A064605
Numbers k such that A064602(k) is divisible by k.
Original entry on oeis.org
1, 2, 8, 74, 146, 150, 158, 307, 526, 541, 16157, 20289, 271343, 953614, 1002122, 2233204, 3015123, 15988923, 48033767, 85110518238
Offset: 1
Summing divisor-square sums for j = 1..8 gives 1+5+10+21+26+50+50+85 = 248, which is divisible by 8, so 8 is a term and the integer quotient is 31.
Cf.
A001157,
A064602,
A050226,
A056650,
A064606,
A064607,
A064610,
A064611,
A064612,
A048290,
A062982,
A045345.
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k = 1; lst = {}; s = 0; While[k < 1000000001, s = s + DivisorSigma[2, k]; If[ Mod[s, k] == 0, AppendTo[lst, k]; Print@ k]; k++]; lst (* Robert G. Wilson v, Apr 25 2011 *)
A064610
Places k where A064608(k) (partial sums of unitary tau) is divisible by k.
Original entry on oeis.org
1, 35, 37, 1015, 27417, 27421, 27449, 27453, 19774739, 530743781, 530743799, 530743807, 530743813
Offset: 1
For n = 37, the sum A064608(37) = 1+2+2+2+2+4+2+...+4+4+4+2 = 111 = 3*37, so 37 is in the sequence.
Analogous "integer-mean" sequences for various arithmetical functions are
A050226,
A056650,
A064605,
A064606,
A064607,
A048290,
A063986,
A063971,
A064911,
A062982,
A045345.
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s[1] = 1; s[n_] := s[n] = s[n - 1] + 2^PrimeNu[n]; Select[Range[30000], Divisible[s[#], #] &] (* Amiram Eldar, Jun 04 2021 *)
A064607
Numbers k such that A064604(k) is divisible by k.
Original entry on oeis.org
1, 2, 7, 151, 257, 1823, 3048, 5588, 6875, 7201, 8973, 24099, 5249801, 9177919, 18926164, 70079434, 78647747, 705686794, 2530414370, 3557744074, 25364328389, 32487653727, 66843959963
Offset: 1
Adding 4th-power divisor-sums for j = 1..7 gives 1+17+82+273+626+1394+2402 = 4795 which is divisible by 7, so 7 is a term and the integer quotient is 655.
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k = 1; lst = {}; s = 0; While[k < 1000000001, s = s + DivisorSigma[4, k]; If[ Mod[s, k] == 0, AppendTo[lst, k]; Print@ k]; k++]; lst (* Robert G.Wilson v, Aug 25 2011 *)
A031439
a(0) = 1, a(n) is the greatest prime factor of a(n-1)^2+1 for n > 0.
Original entry on oeis.org
1, 2, 5, 13, 17, 29, 421, 401, 53, 281, 3037, 70949, 1713329, 1467748131121, 37142837524296348426149, 101591133424866642486477019709, 1650979973845742266714536305651329, 78343914631785958284737, 4029445531112797145738746391569, 350080544438648120162733678636001, 26208090024628793745288451837610346882122253572537, 4717815978577117335515270825550279551117660519482308365269206484133871485221
Offset: 0
a(16)=A006530(a(15)^2+1)=
A006530(101591133424866642486477019709^2+1)=
A006530(10320758390549056348725939119133160378521185060950774444682)=
A006530(2*29*23201*4645528280970018601*1650979973845742266714536305651329)=
1650979973845742266714536305651329, factorization of A006530(a(15)^2+1) by Dario A. Alpern's program (see link).
Cf.
A002144 - Pythagorean primes: primes of form 4n+1;
A002313 - Primes congruent to 1 or 2 modulo 4;
A014442 - Largest prime factor of n^2 + 1.
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gpf[n_] := FactorInteger[n][[-1, 1]]; a[0] = 1; a[n_] := a[n] = gpf[a[n - 1]^2 + 1]; Table[an = a[n]; Print[an]; an, {n, 0, 21}] (* Jean-François Alcover, Nov 04 2011 *)
NestList[FactorInteger[#^2+1][[-1,1]]&,1,21] (* Harvey P. Dale, Jul 04 2013 *)
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gpf(n)=local(pf);pf=factor(n);pf[matsize(pf)[1],1] vector(20,i,r=if(i==1,1,gpf(r^2+1)))
a(17)-a(21) from Richard FitzHugh (fitzhughrichard(AT)hotmail.com), Aug 12 2004
A064612
Partial sum of bigomega is divisible by n, where bigomega(n)=A001222(n) and summatory-bigomega(n)=A022559(n).
Original entry on oeis.org
1, 4, 5, 2178, 416417176, 416417184, 416417185, 416417186, 416417194, 416417204, 416417206, 416417208, 416417213, 416417214, 416417231, 416417271, 416417318, 416417319, 416417326, 416417335, 416417336, 416417338, 416417339, 416417374
Offset: 1
Sum of bigomega values from 1 to 5 is: 0+0+1+1+2+1=5, which is divisible by n=5, so 5 is here, with quotient=1. For the last value,2178,below 1000000 the quotient is only 3.
A064606
Numbers k such that A064603(k) is divisible by k.
Original entry on oeis.org
1, 2, 7, 45, 184, 210, 267, 732, 1282, 3487, 98374, 137620, 159597, 645174, 3949726, 7867343, 13215333, 14153570, 14262845, 317186286, 337222295, 2788845412, 10937683400, 72836157215, 95250594634
Offset: 1
Adding divisor-cube sums for j = 1..7 gives 1+9+28+73+126+252+344 = 833 = 7*119, which is divisible by 7, so 7 is a term and the integer quotient is 119.
Showing 1-6 of 6 results.
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