A216868
Nicolas's sequence whose positivity is equivalent to the Riemann hypothesis.
Original entry on oeis.org
3, 4, 13, 67, 560, 6095, 87693, 1491707, 30942952, 795721368, 22614834943, 759296069174, 28510284114397, 1148788714239052, 50932190960133487, 2532582753383324327, 139681393339880282191, 8089483267352888074399, 512986500081861276401709, 34658318003703434434962860
Offset: 1
prime(2)# = 2*3 = 6 and phi(6) = 2, so a(2) = 6 - floor(2*log(log(6))*e^gamma) = 6 - floor(2*0.58319...*1.78107...) = 6 - floor(2.07...) = 6 - 2 = 4.
- J.-L. Nicolas, Petites valeurs de la fonction d'Euler et hypothèse de Riemann, in Seminar on Number Theory, Paris 1981-82 (Paris 1981/1982), Birkhäuser, Boston, 1983, pp. 207-218.
- Amiram Eldar, Table of n, a(n) for n = 1..350
- J.-L. Nicolas, Petites valeurs de la fonction d'Euler, J. Number Theory, Vol. 17, No.3 (1983), pp. 375-388.
- J.-L. Nicolas, Small values of the Euler function and the Riemann hypothesis, arXiv:1202.0729 [math.NT], 2012; Acta Arith., Vol. 155 (2012), pp. 311-321.
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primorial[n_] := Product[Prime[k], {k, n}]; Table[With[{p = primorial[n]}, p - Floor[EulerPhi[p]*Log[Log[p]]*Exp[EulerGamma]]], {n, 1, 20}]
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nicolas(n) = {p = 2; pri = 2;for (i=1, n, print1(pri - floor(eulerphi(pri)*log(log(pri))*exp(Euler)), ", ");p = nextprime(p+1);pri *= p;);} \\ Michel Marcus, Oct 06 2012
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A216868(n)={(n=prod(i=1,n,prime(i)))-floor(eulerphi(n)*log(log(n))*exp(Euler))} \\ M. F. Hasler, Oct 06 2012
A209079
Integer part of sigma(m)*phi(m)/m for colossally abundant numbers m.
Original entry on oeis.org
1, 4, 9, 44, 96, 312, 2139, 4421, 48234, 623336, 1266781, 3897787, 20138571, 341171088, 6464294306, 148397712765, 299150944780, 8665061848812, 268337399189042, 1911903969221925, 5783509506896323, 213833540687410017
Offset: 1
1 = [3*1/2]
4 = [12*2/6]
9 = [28*4/12]
44 = [168*16/60]
96 = [360*32/120]
312 = [1170*96/360]
2139 = [9360*576/2520]
4421 = [19344*1152/5040]
48234 = [232128*11520/55440]
623336 = [3249792*138240/720720]
1266781 = [6604416*276480/1441440]
3897787 = [20321280*829440/4324320]
20138571 = [104993280*4147200/21621600]
341171088 = [1889879040*66355200/367567200]
6464294306 = [37797580800*1194393600/6983776800]
148397712765 = [907141939200*26276659200/160626866400]
299150944780 = [1828682956800*52553318400/321253732800]
8665061848812 = [54860488704000*1471492915200/9316358251200]
268337399189042 = [1755535638528000*44144787456000/288807105787200]
1911903969221925 = [12508191424512000*309013512192000/2021649740510400]
5783509506896323 = [37837279059148800*927040536576000/6064949221531200]
- G. H. Hardy and E. M. Wright, An introduction to the theory of numbers, 6th edition, Oxford University Press (2008), 350-353.
- G. Robin, Grandes valeurs de la fonction somme des diviseurs et hypothèse de Riemann. J. Math. Pures Appl. 63 (1984), 187-213.
- Amiram Eldar, Table of n, a(n) for n = 1..382
- L. Alaoglu and P. Erdos, On highly composite and similar numbers, Trans. Amer. Math. Soc., 56 (1944), 448-469. Errata
- Keith Briggs, Abundant numbers and the Riemann Hypothesis, Experimental Math., Vol. 16 (2006), p. 251-256.
- J.-L. Nicolas, Petites valeurs de la fonction d'Euler, J. Number Theory 17, no.3 (1983), 375-388.
- S. Ramanujan, Highly composite numbers, Annotated and with a foreword by J.-L. Nicolas and G. Robin, Ramanujan J., 1 (1997), 119-153.
A216920
m such that the integer part of sigma(m)/phi(m) is not attained by any integer less than m.
Original entry on oeis.org
1, 2, 3, 6, 10, 12, 20, 30, 42, 60, 120, 210, 420, 630, 840, 2520, 9240, 10080, 27720, 55440, 120120, 360360, 720720, 2162160, 6126120, 12252240, 36756720, 116396280, 232792560, 698377680, 2677114440, 5354228880, 26771144400, 155272637520, 465817912560
Offset: 1
a(22) = 360360 is in this list because sigma(360360)/phi(360360) = 22.75 and floor(sigma(k)/phi(k)) != 22 for all k < 360360.
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A216920_list := proc(searchlimit)
local p, q, P, R; with(numtheory):
P := {}; R := NULL; p := 1;
while p < searchlimit do
q := iquo(sigma(p), phi(p));
if not member(q, P) then
P := {q} union P; R := R,p fi;
p := p+1 od:
R end:
A216920_list(1000);
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def A216920_list(searchlimit):
P = {}
for p in (1..searchlimit):
q = sigma(p)//euler_phi(p)
if q not in P: P[q] = p
return sorted(P.values())
A216920_list(1000)
Showing 1-3 of 3 results.
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