|
|
A335207
|
|
Numbers L such that there is a prime p <= L for which v_p(H_L - 1) > 1, where v_p(x) is the p-adic valuation of x and H_L is the L-th harmonic number.
|
|
1
|
|
|
|
OFFSET
|
1,1
|
|
COMMENTS
|
This is a subset of A335189. All numbers in this list were copied from one of the links below by Krattenthaler and Rivoal.
For all L in this list (up to 904332), we have v_p(H_L - 1) = 2 with corresponding primes as follows: p(1) = 7, p(2) = 13, p(3) = 7, p(4) = p(5) = 11, p(6) = 41, p(7) = p(8) = 11, p(9) = 53, and p(10) = 97.
The calculation of v_p(H_L-1) and v_p(H_L) for all primes p <= L is related to some results about the integrality of the Taylor coefficients of mirror maps. See Theorems 3 and 4 in Krattenthaler and Rivoal (2007-2009, 2009) and sequences A007757, A131657, and A131658.
|
|
LINKS
|
|
|
MAPLE
|
A335207_list := proc(bound) local p, h, H, L, n;
L := NULL; h := 0;
for n from 1 to bound do
h := h + 1/n; H := h - 1; p:= 2;
while p <= n do
if padic:-ordp(H, p) <= 1
then p := nextprime(p);
else L := L, n; break;
fi
od;
od; L end:
|
|
PROG
|
(PARI) list(nn) = {my(h=-1); for (n=1, nn, h += 1/n; forprime(p=1, n-1, if(valuation(h, p) > 1, print1(n, ", "); break)); ); } \\ Petros Hadjicostas, May 26 2020, courtesy of Michel Marcus
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn,more,hard
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|