cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A225710 Composite squarefree numbers n such that p(i)-10 divides n+10, where p(i) are the prime factors of n.

Original entry on oeis.org

14, 22, 35, 55, 65, 77, 102, 110, 143, 165, 182, 221, 455, 494, 665, 935, 1001, 1173, 1430, 2717, 2795, 4505, 4526, 4862, 5957, 6479, 11526, 27521, 30485, 34661, 35126, 45917, 49715, 52910, 53846, 81686, 90574, 106865, 113477, 118745, 139073, 140822, 147095
Offset: 1

Views

Author

Paolo P. Lava, May 13 2013

Keywords

Examples

			Prime factors of 34661 are 11, 23 and 137. We have that (34661+10)/(11-10) = 34671, (34661+10)/(23-10) = 2667 and (34661+10)/(137-10) = 273.

Crossrefs

Cf. A208728, A225702-A225709, A225711-A225720.

Programs

  • Maple
    with(numtheory); A225710:=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: A225710(10^9,10);
  • Mathematica
    t = {}; n = 0; len = -2; While[len <= 262, n++; {p, e} = Transpose[FactorInteger[n]]; If[Length[p] > 1 && Union[e] == {1} && Union[Mod[n + 10, p - 10]] == {0}, AppendTo[t, n]; len = len + Length[IntegerDigits[n]] + 2]]; t (* T. D. Noe, May 17 2013 *)

Extensions

Extended by T. D. Noe, May 17 2013

A225712 Composite squarefree numbers n such that p(i)+2 divides n-2, where p(i) are the prime factors of n.

Original entry on oeis.org

182, 21827, 32942, 46055, 84502, 151202, 191522, 361802, 532247, 780626, 1368642, 1398377, 1425230, 1556258, 1751927, 1932338, 2209727, 3496502, 4078802, 4216862, 4438709, 5191562, 5991477, 7413002, 8385365, 8797502, 11749127, 13634138, 15921677, 16772177
Offset: 1

Views

Author

Paolo P. Lava, May 13 2013

Keywords

Examples

			Prime factors of 151202 are 2, 19, 23 and 173. We have that (151202-2)/(2+2) = 37800, (151202-2)/(19+2) = 7200, (151202-2)/(23+2) = 6048 and (151202-2)/(173+2)= 864.

Crossrefs

Cf. A208728, A225702-A225711, A225713-A225720.

Programs

  • Maple
    with(numtheory); A225712:=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: A225712(10^9,-2);
  • Mathematica
    t = {}; n = 0; len = -2; While[len <= 262, n++; {p, e} = Transpose[FactorInteger[n]]; If[Length[p] > 1 && Union[e] == {1} && Union[Mod[n - 2, p + 2]] == {0}, AppendTo[t, n]; len = len + Length[IntegerDigits[n]] + 2]]; t (* T. D. Noe, May 17 2013 *)

A306685 Composite squarefree numbers k such that k^2-1 is divisible by p-1 and p+1, where p are all the prime factors of k.

Original entry on oeis.org

35, 1189, 3059, 6479, 8569, 30889, 39689, 51271, 84419, 133399, 272251, 321265, 430199, 544159, 564719, 569449, 585311, 608399, 1033241, 1212751, 1930499, 3131029, 7056721, 7110179, 7639919, 8740601, 11255201, 15857855, 17966519, 18996769, 22427999, 32871761, 34966009
Offset: 1

Views

Author

Paolo P. Lava, Mar 05 2019

Keywords

Examples

			Prime factors of 35 are 5, 7 and 35^2-1 = 1224, 1124/4 = 306, 1124/6 = 204, 1124/8 = 153.
Prime factors of 1189 are 29, 41 and 1189^2-1 = 1413720, 1413720/28 = 50490, 1413720/30 = 47124, 1413720/40 = 35343, 1413720/42 = 33660.

Links

Crossrefs

Cf. A002997, A006972, A208728, A225711, A304291, A306723.

Programs

  • Maple
    with(numtheory): P:=proc(q) local a,k,ok,n; for n from 2 to q do
if not isprime(n) and issqrfree(n) then a:=factorset(n); ok:=1;
for k from 1 to nops(a) do if frac((n^2-1)/(a[k]+1))>0 or frac((n^2-1)/(a[k]-1))>0 then ok:=0; break; fi; od; if ok=1 then print(n); fi; fi; od; end: P(10^9);
  • Mathematica
    csfQ[n_]:=CompositeQ[n]&&SquareFreeQ[n]&&Union[Mod[n^2-1,Flatten[{#+1, #-1}&/@ FactorInteger[n][[All,1]]]]]=={0}; Select[Range[35*10^6],csfQ] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Apr 07 2020 *)
  • PARI
    isok(n) = {if (issquarefree(n) && !isprime(n) && (n>1), my(f = factor(n)[,1], x = n^2-1); for (k=1, #f, if ((x % (f[k]-1)) || (x % (f[k]+1)), return (0));); return (1);); return (0);} \\ Michel Marcus, Mar 12 2019

Extensions

More terms from Giovanni Resta, Mar 06 2019

A306723 Composite squarefree numbers k such that k^2+1 is divisible by p-1, where p are all the prime factors of k.

Original entry on oeis.org

33, 36003, 426747, 220067817
Offset: 1

Views

Author

Paolo P. Lava, Mar 06 2019

Keywords

Comments

Tested up to 3*10^10. - Giovanni Resta, Mar 06 2019

Examples

			Prime factors of 33 are 3, 11 and 33^2+1 = 1090, 1090/2 = 545, 1090/10 = 109.
Prime factors of 220067817 are 3, 59, 131, 9491 and 220067817^2+1 = 48429844079145490, 48429844079145490/2 = 24214922039572745, 48429844079145490/58 = 834997311709405, 48429844079145490/130 = 372537262147273, 48429844079145490/9490 = 5103250166401.

Crossrefs

Cf. A002997, A006972, A208728, A225711, A304291, A306685.

Programs

  • Maple
    with(numtheory): P:=proc(q) local a,k,ok,n;
for n from 1 to q do if not isprime(n) and issqrfree(n) then a:=factorset(n); ok:=1; for k from 1 to nops(a) do if frac((n^2+1)/(a[k]+1))>0 then ok:=0; break; fi; od; if ok=1 then print(n); fi; fi; od; end: P(10^20);
  • PARI
    isok(n) = {if (issquarefree(n) && !isprime(n) && (n>1), my(f = factor(n)[,1], x = n^2+1); for (k=1, #f, if ((x % (f[k]-1)), return (0));); return (1);); return (0);} \\ Michel Marcus, Mar 12 2019

Extensions

a(4) from Giovanni Resta, Mar 06 2019

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

Views

Author

Paolo P. Lava, Mar 29 2025

Keywords

Examples

			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.

Crossrefs

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

Programs

  • Maple
    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);
  • PARI
    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

Extensions

More terms from Michel Marcus, Mar 30 2025
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.