|
%I #26 May 28 2015 03:38:55
%S 49,132,1070,1140,2862,40652,158170,204252,365859,656092,806526,
%T 812571,861444,1031941,4017612,4227164,8045675,15843252,16298931,
%U 48625784,81869208,129071545,142516026,219039320,266299218,520700301,537506243,590578292,600500937,915352703
%N Composite numbers equal to the sum of the prime factors, with multiplicity, of the next k numbers, for some k.
%C Values of k are 3, 4, 8, 5, 9, 9, 17, 25, 22, 18, 11, 15, 9, 20, 10, 12, 21, 26, 30, 25, 15, 14, 21, 30, 22, 26, 20, 13, 19, 11, ...
%e For 49, consider the prime factors of the next 3 numbers, 50, 51, 52: 2, 5, 5; 3, 17; 2, 2, 13. Their sum is 2 + 5 + 5 + 3 + 17 + 2 + 2 + 13 = 49.
%e For 132, consider the prime factors of the next 4 numbers, 133, 134, 135, 136: 7, 19; 2, 67; 3, 3, 3, 5; 2, 2, 2, 17. Their sum is 7 + 19 + 2 + 67 + 3 + 3 + 3 + 5 + 2 + 2 + 2 + 17 = 132.
%p with(numtheory): P:= proc(q) local a,d,j,k,n;
%p for n from 2 to q do if not isprime(n) then a:=0; k:=0;
%p while a<n do k:=k+1; d:=ifactors(n+k)[2];
%p d:=add(d[j][1]*d[j][2],j=1..nops(d));
%p a:=a+d; od; if a=n then print(n);
%p fi; fi; od; end: P(10^9);
%o (PARI) sopfr(n) = my(f=factor(n)); sum(k=1, #f~, f[k, 1]*f[k, 2]);
%o isok(n) = {my(s = 0); my(k = 1); while (s < n, s += sopfr(n+k); k++); s == n;}
%o lista(nn) = {forcomposite(n=2, nn, if (isok(n), print1(n, ", ")););} \\ _Michel Marcus_, May 27 2015
%Y Cf. A257367, A257525, A257929, A257930.
%K nonn
%O 1,1
%A _Paolo P. Lava_, Apr 28 2015
|
