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

1,1

Comments

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, ..., .

Links

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

Example

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.

Mathematica

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]

Prog

(PARI) 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

Crossrefs

Cf. A000005, A027423, A069784, A286872.

Sequence in context: A164094 A246319 A302649 * A112269 A314527 A314528

Adjacent sequences: A286832 A286833 A286834 * A286836 A286837 A286838

Keyword

nonn

Author

Robert G. Wilson v, Jul 31 2017

Status

approved