A072155
Denominator of Sum_{k=1..n} phi(k)/k.
Original entry on oeis.org
1, 2, 6, 3, 15, 5, 35, 70, 210, 210, 2310, 770, 10010, 10010, 30030, 15015, 255255, 255255, 4849845, 4849845, 4849845, 4849845, 111546435, 37182145, 37182145, 37182145, 111546435, 111546435, 3234846615, 3234846615, 100280245065, 200560490130, 200560490130
Offset: 1
1, 3/2, 13/6, 8/3, 52/15, 19/5, 163/35, 361/70, 1223/210, ...
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List([1..35], n-> DenominatorRat( Sum([1..n], k-> Phi(k)/k) ) ); # G. C. Greubel, Aug 25 2019
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[Denominator( &+[EulerPhi(k)/k: k in [1..n]] ): n in [1..35]]; // G. C. Greubel, Aug 25 2019
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with(numtheory); seq(denom( add(phi(k)/k, k=1..n)), n =1..35); # G. C. Greubel, Aug 25 2019
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Table[Sum[EulerPhi[k]/k,{k,n}],{n,40}]//Denominator (* Harvey P. Dale, Jun 08 2017 *)
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a(n) = denominator(sum(k=1, n, eulerphi(k)/k)); \\ Michel Marcus, Jan 26 2015
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[denominator( sum(euler_phi(k)/k for k in (1..n)) ) for n in (1..35)] # G. C. Greubel, Aug 25 2019
A250031
a(n) is the numerator of the density of natural numbers m such that gcd(m,floor(m/n))=1.
Original entry on oeis.org
0, 1, 1, 13, 8, 26, 19, 163, 361, 1223, 1307, 16477, 5749, 83977, 88267, 280817, 147916, 1377406, 2839897, 58552633, 60492571, 63263911, 65468386, 403117367, 549883871, 579629587, 596790577, 1864736021, 1912541636, 29293503812, 59449633388, 969992016739
Offset: 1
When n=10, the density of numbers m that are coprime to floor(m/10) turns out to be 1223/2100. Hence a(10) = 1223/2100.
When n=2, all odd numbers qualify, but only the m=2 among even numbers does; hence the density is 1/2 and therefore a(2)=1.
When n=1, only m=1 qualifies, so that the density is 0, and a(1) = 0.
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s_aux(n,p0,inp)={my(t=0/1,tt=0/1,in=inp,pp);while(1,pp=p0*prime(in);tt=n\pp;if(tt==0,break,t+=tt/pp-s_aux(n,pp,in++)));return(t)};
s(n)=1+s_aux(n,1,1);
a=vector(1000,n,numerator(1-s(n-1)/n))
A250032
a(n) is the numerator of the density of natural numbers m such that gcd(m,floor(m/n))>1.
Original entry on oeis.org
1, 1, 1, 11, 7, 19, 16, 117, 269, 877, 1003, 11243, 4261, 56163, 61883, 199663, 107339, 919889, 2009948, 38444267, 41354174, 43432679, 46078049, 266161243, 379669754, 387106183, 407127338, 1258564159, 1322304979, 19229195413, 40830611677, 634491904301, 2638247862269, 2717256540199, 2823435623209, 2886468920107, 1006725304509
Offset: 1
When n=1, S includes all natural numbers except 1, so d(1)=1. Hence a(1)=1 and A250033(1)=1.
When n=2, S includes all even numbers greater than 2, so d(2)=1/2. Hence a(2)=1 and A250033(2)=2.
When n=10, the subset S is A248500 and d(10)=877/2100. Hence a(10)=877 and A250033(10)=2100.
When n=16, S is A248502 and d(16)=199663/480480. Hence a(16)=199663 and A250033(16)=480480.
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s_aux(n,p0,inp)={my(t=0/1,tt=0/1,in=inp,pp);while(1,pp=p0*prime(in);tt=n\pp;if(tt==0,break,t+=tt/pp-s_aux(n,pp,in++)));return(t)};
s(n)=1+s_aux(n,1,1);
a=vector(1000,n,numerator(s(n-1)/n))
Showing 1-3 of 3 results.
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