A050781
Numbers k such that sopfr(k) = sopfr(k - sopfr(k)).
Original entry on oeis.org
55, 72, 84, 119, 143, 256, 2106, 2211, 2279, 3024, 3150, 3551, 4284, 6360, 6500, 9350, 10200, 10285, 10919, 13560, 14279, 14351, 15606, 16463, 17063, 23595, 25011, 27208, 28208, 28600, 31460, 33096, 42180, 44330, 52320, 53053, 53824
Offset: 0
sopfr(72) = 2+2+2+3+3 = 12 = 2+2+3+5 = sopfr(72 - sopfr(72)), so 72 is in the sequence.
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sopf[n_]:=Module[{sopfn=Total[Times@@@FactorInteger[n]],m},m=n+sopfn;If[n==m-Total[Times@@@FactorInteger[m]],m,0]]; DeleteCases[Table[ sopf[n],{n,55000}],?(#==0&)] (* _Harvey P. Dale, Jun 15 2011 *)
A279935
Numbers n such that n + sopf(n) + rad(n) = m and m - sopf(m) - rad(m) = n, where sopf(n) is the sum of the distinct primes dividing n and rad(n) is the squarefree kernel of n.
Original entry on oeis.org
3, 4, 75, 112, 2057, 9178, 29818, 73813, 138992, 240469, 531002, 661489, 716856, 763648, 905474, 1033909, 1395554, 1572001, 1605519, 1643372, 1661030, 1692277, 1705724, 2312593, 2864773, 2911839, 2928193, 2977676, 3114366, 3744951, 4035647, 4122178, 4227036, 5716177
Offset: 1
Prime factors of 9178 are 2, 13, 353:
sopf(9178) = 2 + 13 + 353 = 368, rad(9178) = 2 * 13 * 353 = 9178 and 9178 + 368 + 9178 = 18724.
Prime factors of 18724 are 2, 2, 31, 151:
sopf(18724) = 2 + 31 + 151 = 184, rad(18724) = 2 * 31 * 151 = 9362 and 18724 - 184 - 9362 = 9178.
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with(numtheory): P:=proc(q) local a,b,c,k,n; for n from 1 to q do
a:=ifactors(n)[2]; b:=mul(a[k][1],k=1..nops(a))+add(a[k][1],k=1..nops(a));
c:=n+b; a:=ifactors(c)[2]; b:=mul(a[k][1],k=1..nops(a))+add(a[k][1],k=1..nops(a));
d:=c-b; if d=n then print(n); fi; od; end: P(10^9);
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f[n_] := Block[{pd = First@# & /@ FactorInteger@n}, Times @@ pd + Plus @@ pd]; fQ[n_] := n + f[n] - f[n + f[n]] == n; Select[ Range@ 1000000, fQ] (* Robert G. Wilson v, Dec 24 2016 *)
A057874
Sets of three composites in bidirectional 'sum of prime factors' progression/retrogression.
Original entry on oeis.org
95, 119, 143, 174191, 175031, 175871, 298687992, 298688708, 298689424
Offset: 0
First set is (95,119,143) all terms having 'sum of prime factors' = 24. So '95' + (5+19) = '119' + (7+17) = '143' AND '143' - (11+13) = '119' - (7+17) = '95'.
This really seems to be three sequences, not one! Should be split. -
N. J. A. Sloane
Showing 1-3 of 3 results.
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