A092579 -id:A092579 - cp's OEIS frontend

A119788 Ratio of the numerator of the product of n and the n-th alternating harmonic number nH'(n) to the numerator of the n-th alternating harmonic number H'(n) = Sum_{k=1..n} (-1)^(k+1)1/k.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 13

Offset: 1

Alexander Adamchuk, Jun 26 2006, Sep 21 2006

Indices n such that a(n) is not equal to 1 are listed in A121594.
It appears that most a(n) > 1 are a prime divisor of their corresponding indices A121594(n). The first and only composite term up to a(6000) is a(1470) = 49 that also divides its index.
A compressed version of this sequence (all 1 entries are excluded) is A121595.

Alexander Adamchuk, Table of n, a(n) for n = 1..400

Cf. A058313, A092579, A119787, A121594, A121595.

Numerator[Table[n*Sum[(-1)^(i+1)*1/i, {i, 1, n}],{n,1,600}]]/Numerator[Table[Sum[(-1)^(i+1)*1/i, {i, 1, n}], {n,1,600}]]

a(n) = numerator(n*Sum_{i=1..n} (-1)^(i+1)*1/i) / numerator(Sum_{i=1..n}(-1)^(i+1)*1/i).

a(n) = A119787(n) / A058313(n).

A121595 Compressed version of A119788 (all entries equal to 1 are excluded).

Original entry on oeis.org

5, 7, 5, 11, 13, 17, 7, 29, 7, 37, 19, 47, 119, 41, 23, 5, 29, 31, 11, 37, 37, 41, 43, 71, 13, 7, 13, 13, 47, 13, 49, 7, 7, 7, 53, 5, 79, 59, 97, 61, 71, 103, 67, 17, 71, 61, 73, 139, 17, 17, 79, 19, 19, 19, 83, 19, 151, 89, 29, 29, 263, 97

Offset: 1

Alexander Adamchuk, Aug 09 2006

nonn

Also the ratio of the numerators of n*H'(n) = A119787(n) and H'(n) = A058313(n) when they are different. (H'(n) is the alternating harmonic number H'(n) = Sum_{k=1..n} (-1)^(k+1)*1/k.)
The ratio of numerators A119787(n)/A058313(n) for n = 1..400 is given in A119788(n).
It appears that most a(n) are prime divisors of the corresponding indices A121594(n).
The first and only composite a(n) up to A119788(6000) is a(31) = 49 corresponding to A119788(1470).
It appears that all a(n) belong to A092579(n), which is a sieve using the Fibonacci sequence over the integers >= 2.  [Edited by Petros Hadjicostas, May 11 2020]

Cf. A058313, A092579, A119787, A119788, A121594.

Do[H=Sum[(-1)^(i+1)*1/i, {i, 1, n}]; a=Numerator[n*H]; b=Numerator[H]; If[ !Equal[a,b],Print[{n,a/b}]], {n,1,6000}]

a(n) = A119788(A121594(n)), while the corresponding indices are given in A121594(n).

A147956 All positive integers that are not multiples of any Fibonacci numbers >= 2.

Original entry on oeis.org

1, 7, 11, 17, 19, 23, 29, 31, 37, 41, 43, 47, 49, 53, 59, 61, 67, 71, 73, 77, 79, 83, 97, 101, 103, 107, 109, 113, 119, 121, 127, 131, 133, 137, 139, 149, 151, 157, 161, 163, 167, 173, 179, 181, 187, 191, 193, 197, 199, 203, 209, 211, 217, 223, 227, 229, 239, 241

Offset: 1

Leroy Quet, Nov 17 2008

nonn

This sequence contains a 1 and all terms of sequence A092579 that are not prime Fibonacci numbers.

			77 has the divisors 1,7,11,77. None of these divisors is a Fibonacci number >= 2. So 77 is included in the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A092579.

q:= n-> not ormap(d-> (t-> issqr(t+4) or issqr(t-4)
        )(5*d^2), numtheory[divisors](n) minus {1}):
select(q, [$1..250])[];  # Alois P. Heinz, Jul 15 2022

fibQ[n_] := IntegerQ @ Sqrt[5 n^2 - 4] || IntegerQ @ Sqrt[5 n^2 + 4]; aQ[n_] := !AnyTrue[Rest[Divisors[n]], fibQ]; Select[Range[250], aQ] (* Amiram Eldar, Oct 06 2019 *)

isfib1(n) = if (n>1, my(k=n^2); k+=(k+1)<<2; (issquare(k) || issquare(k-8)));
isok(k) = fordiv(k, d, if (isfib1(d), return(0))); 1; \\ Michel Marcus, Jul 15 2022

Extended by Ray Chandler, Nov 24 2008

cp's OEIS Frontend

A119788 Ratio of the numerator of the product of n and the n-th alternating harmonic number nH'(n) to the numerator of the n-th alternating harmonic number H'(n) = Sum_{k=1..n} (-1)^(k+1)1/k.

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A121595 Compressed version of A119788 (all entries equal to 1 are excluded).

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A147956 All positive integers that are not multiples of any Fibonacci numbers >= 2.

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cp's OEIS Frontend

A119788 Ratio of the numerator of the product of n and the n-th alternating harmonic number n*H'(n) to the numerator of the n-th alternating harmonic number H'(n) = Sum_{k=1..n} (-1)^(k+1)*1/k.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A121595 Compressed version of A119788 (all entries equal to 1 are excluded).

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Formula

A147956 All positive integers that are not multiples of any Fibonacci numbers >= 2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Extensions

A119788 Ratio of the numerator of the product of n and the n-th alternating harmonic number nH'(n) to the numerator of the n-th alternating harmonic number H'(n) = Sum_{k=1..n} (-1)^(k+1)1/k.