A069784 -id:A069784 - cp's OEIS frontend

A286835 a(n) is the number of terms m such that d(m!) divides d((m!)^(2n+1)), where d is A000005.

5, 8, 15, 6, 29, 27, 5, 54, 60, 6, 63, 7, 6, 54, 75, 6, 12, 52, 7, 76, 69, 5, 74, 27, 6, 78, 12, 6, 97, 33, 6, 15, 85, 5, 99, 46, 5, 15, 95, 6, 56, 13, 6, 82, 20, 5, 7, 81, 6, 126, 141, 5, 130, 67, 6, 52, 13, 5, 17, 38, 5, 8, 55, 6, 85, 15, 5, 106, 143, 5, 22, 12, 6, 95, 94, 6

Offset: 1

Robert G. Wilson v, Jul 31 2017

nonn

Inspired by A069784.
Bisection of A286872.
d(m!) divides d((m!)^(2n)) only when m = 1.
The largest m for a(n) is: 5, 15, 91, 9, 275, 488, 5, 655, 1205, 21, 1687, 14, 9, 1462, 2313, 21, 35, 3436, 21, 7447, 4687, 5, 2555, 220, 9, 4627, 38, 9, 5114, 2606, 21, 65, 6071, 5, 4935, 5509, 5, 77, 10173, 9, 1646, 39, 9, 6715, 95, 5, 65, 2321, 9, 3786, 7059, 5, 7014, 1264, 9, 6272, 35, 5, 33, 215, 5, 27, 4283, 9, 2471, ..., ; and their records: 5, 15, 91, 275, 488, 655, 1205, 1687, 2313, 3436, 7447, 10173, 15464, 20004, 38539, 40605, 49143, ..., .

			a(1) = 5 since d(m!) divides d(m!^3) only for m = {1, 2, 3, 4, 5};
a(2) = 8 since d(m!) divides d(m!^5) only for m = {1, 2, 3, 4, 5, 12, 13, 15};
a(3) = 15 since d(m!) divides d(m!^7) only for m = {1, 2, 3, 4, 5, 6, 7, 8, 9, 14, 32, 33, 34, 35, 91};
a(4) = 6 since d(m!) divides d(m!^9) only for m = {1, 2, 3, 4, 5, 9};
a(5) = 29 since d(m!) divides d(m!^11) only for m = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, ..., 274, 275}; etc.

Robert G. Wilson v, Table of n, a(n) for n = 1..625

Cf. A000005, A027423, A069784, A286872.

factExpLst[nbr_] := factExpLst[nbr] = Table[Plus @@ Rest@ NestWhileList[ Floor[#/prm] &, nbr, # > 0 &], {prm, Prime@ Range@ PrimePi@ nbr}] (* which is the same as Transpose[ FactorInteger[ nbr!]][[2]] *);
ds0[nbr_, exp_] := Times @@ (1 + exp*factExpLst[ nbr]);
fQ[nbr_, exp_] := Mod[ds0[nbr, exp], ds0[nbr, 1]] == 0;
f[n_] := f[n] = If[EvenQ@ n, {1}, Select[Range@ 100000, fQ[#, n] &]];
f[1] = {};
Array[ Length@ f[2# +1] &, 60]

valp(n,p)=my(s); while(n\=p, s+=n); s
f(m,e)=my(s=1); forprime(p=2,m\2, s*=e*valp(m,p)+1); s*(e+1)^(primepi(m)-primepi(m\2))
search(n,lim=100*n^2)=my(v=List(),e=2*n+1); for(m=1,lim, if(f(m,e)%f(m,1)==0, listput(v,m))); Vec(v) \\ N.B., empirical upper bound
a(n)=#search(n) \\ Charles R Greathouse IV, Aug 01 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

A286835 a(n) is the number of terms m such that d(m!) divides d((m!)^(2n+1)), where d is A000005.

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