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.

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A274445 a(n) is the smallest composite squarefree number k such that (p+n) | (k-1) for every prime p dividing k.

Original entry on oeis.org

385, 91, 65, 451, 33, 170171, 145, 1261, 161, 78409, 469, 294061, 649, 13051, 1921, 5251, 721, 8453501, 145, 300243, 1121, 47611, 3601, 1915801, 1057, 41311, 545, 5671, 1261, 19723133, 4321, 37759, 6913, 451, 4033, 102821, 1513, 40891, 11521, 1259497, 721, 364781, 145
Offset: 1

Views

Author

Paolo P. Lava, Jun 23 2016

Keywords

Examples

			For n=1, prime factors of 385 are 5, 7 and 11. (385 - 1)/(5 + 1) = 384/6 = 64, (385 - 1)/(7 + 1) = 384/8 = 48 and (385 - 1)/(11 + 1) = 384/12 = 32.
For n=2, prime factors of 91 are 7 and 13. (91 - 1)/(7 + 2) = 90/9 = 10 and (91 - 1)/(13 + 2) = 90/15 = 6.

Links

Crossrefs

Cf. A208728, A225702-A225720, A226020, A226111-A226114, A226364, A226448, A228299-A228302, A229273-A229276, A229321-A229324, A274443, A274444, A274446.

Programs

  • Maple
    with(numtheory); P:=proc(q) local d,k,n,ok,p;
for k from 1 to q do for n from 2 to q do
if not isprime(n) and issqrfree(n) then p:=ifactors(n)[2]; ok:=1;
for d from 1 to nops(p) do
if not type((n-1)/(p[d][1]+k),integer) then ok:=0; break; fi; od;
if ok=1 then print(n); break; fi; fi; od; od; end: P(10^9);
  • Mathematica
    t = Select[Range[10^6], SquareFreeQ@ # && CompositeQ@ # &]; Table[ SelectFirst[t, Function[k, AllTrue[First /@ FactorInteger@ k, Divisible[k - 1, # + n] &]]], {n, 17}] (* Michael De Vlieger, Jun 24 2016, Version 10 *)
  • PARI
    isok(k,n)=if (! issquarefree(k), return (0)); vp = factor(k) [,1]; if (#vp == 1, return (0)); for (i=1, #vp, if ((k-1) % (n+vp[i]), return (0));); 1;
a(n) = my(k=2); while (! isok(k,n), k++); k; \\ Michel Marcus, Jun 28 2016

Extensions

a(18), a(24), a(30) added by Giovanni Resta, Jun 23 2016
More terms from Michel Marcus, Jun 28 2016

A225703 Composite squarefree numbers n such that p(i)-3 divides n+3, where p(i) are the prime factors of n.

Original entry on oeis.org

77, 2717, 3245, 18221, 30797, 37177, 46397, 51997, 56573, 61997, 111757, 128573, 149765, 158197, 263117, 264517, 314717, 437437, 475157, 617437, 667573, 683537, 701005, 718333, 834197, 864497, 902957, 904397, 929005, 945277, 1030237, 1096205, 1139653, 1188317
Offset: 1

Views

Author

Paolo P. Lava, May 13 2013

Keywords

Examples

			Prime factors of 37177 are 7, 47 and 113. We have that (37177+3)/(7-3) = 9295, (37177+3)/(47-3) = 845 and (37177+3)/(113-3) = 338.

Links

Crossrefs

Cf. A208728, A225702, A225704-A225720.

Programs

  • Maple
    with(numtheory); A225703:=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: A225703(10^9,3);
  • Mathematica
    t = {}; n = 0; len = -2; While[len <= 262, n++; {p, e} = Transpose[FactorInteger[n]]; If[Length[p] > 1 && Union[e] == {1} && Mod[n, 3] > 0 && Union[Mod[n + 3, p - 3]] == {0}, AppendTo[t, n]; len = len + Length[IntegerDigits[n]] + 2]]; t (* T. D. Noe, May 17 2013 *)

A225704 Composite squarefree numbers n such that p(i)-4 divides n+4, where p(i) are the prime factors of n.

