[math-fun] Patchy stretches of Primes
the denominator of the sum of the first n Harmonic numbers, [[see PS below]] equals LCM(1,2,..,n) for values of n of the form n=(p-1) , p prime. Not all primes qualify for the converse statement. Those that do (listed below by prime_index PrimePi(p) 1,2,3; 5,6,7; 10,11; 16; 24,25; 31,32,33,34,35,36; ... or as {start index of patch, length of patch} {{1, 3}, {5, 3}, {10, 2}, {16, 1}, {24, 2}, {31, 6}, {54, 5}, {62, 1}, {164, 16}, {220, 12}, {358, 10}, {446, 31}, {488, 24}, {520, 31}, {887, 10}, {910, 51}, {986, 5}, {1001, 7},... Who can explain this Patchy behaviour? PS. [[ (-n+(1+n)HarmonicNumber[n]) or -n +(1+n)(EulerGamma+PolyGamma[0,n+1]) since Mma burps on HarmonicNumber[k] with k>1000. ]] Wouter. =============================== This email is confidential and intended solely for the use of the individual to whom it is addressed. If you are not the intended recipient, be advised that you have received this email in error and that any use, dissemination, forwarding, printing, or copying of this email is strictly prohibited. You are explicitly requested to notify the sender of this email that the intended recipient was not reached.
Wouter, et al:
the denominator of the sum of the first n Harmonic numbers, equals LCM(1,2,..,n) for values of n of the form n=(p-1) , p prime.
Not all primes qualify for the converse statement.
Wouter then asks, why do the qualifying and non-qualifying primes come in runs. --- Let's call the N'th harmonic number H(n), and its reduced denominator D(n). Sometimes, D(n) is less than D(n-1). It happens when N is a multiple of some prime P, and the addition of 1/n makes the numerator a multiple of P: The P's cancel. Generally, the next multiple of P restores P to the denominator. Between those multiples of P, D(n) cannot equal LCM(1..n). As an example, let's start with D(31). LCM(1..31) = D(31) = 72201776446800. H(32) introduces a new power of two in both cases, and LCM(1..32) = D(32). H(33) makes the numerator a multiple of 11, and so the factor 11 vanishes in D(33). It is not restored until H(44). In particular, D(37), D(41), and D(43) have no factor 11, and are unequal to LCM(1..37), LCM(1..41), and LCM(1..43). Meanwhile, the factor 7 was lost at H(42), and D(47) != LCM(1..47). But then H(49) restores 7, and D(53) = LCM(1..53). -- Don Reble djr@nk.ca
participants (2)
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Don Reble -
Meeussen Wouter (bkarnd)