Karttunen_'s IntSeq-library)
%o A276937 (define A276937 (MATCHING-POS 1 1 (lambda (n) (and (not (zero? (A276935 n))) (zero? (A276077 n))))))
%o A276937 (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 A276937 Intersection of A276078 and A276936.
%Y A276937 Topmost row of A276941 (leftmost column in A276942).
%Y A276937 Cf. A276935, A276077.
%K A276937 nonn,easy
%O A276937 1,1
%A A276937 _Antti Karttunen_, Sep 24 2016