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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A225714 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

1054, 9541, 91039, 371074, 985054, 1043959, 1063003, 1107754, 1162498, 1357339, 1786054, 4018018, 5368549, 5820154, 8725747, 9994954, 12402709, 17138503, 17914054, 20855839, 23116009, 25077199, 26545054, 29247229, 30308359, 31424419, 33892759, 44141629
Offset: 1

Views

Author

Paolo P. Lava, May 13 2013

Keywords

Examples

			Prime factors of 1043959 are 7, 293 and 509. We have that (1043959-4)/(7+4) = 94905, (1043959-4)/(293+4) = 3515 and (1043959-4)/(509+4) = 2035.

Crossrefs

Cf. A208728, A225702-A225713, A225715-A225720.

Programs

  • Maple
    with(numtheory); A225714:=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: A225714(10^9,-4);

Extensions

a(20)-a(28) from Donovan Johnson, Nov 15 2013

A225715 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

165, 1085, 3965, 4085, 5621, 7733, 8645, 14405, 19877, 23405, 33269, 40397, 45365, 66929, 88949, 110885, 114917, 135005, 243941, 275621, 280085, 421085, 439565, 455285, 460229, 474677, 480245, 496589, 505517, 518081, 570245, 706805, 709973, 900581, 912021
Offset: 1

Views

Author

Paolo P. Lava, May 13 2013

Keywords

Examples

			Prime factors of 7733 are 11, 19 and 37. We have that (7733-5)/(11+5) = 483, (7733-5)/(19+5) = 322 and (7733-5)/(37+5) = 184.

Links

Crossrefs

Cf. A208728, A225702-A225714, A225716-A225720.

Programs

  • Maple
    with(numtheory); A225715:=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: A225715(10^9,-5);
  • 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 - 5, p + 5]] == {0}, AppendTo[t, n]; len = len + Length[IntegerDigits[n]] + 2]]; t (* T. D. Noe, May 17 2013 *)

A225716 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, 26781, 120791, 5099531, 5720435, 14637451, 24110358, 31552261, 33792198, 57804181, 71925054, 88324781, 92849126, 441031331, 650715071, 924029951, 1425902869, 2093676486, 2336689491, 3273172441, 4533042611, 4711366831, 5162021871, 5502040431, 6427899582
Offset: 1

Views

Author

Paolo P. Lava, May 13 2013

Keywords

Examples

			Prime factors of 14637451 are 41, 229 and 1559. We have that (14637451-6)/(41+6) = 311435, (14637451-6)/(229+6) = 62287 and (14637451-6)/(1559+6) = 9353.

Crossrefs

Cf. A208728, A225702-A225715, A225717-A225720.

Programs

  • Maple
    with(numtheory); A225716:=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: A225716(10^9,-6);

Extensions

a(14)-a(25) from Donovan Johnson, Nov 15 2013

A225717 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

1015, 4147, 7567, 9367, 13447, 15847, 25543, 29127, 33847, 39319, 40807, 58327, 80647, 87607, 116071, 135439, 139867, 145915, 177415, 186667, 190747, 203287, 222343, 253897, 321127, 356167, 380887, 384391, 391207, 403495, 453607, 470587, 501607, 602167, 606535
Offset: 1

Views

Author

Paolo P. Lava, May 13 2013

Keywords

Examples

			Prime factors of 15847 are 13, 23 and 53. We have that (15847-7)/(13+7) = 792, (15847-7)/(23+7) = 528 and (15847-7)/(53+7) = 264.

Links

Crossrefs

Cf. A208728, A225702-A225716, A225718-A225720.

Programs

  • Maple
    with(numtheory); A225717:=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: A225717(10^9,-7);
  • 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 - 7, p + 7]] == {0}, AppendTo[t, n]; len = len + Length[IntegerDigits[n]] + 2]]; t (* T. D. Noe, May 17 2013 *)

A225718 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

4958, 51653, 55583, 271358, 291338, 789173, 1379438, 5430797, 5785073, 6350885, 7159958, 10532333, 12822818, 13892243, 14809517, 23831423, 24547058, 26734058, 27391073, 32079671, 32673383, 36126098, 42560693, 51346358, 52177658, 54949958
Offset: 1

Views

Author

Paolo P. Lava, May 13 2013

Keywords

Examples

			Prime factors of 789173 are 7, 11, 37 and 277. We have that (789173-8)/(7+8) = 52611, (789173-8)/(11+8) = 41535, (789173-8)/(37+8) = 17537 and (789173-8)/(277+8) = 2769.

Crossrefs

Cf. A208728, A225702-A225717, A225719, A225720.

Programs

  • Maple
    with(numtheory); A225718:=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: A225718(10^9,-8);

A225719 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

2193, 3705, 9889, 12749, 23529, 116265, 283929, 514569, 641769, 661789, 772809, 950609, 1144313, 1241561, 1452009, 1708233, 1797609, 1878569, 2244901, 2255689, 2675409, 2792937, 3426089, 4021369, 4187465, 5242569, 7327329, 7942209, 8601329, 8668921, 9608729
Offset: 1

Views

Author

Paolo P. Lava, May 13 2013

Keywords

Examples

			Prime factors of 116265 are 3, 5, 23 and 337. We have that (116265-9)/(3+9) = 9688, (116265-9)/(5+9) = 8304, (116265-9)/(23+9) = 3633 and (116265-9)/(337+9) = 336.

