A225710 -id:A225710 - cp's OEIS frontend

A225711 Composite squarefree numbers n such that p(i)+1 divides n-1, where p(i) are the prime factors of n.

385, 2737, 6061, 6721, 17641, 24769, 25201, 31521, 34561, 49105, 66385, 76609, 79401, 113221, 136081, 180481, 194833, 199801, 254881, 268801, 311905, 321937, 328321, 362881, 436645, 469201, 506521, 545905, 547561, 558145, 628705, 642505, 649153, 778261, 884305

Offset: 1

Paolo P. Lava, May 13 2013

nonn

			Prime factors of 24769 are 17, 31 and 47. We have that (24769-1)/(17+1) = 1376, (24769-1)/(31+1) = 774 and (24769-1)/(47+1) = 516.

Paolo P. Lava, Table of n, a(n) for n = 1..150
Qi-Yang Zheng, There are infinitely many (-1,1)-Carmichael numbers, arXiv:2207.08641 [math.NT], 2022.

Cf. A208728, A225702-A225710, A225712-A225720.

with(numtheory); A225711:=proc(i,j) local c, d, n, ok, p, t;
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 p[d][1]=j then ok:=0; break; fi;
if  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: A225711(10^9,-1);

t = {}; n = 0; len = -2; While[len <= 262, n++; {p, e} = Transpose[FactorInteger[n]]; If[Length[p] > 1 && Union[e] == {1} && Union[Mod[n - 1, p + 1]] == {0}, AppendTo[t, n]; len = len + Length[IntegerDigits[n]] + 2]]; t (* T. D. Noe, May 17 2013 *)

A225709 Composite squarefree numbers n such that p(i)-9 divides n+9, where p(i) are the prime factors of n.

Original entry on oeis.org

15, 21, 33, 35, 39, 55, 77, 91, 119, 143, 195, 231, 255, 299, 455, 551, 651, 663, 715, 935, 1131, 1155, 1419, 2015, 2035, 2431, 3003, 3111, 3927, 4611, 5451, 7215, 7735, 8151, 8671, 9191, 10455, 11571, 15015, 15477, 16511, 18343, 18615, 23541, 24871, 25415, 28391

Offset: 1

Paolo P. Lava, May 13 2013

nonn

			Prime factors of 16511 are 11, 19 and 79. We have that (16511+9)/(11-9) = 8260, (16511+9)/(19-9) = 1652 and (16511+9)/(79-9) = 236.

Cf. A208728, A225702-A225708, A225710-A225720.

with(numtheory); A225709:=proc(i,j) local c, d, n, ok, p, t;
for n from 1 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 p[d][1]=j then ok:=0; break; fi;
if  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: A225709(10^9,9);

t = {}; n = 0; While[Length[t] < 50, n++; {p, e} = Transpose[FactorInteger[n]]; If[Length[p] > 1 && Union[e] == {1} && Union[Mod[n + 9, p - 9]] == {0}, AppendTo[t, n]]]; t (* T. D. Noe, May 17 2013 *)

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

Paolo P. Lava, Mar 29 2025

nonn

			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.

Cf. A208728, A225702-A225710, A225711-A225720.

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);

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

More terms from Michel Marcus, Mar 30 2025

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A225711 Composite squarefree numbers n such that p(i)+1 divides n-1, where p(i) are the prime factors of n.

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A225709 Composite squarefree numbers n such that p(i)-9 divides n+9, where p(i) are the prime factors of n.

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A382484 Least composite squarefree numbers k > n such that p + n divides k - n, for each prime p dividing k.

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