A100605 -id:A100605 - cp's OEIS frontend

A101638 Number of distinct 4-almost primes A014613 which are factors of n.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 2, 0, 0, 0, 1

Offset: 1

Jonathan Vos Post, Dec 10 2004

This is the inverse Moebius transform of A101637. If we take the prime factorization of n = (p1^e1)*(p2^e2)* ... * (pj^ej) then a(n) = |{k: ek>=4}| + ((j-1)/2)*|{k: ek>=3}| + C(|{k: ek>=2}|,2) + C(j,4). The first term is the number of distinct 4th powers of primes in the factors of n (the first way of finding a 4-almost prime). The second term is the number of distinct cubes of primes, each of which can be multiplied by any of the other distinct primes, halved to avoid double-counts (the second way of finding a 4-almost prime). The third term is the number of distinct pairs of squares of primes in the factors of n (the third way of finding a 4-almost prime). The 4th term is the number of distinct products of 4 distinct primes, which is the number of combinations of j primes in the factors of n taken 4 at a time, A000332(j), (the 4th way of finding a 4-almost prime).

			a(96) = 2 because 96 = 16 * 6 hence divisible by the 4-almost prime 16 and also 96 = 24 * 4 hence divisible by the 4-almost prime 24.

Hardy, G. H. and Wright, E. M. Section 17.10 in An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, 1979.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
E. A. Bender and J. R. Goldman, On the Applications of Moebius Inversion in Combinatorial Analysis, Amer. Math. Monthly 82, 789-803, 1975.
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, arXiv:math/0205301 [math.CO], 2002; Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
Eric Weisstein's World of Mathematics, Almost Prime.
Eric Weisstein's World of Mathematics, Moebius Transform.

Cf. A101638, A014613, A000332, A086971, A100605, A000040, A001358, A014612, A014614.

a(n)=my(f=factor(n)[,2], v=apply(k->sum(i=1,#f,f[i]>k), [0..3])); v[4] + v[3]*(v[1]-1) + binomial(v[2],2) + v[2]*binomial(v[1]-1,2) + binomial(v[1],4) \\ Charles R Greathouse IV, Sep 14 2015

A100604 Numbers k such that (prime(k)-1)! + prime(k)^3 is prime.

Original entry on oeis.org

2, 3, 4, 28, 894

Offset: 1

Jonathan Vos Post, Nov 30 2004

k = {2, 3, 4, 28} yields primes p(k) = {3, 5, 7, 107}. There are no more such k up to k=100. Verified by Ray Chandler.
a(5) > 600. - Jinyuan Wang, Apr 10 2020
a(6) > 2500. - Michael S. Branicky, Jul 02 2024

			a(3) = 4 because (prime(4)-1)! + prime(4)^3 = (7-1)! + 7^3 = 720 + 343 = 1063 is the 3rd prime of this form.

Cf. A100605, A100858.

Select[Range[30],PrimeQ[(Prime[#]-1)!+Prime[#]^3]&] (* Harvey P. Dale, Jul 13 2022 *)

is(k) = ispseudoprime((prime(k)-1)! + prime(k)^3); \\ Jinyuan Wang, Apr 10 2020

Numbers k such that (prime(k)-1)! + prime(k)^3 is prime, where prime(k) is the k-th prime.

a(5) from Michael S. Branicky, Jul 01 2024

cp's OEIS Frontend

A101638 Number of distinct 4-almost primes A014613 which are factors of n.

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