A290348 -id:A290348 - cp's OEIS frontend

A330719 a(n) = denominator(Sum_{k=1..n} (2^(k-1) - 1)/k).

1, 2, 2, 4, 4, 12, 12, 24, 8, 40, 40, 120, 120, 840, 840, 1680, 1680, 5040, 5040, 5040, 5040, 55440, 55440, 55440, 11088, 144144, 48048, 48048, 48048, 80080, 80080, 160160, 160160, 2722720, 544544, 4900896, 4900896, 93117024, 93117024, 465585120, 465585120, 465585120

Offset: 1

Amiram Eldar and Thomas Ordowski, Dec 28 2019

Conjecture: if p is an odd prime, then p | A330718(p+1) - a(p+1).
Below 10^6 there is only one pseudoprime, namely 25. Are there others?
Primes p such that p^2 | A330718(p+1) - a(p+1) are 3, 5, 45827, ...

			Denominators of 0, 1/2, 3/2, 13/4, 25/4, 137/12, 245/12, ...

Metin Sariyar, Table of n, a(n) for n = 1..500

Cf. A000265, A002805, A064169, A279683, A290348, A330718.

[Denominator( &+[(2^(k-1)-1)/k: k in [1..n]] ): n in [1..45]]; // G. C. Greubel, Dec 28 2019

seq(denom(add((2^(k-1)-1)/k, k = 1..n)), n = 1..45); # G. C. Greubel, Dec 28 2019

Denominator@Accumulate@Array[(2^(#-1) -1)/# &, 45]
Table[Denominator[-(2^n*LerchPhi[2, 1, n+1] +Pi*I/2 +HarmonicNumber[n])], {n, 45}] (* G. C. Greubel, Dec 28 2019 *)

a(n) = denominator(sum(k=1, n, (2^(k-1)-1)/k)); \\ Michel Marcus, Dec 28 2019

[denominator( sum((2^(k-1)-1)/k for k in (1..n)) ) for n in (1..45)] # G. C. Greubel, Dec 28 2019

a(n) = denominator(-(2^n*LerchPhi(2,1,n+1) + Pi*i/2 + HarmonicNumber(n))). - G. C. Greubel, Dec 28 2019
a(n) = denominator(A279683(n)/n!) for n > 0. - Amiram Eldar and Thomas Ordowski, Jan 15 2020
A000265(a(n)) = A290348(n). - Thomas Ordowski, Mar 29 2025

A290347 Numerators of the Harary index for the n-halved cube graph.

Original entry on oeis.org

0, 1, 6, 26, 100, 1096, 3920, 13936, 16544, 296256, 1068672, 11652352, 42658304, 1100471296, 4079876096, 15205967872, 56939270144, 642281037824, 2423854317568, 9177027411968, 34846713511936, 1459319692460032, 5568939824513024, 21297365878571008

Offset: 1

Eric W. Weisstein, Jul 28 2017

For p > 3, if p is prime, then p^2 divides a(p). Conjecture: for n > 3, if n^2 divides a(n), then n is prime. Primes p such that p^3 diviedes a(p) are probably A088164. - Thomas Ordowski, Mar 30 2025

			First few terms are 0, 1, 6, 26, 100, 1096/3, 3920/3, 13936/3, 16544, 296256/5, ....

Eric Weisstein's World of Mathematics, Harary Index
Eric Weisstein's World of Mathematics, Halved Cube Graph

Cf. A000265, A088164, A290348 (denominators), A330718.

Table[-2^(n - 1) HarmonicNumber[n] - 2^(2 n - 1) Re[LerchPhi[2, 1, n + 1]], {n, 20}] // Numerator

a(n) = -2^(n-1)*HarmonicNumber(n)-2^(2*n-1)*Re(LerchPhi(2,1,n+1)).

For n > 1, A000265(a(n)) = A330718(n). - Thomas Ordowski, Mar 30 2025

cp's OEIS Frontend

A330719 a(n) = denominator(Sum_{k=1..n} (2^(k-1) - 1)/k).

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