A087719 Least number m such that the number of numbers k <= m with k > spf(k)^n exceeds the number of numbers with k <= spf(k)^n.

15, 27, 57, 135, 345, 927, 2577, 7335, 21225, 62127, 183297, 543735, 1618905, 4832127, 14447217, 43243335, 129533385, 388206927, 1163834337, 3489930135, 10466644665, 31393642527, 94168344657, 282479868135

Offset

1,1

Comments

m<a(n): #{k: k>spf(k)^n & 1<=k<=m} <= m/2;

m>=a(n): #{k: k>spf(k)^n & 1<=k<=m} > m/2.

a(n) = 3^n + 3*2^n + 6. Proof: Observe that for any composite k with spf(k)>=5, we have k>=p^2. For the condition k>p^n to hold, k must be very large. No numbers with spf >= 5 contribute to the "exceeding" count in the range 1...a(n). The count of satisfying numbers reduces to counting integers k from 1...m such that (2|k and k>2^n) or (3|k and k odd and k>3^n). This count can be expressed using floor functions: C(m) = floor(m/2) - 2^(n-1) + floor((m+3)/6) - floor((3^n+3)/6). The condition C(m) > m-C(m) leads to 2C(m) > m, and m = 3^n + 3*2^n + 6 is the smallest integer satisfying this inequality. - Ralf Stephan, Jan 27 2026

Links

Table of n, a(n) for n=1..24.

Formula

a(n) = 3^n + 3*2^n + 6.

G.f.: 3*x*(5-21*x+20*x^2)/(1-x)/(1-2*x)/(1-3*x). - Colin Barker, Feb 22 2012

Empirical: a(n) = 3*A183624(n-2), n>=3. - R. J. Mathar, Apr 16 2026

Crossrefs

Cf. A088382, A088382, A088380, A088381, A000430, A080257.

Sequence in context: A227804 A365451 A343139 * A174216 A249874 A116070

Adjacent sequences: A087716 A087717 A087718 * A087720 A087721 A087722

Keyword

nonn

Author

Reinhard Zumkeller, Sep 29 2003

Extensions

a(14)-a(24) from Giovanni Resta, May 23 2013

Status

approved