Crossrefs

Cf. A208728, A225702-A225718, A225720.

Programs

  • Maple
    with(numtheory); A225719:=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: A225719(10^9,-9);

A226112 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

133653, 1280533, 193638133, 514276565, 1421486733, 1567953933, 3857178453, 3973200933, 5411272533, 7694639213, 8021152533, 8469827669, 9820706133, 15832804533, 18238619373, 22356801133, 23037766613, 25136796813, 27315827733, 32434329685, 39817016633
Offset: 1

Views

Author

Paolo P. Lava, May 29 2013

Keywords

Comments

Also composite squarefree numbers n such that (3*p(i)+1) | (3*n+1).

Examples

			The prime factors of 133653 are 3, 13, 23 and 149. We see that (133653 + 1/3)/(3 + 1/3) = 40096, (133653 + 1/3)/(13 + 1/3) = 10024, (133653 + 1/3)/(23 + 1/3) = 5728 and (133653 + 1/3)/(149 + 1/3) = 895. Hence 133653 is in the sequence.
The prime factors of 1127749 are 7, 31 and 5197. We see that
(1127749 + 1/3)/(7 + 1/3) = 153784, (1127749 + 1/3)/(31 + 1/3) = 35992 but (1127749 + 1/3)/(5197 + 1/3) = 422906/1949. Hence 1127749 is not in the sequence.

Links

Crossrefs

Cf. A208728, A225702-A225720, A226020, A226111, A226113, A226114.

Programs

  • Maple
    with(numtheory); A226112:=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: A226112(10^9,1/3);

Extensions

a(4)-a(21) from Giovanni Resta, Jun 02 2013

A227973 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

15, 273, 77, 6, 21, 6, 33, 10, 15, 14, 21, 33, 35, 22, 33, 26, 39, 57, 65, 34, 51, 38, 57, 551, 95, 46, 69, 203, 115, 145, 161, 58, 87, 62, 93, 629, 155, 697, 217, 74, 111, 518, 185, 82, 123, 86, 129, 2537, 215, 94, 141, 689, 235, 4366, 329, 106, 159, 1247, 265
Offset: 1

Views

Author

Paolo P. Lava, Aug 02 2013

Keywords

Examples

			For n=185 the minimum k is 543. Prime factors of 543 are 3 and 181. We have: 543 + 185 = 728, 3 - 185 = -182 and 728 / (-182) = -4, 181 - 185 = -4 and 728 / (-4) = 182.

Links

Crossrefs

Cf. A208728, A225702-A225720, A227974-A227976.

Programs

  • Maple
    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 or p[d][1]=k 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);

A227976 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

561, 1595, 35, 6, 21, 6, 15, 14, 21, 10, 35, 22, 33, 14, 15, 133, 65, 34, 51, 38, 21, 22, 95, 46, 69, 26, 115, 217, 161, 30, 87, 62, 33, 34, 35, 1247, 217, 38, 39, 817, 185, 42, 123, 86, 129, 46, 215, 94, 141, 1247, 51, 1802, 329, 106, 55, 1541, 57, 58, 371
Offset: 1

Views

Author

Paolo P. Lava, Aug 06 2013

Keywords

Examples

			For n=2 the minimum k is 1595. Prime factors of 1595 are 5, 11, and 29. We have 1595 - 2 = 1593, 5 - 2 = 2 and 1593 / 3 = 531, 11 - 2 = 9 and 1593 / 9 = 177, 29 - 2 = 27 and 1593 / 27 = 59.

Links

Crossrefs

Cf. A208728, A225702-A225720, A227973-A227974.

Programs

  • Maple
    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 or p[d][1]=k 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);

A274446 a(n) is the smallest composite squarefree number k such that (p+n) | (k+1) for all primes dividing k.

Original entry on oeis.org

399, 299, 55, 611, 143, 5549, 39, 155, 493, 615383, 713, 3247, 119, 1304489, 1333, 31415, 2599, 749, 2183, 440153, 155, 75499, 119, 168600949, 4223, 223649, 559, 66299, 6407, 15157, 3431, 85499, 799, 31589, 7313
Offset: 1

Views

Author

Paolo P. Lava, Jun 23 2016

Keywords

Examples

			Prime factors of 399 are 3, 7 and 19. (399 + 1) / (3 + 1) = 400 / 4 = 100, (399 + 1) / (7 + 1) = 400 / 8 = 50 and (399 + 1) / (19 + 1) = 400 / 20 = 20.
Prime factors of 299 are 13 and 23. (399 + 1) / (13 + 2) = 300 / 15 = 20 and (399 + 1) / (23 + 2) = 300 / 25 = 12.

Crossrefs

Cf. A208728, A225702-A225720, A226020, A226111-A226114, A226364, A226448, A228299-A228302, A229273-A229276, A229321-A229324, A274443, A274444, A274445.

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[2000000], SquareFreeQ@ # && CompositeQ@ # &]; Table[SelectFirst[t, Function[k, AllTrue[First /@ FactorInteger@ k, Divisible[k + 1, # + n] &]]], {n, 23}] (* 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(24) from Giovanni Resta, Jun 23 2016
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