Original entry on oeis.org

6, 10, 14, 15, 30, 35, 66, 266, 455, 806, 4154, 4686, 6665, 10370, 16646, 22781, 31146, 36305, 72086, 205871, 246506, 473711, 570011, 653666, 733586, 900581, 904046, 1422410, 1941971, 1969565, 2023010, 2807255, 2821269, 3009821, 3043274, 3355271, 3880301
Offset: 1

Views

Author

Paolo P. Lava, May 13 2013

Keywords

Examples

			Prime factors of 205871 are 29, 31 and 229. We have that (205871+4)/(29-4) = 8235, (205871+4)/(31-4) = 7625 and (205871+4)/(229-4) = 915.

Crossrefs

Cf. A208728, A225702, A225703, A225705-A225720.

Programs

  • Maple
    with(numtheory); A225704:=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: A225704(10^9,4);
  • 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 + 4, p - 4]] == {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

A225705 Composite squarefree numbers n such that p(i)-5 divides n+5, where p(i) are the prime factors of n.

Original entry on oeis.org

21, 91, 187, 391, 3451, 4147, 6391, 7579, 8827, 9499, 9823, 11803, 15283, 21307, 22243, 26887, 29563, 36091, 42763, 49387, 62491, 63427, 84091, 89947, 107707, 116083, 126451, 139867, 155227, 227263, 270391, 287419, 302731, 317191, 320827, 376987, 381667, 433939
Offset: 1

Views

Author

Paolo P. Lava, May 13 2013

Keywords

Examples

			Prime factors of 6391 are 7, 11 and 83. We have that (6391+5)/(7-5)  =3198, (6391+5)/(11-5) = 1066 and (6391+5)/(83-5) = 82.

Links

Crossrefs

Cf. A208728, A225702-A225704, A225706-A225720.

Programs

  • Maple
    with(numtheory); A225705:=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: A225705(10^9,5);
  • Mathematica
    t = {}; n = 0; While[Length[t] < 50, n++; {p, e} = Transpose[FactorInteger[n]]; If[Length[p] > 1 && Union[e] == {1} && Mod[n, 5] > 0 && Union[Mod[n + 5, p - 5]] == {0}, AppendTo[t, n]]]; t (* T. D. Noe, May 17 2013 *)

A225706 Composite squarefree numbers n such that p(i)-6 divides n+6, where p(i) are the prime factors of n.

Original entry on oeis.org

6, 10, 14, 15, 21, 30, 35, 42, 70, 78, 105, 154, 170, 210, 357, 759, 1110, 6195, 42465, 43554, 61755, 94605, 106386, 146910, 189399, 229119, 276914, 453590, 924099, 1239870, 2407119, 3915714, 4404394, 4524074, 5819145, 7396394, 8324869, 23701854, 30242654, 33413919
Offset: 1

Views

Author

Paolo P. Lava, May 13 2013

Keywords

Examples

			Prime factors of 8324869 are 7, 19, 53 and 1181. We have that (8324869+6)/(7-6) = 8324875, (8324869+6)/(19-6) = 640375, (8324869+6)/(53-6) = 177125 and (8324869+6)/(1181-6) = 7085.

Crossrefs

Cf. A208728, A225702-A225705, A225707-A225720.

Programs

  • Maple
    with(numtheory); A225706:=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: A225706(10^9,6);
  • Mathematica
    t = {}; n = 0; While[Length[t] < 50, n++; {p, e} = Transpose[FactorInteger[n]]; If[Length[p] > 1 && Union[e] == {1} && Union[Mod[n + 6, p - 6]] == {0}, AppendTo[t, n]]]; t (* T. D. Noe, May 17 2013 *)

Extensions

Extended by T. D. Noe, May 17 2013

A225707 Composite squarefree numbers n such that p(i)-7 divides n+7, where p(i) are the prime factors of n.

