A378720 -id:A378720 - cp's OEIS frontend

A284411 Least prime p such that more than half of all integers are divisible by n distinct primes not greater than p.

3, 37, 42719, 5737850066077

Offset: 1

Peter Munn, Mar 26 2017

The proportion of all integers that satisfy the divisibility criterion for p=prime(m) is determined using the proportion that satisfy it over any interval of primorial(m)=A002110(m) integers.
a(4) is from De Koninck, 2009; calculation credited to David Grégoire.
a(5) is about 7.887*10^34 assuming the Riemann Hypothesis, and about 7*10^34 unconditionally (De Koninck and Tenenbaum, 2002). - Amiram Eldar, Dec 05 2024

			Exactly half of the integers are divisible by 2, so a(1)>2. Two-thirds of all integers are divisible by 2 or 3, so a(1) = 3.

Jean-Marie De Koninck, Those Fascinating Numbers, American Mathematical Society, 2009, pp. 13, 216 and 368.

Jean-Marie De Koninck and Gérald Tenenbaum, Sur la loi de répartition du k-ième facteur premier d'un entier, Mathematical Proceedings of the Cambridge Philosophical Society, Vol. 133, No. 2 (2002), pp. 191-204.
Gérald Tenenbaum, Some of Erdős' unconventional problems in number theory, thirty-four years later, Erdős Centennial, Janos Bolyai Math. Soc., 2013, 651-681. HAL Id: hal-01281530.

Cf. A002110, A077761, A096294, A194156, A281889.

Cf. A038110, A038111, A342479, A342480, A378720, A378721.

a(n) is least p=prime(m) such that 2*Sum_{k=0..n-1} A096294(m,k) < A002110(m).

log(log(a(n))) = n - b + O(1/sqrt(n)), where b = 1/3 + A077761 (De Koninck and Tenenbaum, 2002). - Amiram Eldar, Dec 05 2024

Definition edited by N. J. A. Sloane, Apr 01 2017

A378721 a(n) is the denominator of the asymptotic density of numbers whose third smallest prime divisor is prime(n).

Original entry on oeis.org

1, 1, 30, 30, 165, 15015, 36465, 62985, 7436429, 11849255, 73465381, 33426748355, 50708377254535, 436092044389001, 1863302371480277, 1086305282573001491, 64092011671807087969, 3909612711980232366109, 8449808119441147371913, 18598027670889965365580513, 3543193335582015099413

Offset: 1

Robert G. Wilson v and Amiram Eldar, Dec 05 2024

See A378720 for more details.

Amiram Eldar, Table of n, a(n) for n = 1..365

Cf. A000040, A038110, A038111, A342479, A342480, A378720 (numerators).

a[n_] := Block[{p, q = Prime@ Range@ n}, p = Fold[Times, 1, q]; q = Most@ q; Plus @@ Times @@@ Subsets[q -1, {n -3}]/p]; a[1] = 0; Denominator@ Array[a, 21]

a(n) = {my(v = primes(n), q = vecextract(apply(x -> x-1, v),"^-1"), p = vecprod(v), prd = vecprod(q)/p, sm = 0, sb); forsubset([#q, 2], s, sb = vecextract(q, s); sm += 1/vecprod(sb)); denominator(prd * sm);}

cp's OEIS Frontend

A284411 Least prime p such that more than half of all integers are divisible by n distinct primes not greater than p.

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