A069783 -id:A069783 - cp's OEIS frontend

A069784 Numbers m such that gcd(d((m!)^3), d(m!)) = 2^k, i.e., is a power of 2; d = A000005.

1, 2, 3, 4, 5, 6, 7, 8, 9, 12, 13, 14, 18, 19, 30, 31, 32, 35, 38, 39

Offset: 1

Labos Elemer, Apr 08 2002

From David A. Corneth, Jul 31 2017: (Start)
Theorem: There are no further terms.
Proof:
Let e_n(n, p) be the exponent of p in n!. The prime p has exponent e_(n, p) = n/p for sqrt(n) < p < n in n!. n/4 <= p < n/3, e_(n, p) = 3 so e_(n, p) * 3 = 9. and for n/5 <= p < n/4, e = 4. The gap g_n between prime(n) and prime(n+1) is about sqrt(n) * log(n). There is a gap of n/4 - n/5 = n/20 between n/5 and n/4. primepi(1000) = 168, so for n > 5*1000, the gap between n/5 and the next prime is about sqrt(168) * log(168) ~= 66. This is much less than n/20. No 40 <= m <= 15000 is in the sequence, which completes the proof. (End)

Cf. A000005, A000142, A069780, A069781, A069782, A069783.

Do[s=GCD[DivisorSigma[0, (n!)^3], DivisorSigma[0, n! ]]; If[IntegerQ[n/100], Print[{n}]]; If[IntegerQ[Log[2, s]], Print[n]], {n, 1, 10000}]

val(n, p) = my(r=0); while(n, r+=n\=p);r
is(n) = {my(p1 = p2 = 1); forprime(p=2, n, v = val(n, p); p1 *= (v + 1); p2 *= (3*v + 1)); g = gcd(p1, p2); g==2^(valuation(g, 2))} \\ David A. Corneth, Jul 31 2017

Keywords fini and full added by David A. Corneth, Jul 31 2017

A069785 a(n) = A061680(n!).

Original entry on oeis.org

1, 1, 1, 1, 1, 15, 15, 3, 5, 135, 135, 99, 99, 9, 63, 21, 21, 459, 459, 135, 19, 15, 15, 15, 21, 189, 189, 585, 585, 18225, 18225, 675, 15, 135, 891, 8505, 25515, 81, 81, 7695, 7695, 1575, 1575, 4725, 6615, 40635, 40635, 945, 1215, 3645, 3645, 151875, 151875

Offset: 1

Labos Elemer, Apr 09 2002

			Observe cases when consecutive terms are equal: n={1,2,3,4,6,10,...,78,80,82,88,96}.

Amiram Eldar, Table of n, a(n) for n = 1..1000

Cf. A000005, A000142, A061680, A061701, A048691, A069780, A069781, A069782, A069783, A069784.

a[n_] := Module[{e = FactorInteger[n!][[;;, 2]]}, GCD[Times @@ (2*e+1), Times @@ (e+1)]]; Array[a, 100] (* Amiram Eldar, Dec 02 2023 *)

a(n) = {my(e = factor(n!)[,2]); gcd(vecprod(apply(x -> 2*x+1, e)), vecprod(apply(x -> x+1, e)));} \\ Amiram Eldar, Dec 02 2023

a(n) = A061680(A000142(n)). - Amiram Eldar, Dec 02 2023

cp's OEIS Frontend

A069784 Numbers m such that gcd(d((m!)^3), d(m!)) = 2^k, i.e., is a power of 2; d = A000005.

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