A226113
Composite squarefree numbers n such that the ratio (n - 1/3)/(p(i) + 1/3) is an integer, where p(i) are the prime factors of n.
Original entry on oeis.org
773227, 13596427, 26567147, 140247467, 525558107, 1390082027, 1847486667, 2514565387, 3699765755, 4060724267, 4520219947, 6185512667, 6480142667, 8328046827, 9951353867, 10268992067, 11720901387, 14149448387, 14913513067, 21926400427, 22367433387, 24260249387
Offset: 1
The prime factors of 773227 are 7, 13, 29 and 293. We see that (773227 - 1/3)/(7 + 1/3) = 231968, (773227 - 1/3)/(13 + 1/3) = 57992, (773227 - 1/3)/(29 + 1/3) = 26360 and (773227 - 1/3)/(293 + 1/3) = 2636. Hence 773227 is in the sequence.
The prime factors of 1128387 are 3, 13 and 28933. We see that
(1128387 - 1/3)/(3 + 1/3) = 338516, (1128387 - 1/3)/(13 + 1/3) = 84629 but (1128387 - 1/3)/(28933 + 1/3) = 84629/2170. Hence 1128387 is not in the sequence.
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with(numtheory); A226113:=proc(i, j) local c, d, n, ok, p;
for n from 2 to i do if not isprime(n) then p:=ifactors(n)[2]; ok:=1;
for d from 1 to nops(p) do if p[d][2]>1 or not type((n-j)/(p[d][1]+j),integer) then ok:=0; break; fi; od;
if ok=1 then print(n); fi; fi; od; end: A226113(10^9,1/3);
A227974
Minimum composite squarefree numbers k such that p(i)+n divides k-n, for n=1, 2, 3, 4,..., where p(i) are the prime factors of k .
Original entry on oeis.org
385, 182, 195, 1054, 165, 6, 1015, 4958, 2193, 10, 5159, 113937, 5593, 14, 15, 196009, 3657, 6318638, 2755, 1227818, 21, 22, 2795, 152358, 12121, 26, 21827, 17578, 36569, 30, 38335, 457907, 33, 34, 35
Offset: 1
For n=3 the minimum k is 195. Prime factors of 195 are 3, 5 and 13. We have: 195 - 3 = 192, 3 + 3 = 6 and 192 / 6 = 32, 5 + 3 = 8 and 192 / 8 = 24, 13 + 3 = 16 and 192 / 16 = 12.
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with(numtheory); P:=proc(i) local c, d, k, n, ok, p; for k from 1 to i do
for n from 2 to i do if not isprime(n) then p:=ifactors(n)[2]; ok:=1;
for d from 1 to nops(p) do if p[d][2]>1 then ok:=0; break; fi;
if not type((n-k)/(p[d][1]+k), integer) then ok:=0; break; fi; od;
if ok=1 then print(n); break; fi; fi; od; od; end: P(10^6);
A382484
Least composite squarefree numbers k > n such that p + n divides k - n, for each prime p dividing k.
Original entry on oeis.org
385, 182, 195, 1054, 165, 26781, 1015, 4958, 2193, 79222, 5159, 113937, 5593, 160937, 6351, 196009, 3657, 6318638, 2755, 1227818, 12669, 41302, 2795, 152358, 12121, 366821, 21827, 17578, 36569, 12677695, 38335, 457907, 2553, 15334, 141155, 69722351, 1045, 14003, 4823, 2943805
Offset: 1
a(20) = 1227818 = 2 * 19 * 79 * 409 and
(1227818 - 20) /(2 + 20) = 55809;
(1227818 - 20) /(19 + 20) = 31482;
(1227818 - 20) /(79 + 20) = 12402;
(1227818 - 20) /(409 + 20) = 2862.
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with(numtheory): P:=proc(q) local d,k,ok,n,p;
for n from 1 to 17 do for k from n+1 to q do
if issqrfree(k) and not isprime(k) then p:=factorset(k); ok:=1;
for d from 1 to nops(p) do if frac((k-n)/(p[d]+n))>0 then ok:=0;
break; fi; od; if ok=1 then lprint(n,k); break; fi; fi; od; od; end: P(10^8);
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isok(k,n) = if (!issquarefree(k) || isprime(k), return(0)); my(f=factor(k)[,1]); for (i=1, #f, if ((k-n) % (f[i]+n), return(0));); return(1);
a(n) = my(k=n+1); while (!isok(k, n), k++); k; \\ Michel Marcus, Mar 30 2025
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