Original entry on oeis.org

33, 65, 165, 209, 345, 713, 1353, 2717, 2945, 4433, 4745, 6149, 7733, 9785, 11297, 16985, 21593, 25265, 26273, 28545, 32357, 35673, 47945, 49913, 55913, 61013, 69113, 69513, 88913, 95465, 106913, 116513, 119009, 121785, 133433, 159185, 167765, 201773
Offset: 1

Views

Author

Paolo P. Lava, May 13 2013

Keywords

Examples

			Prime factors of 7733 are 11, 19 and 37. We have that (7733+7)/(11-7) = 1935, (7733+7)/(19-7) = 645 and (7733+7)/(37-7) = 258.

Links

Crossrefs

Cf. A208728, A225702-A225706, A225708-A225720.

Programs

  • Maple
    with(numtheory); A225707:=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: A225707(10^9,7);
  • Mathematica
    t = {}; n = 0; While[Length[t] < 50, n++; {p, e} = Transpose[FactorInteger[n]]; If[Length[p] > 1 && Union[e] == {1} && Mod[n, 7] > 0 && Union[Mod[n + 7, p - 7]] == {0}, AppendTo[t, n]]]; t (* T. D. Noe, May 17 2013 *)

A225708 Composite squarefree numbers n such that p(i)-8 divides n+8, where p(i) are the prime factors of n.

Original entry on oeis.org

10, 22, 55, 70, 154, 190, 322, 385, 442, 595, 682, 2002, 2737, 3619, 5530, 14986, 23782, 24817, 25102, 26767, 30430, 31042, 34762, 37810, 85462, 106582, 141427, 171790, 189727, 225910, 243217, 248482, 255142, 272782, 307090, 381547, 388102, 471262, 637849, 798490
Offset: 1

Views

Author

Paolo P. Lava, May 13 2013

Keywords

Examples

			Prime factors of 381547 are 23, 53 and 313. We have that (381547+8)/(23-8)=25437, (381547+8)/(53-8)=8479 and (381547+8)/(313-8)=1251.

Crossrefs

Cf. A208728, A225702-A225707, A225709-A225720.

Programs

  • Maple
    with(numtheory); A225708:=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: A225708(10^9,8);
  • Mathematica
    t = {}; n = 0; While[Length[t] < 40, n++; {p, e} = Transpose[FactorInteger[n]]; If[Length[p] > 1 && Union[e] == {1} && Union[Mod[n + 8, p - 8]] == {0}, AppendTo[t, n]]]; 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

Views

Author

Paolo P. Lava, May 13 2013

Keywords

Examples

			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.

Crossrefs

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

Programs

  • Maple
    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);
  • Mathematica
    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 *)

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 *)

A225713 Composite squarefree numbers n such that p(i)+3 divides n-3, where p(i) are the prime factors of n.

Original entry on oeis.org

195, 1235, 1443, 2915, 4403, 5883, 35203, 37635, 54723, 66563, 77503, 97555, 157403, 158403, 188355, 200203, 265411, 273003, 299715, 317203, 358179, 376995, 380373, 438243, 476003, 492803, 506883, 511683, 567633, 630203, 636803, 654951, 742269, 764463, 827203
Offset: 1

Views

Author

Paolo P. Lava, May 13 2013

Keywords

Examples

			Prime factors of 5883 are 3, 37 and 53. We have that (3+3)/(5883-3) = 980, (37+3)/(5883-3) = 147 and (53+3)/(5883-3) = 105.

Links

Crossrefs

Cf. A208728, A225702-A225712, A225714-A225720.

Programs

  • Maple
    with(numtheory); A225713:=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: A225713(10^9,-3);
  • 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 - 3, p + 3]] == {0}, AppendTo[t, n]; len = len + Length[IntegerDigits[n]] + 2]]; t (* T. D. Noe, May 17 2013 *)
Previous Showing 21-30 of 46 results. Next

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