A134652 - OEIS

%I #13 Apr 08 2019 09:52:02

%S 59,1519,7814,17225,39079,950619,977019,1280699

%N Indices for which A097344 differs from A097345.

%C The terms 59 and 1519 were found by Daniel Glasscock (glasscock(AT)rice.edu), Jan 04 2008.

%C a(6) > 10^5.

%C These are the numbers m such that f(m) = Sum_{k=0..m} binomial(m,k)/(k+1)^2 (binomial transform of 1/(k+1)^2) has the same numerator as g(m) = Sum_{k=0..m} (2^(k+1) - 1)/(k+1) (which are also the partial sums of the binomial transformation of 1/(k+1)).

%C Obviously, f(m) = Sum_{k=0..m} binomial(m+1, k+1)/((k+1)*(m+1)) and since g(m) = (m+1) f(m) (cf. notes by _R. J. Mathar_ on A097345), g(m) = Sum_{k=1..m+1} binomial(m+1,k)/k.

%C We have the equivalences: numerator(g(n)) = numerator(f(n)) <=> (n+1) | denominator(f(n)) <=> gcd(numerator(g(n)), n+1) = 1.

%C Therefore this sequence can be alternatively defined in either of the following two ways: numbers n such that the denominator of f(n) is not divisible by (n+1); numbers n such that the numerator of g(n) is not coprime to (n+1).

%C In terms of M = m+1, the characterization reads: a(n)+1 = numbers M such that denominator(Sum_{k=1..M} binomial(M-1, k-1)/k^2) is not a multiple of M = numbers M such that numerator(Sum_{k=1..M} (2^k - 1)/k) is not coprime to M.

%t Reap[ For[n = 1, n < 10^5, n++, If[ !Divisible[ Denominator[ HypergeometricPFQ[{1, 1, -n}, {2, 2}, -1]], n+1], Print[n]; Sow[n] ] ] ][[2, 1]] (* _Jean-François Alcover_, Oct 15 2013 *)

%o (PARI) t=1; for( n=2,10^5, gcd( numerator(t+=(1<<n-1)/n),n)>1 & print(n-1))

%Y Cf. A097344, A097345.

%K nonn,hard,more

%O 1,1

%A _M. F. Hasler_, Jan 25 2008

%E a(6)-a(8) from _Amiram Eldar_, Apr 08 2019