A134652 Indices for which A097344 differs from A097345.

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

Offset: 1

Author

M. F. Hasler, Jan 25 2008

Keywords

• nonn
• hard
• more

Comments

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

a(6) > 10^5.

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)).

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.

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

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).

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.

Crossrefs

Cf. A097344, A097345.

Programs

Mathematica

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 *)

PARI

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

Extensions

a(6)-a(8) from Amiram Eldar, Apr 08 2019