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 1-10 of 10 results.

A226114 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

1045, 1639605, 7343133, 7938133, 25615893, 282388773, 296251293, 346148733, 895445173, 1217200533, 1584568533, 2578055893, 3604398933, 4078150853, 5181367893, 5621460973, 7591692693, 8199401613, 9393224533, 9489314501, 12671984033, 12723857813, 14057815893
Offset: 1

Views

Author

Paolo P. Lava, May 27 2013

Keywords

Comments

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

Examples

			The prime factors of 1045 are 5, 11 and 19. We see that (1045 + 1/3)/(5 - 1/3) = 224, (1045 + 1/3)/(11 - 1/3) = 98 and (1045 + 1/3)/(19 - 1/3) = 56. Hence 1045 is in the sequence.
The prime factors of 1639605 are 3, 5, 11, 19 and 523. We see that (1639605 + 1/3)/(3 - 1/3) = 614852, (1639605 + 1/3)/(5 - 1/3) = 351344, (1639605 + 1/3)/(11 - 1/3) = 153713, (1639605 + 1/3)/(19 - 1/3) = 87836 and (1639605 + 1/3)/(523 - 1/3) = 3137. Hence 1639605 is in the sequence.
The prime factors of 1117965 are 3, 5 and 74531. We see that (1117965 + 1/3)/(3 - 1/3) = 419237, (1117965 + 1/3)/(5 - 1/3) = 239564 but (1117965 + 1/3)/(74531 - 1/3) = 419237/27949. Hence 1117965 is not in the sequence.

Links

Crossrefs

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

Programs

  • Maple
    with(numtheory); A226114:=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: A226114(10^9,1/3);

Extensions

a(6)-a(23) from Giovanni Resta, Jun 02 2013

A226020 Composite squarefree numbers n such that the ratio (n + 1/2)/(p(i) + 1/2) is an integer, where p(i) are the prime factors of n.

Original entry on oeis.org

13702, 42997, 1004062, 1684462, 38447662, 40243549, 70801087, 107728582, 409055062, 594021862, 760767262, 1045475437, 1104435202, 1471700587, 1686747562, 1920806662, 3136180162, 3469071937, 5291041297, 7239716347, 7903353667, 12738885862, 22711489762
Offset: 1

Views

Author

Paolo P. Lava, May 23 2013

Keywords

Comments

Also composite squarefree numbers n such that (2*p(i)+1) | (2*n+1).

Examples

			The prime factors of 13702 are 2, 13, 17 and 31. We see that (13702 + 1)/(2 + 1/2) = 5481, (13702 + 1/2)/(13 + 1/2) = 1015, (13702 + 1)/(17 + 1/2) = 783 and ( 13702 + 1/2)/(31 + 1/2) = 435. Hence 13702 is in the sequence.
The prime factors of 1123545 are 3, 5 and 74903. We see that
(1123545 + 1/2)/(3 + 1/2) = 321013, (1123545 + 1/2)/(5 + 1/2) = 204281 but (1123545 + 1/2)/(74903+ 1/2) = 321013/21401. Hence 1123545 is not in the sequence.

Links

Crossrefs

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

Programs

  • Maple
    with(numtheory); A226020:=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: A226020(10^9,1/2);

Extensions

a(9)-a(23) from Giovanni Resta, Jun 02 2013

A226364 Composite squarefree numbers n such that the ratios (n - 1/3)/(p - 1/3) are integers for each prime p dividing n.

Original entry on oeis.org

308267, 1420467, 1445995, 46874667, 153810067, 324218667, 355724747, 393253747, 471957547, 618729307, 886489707, 901990059, 1062803467, 1525582667, 1735517355, 4306362667, 4815895467, 6528285867, 6634856107, 11460166667, 12364885867, 13330858667, 20628538667
Offset: 1

Views

Author

Paolo P. Lava, Jun 05 2013

Keywords

Comments

Also composite squarefree numbers n such that (3p - 1) | (3n - 1).

Examples

			The prime factors of 1445995 are 5, 19, 31 and 491. We see that (1445995 - 1/3)/(5 - 1/3) = 309856, (1445995 - 1/3)/(19 - 1/3) = 77464, (1445995 - 1/3)/(31 - 1/3) = 47152 and (1445995 - 1/3)/(491 - 1/3) = 2947. Hence 1445995 is in the sequence.
The prime factors of 1112307 are 3, 7 and 52967. We see that (1112307 - 1/3)/(3 - 1/3) = 417115, (1112307 - 1/3)/(7 - 1/3) = 166846 but (1112307 - 1/3)/(52967 - 1/3) = 166846/7945. Hence 1112307 is not in the sequence.

Links

Crossrefs

Cf. A208728, A225702-A225720, A226020, A226111-A226114, A226448.

Programs

  • Maple
    with(numtheory); ListA226364:=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: ListA226364(10^9,1/3);
  • PARI
    is(n,P)=n=3*n-1; for(i=1,#P,if(n%(3*P[i]-1),return(0))); 1
list(lim,P=[],n=1,mx=lim\2)=my(v=[],t);if(#P>1&&is(n,P), v=[n]); P=concat(P,0); forprime(p=2,min(lim,mx),P[#P]=p;t=list(lim\p,P,n*p,p-1);if(#t,v=concat(v,t))); v \\ Charles R Greathouse IV, Jun 07 2013

Extensions

a(5)-a(23) from Giovanni Resta, Jun 07 2013

A226448 Composite squarefree numbers k such that the ratios (k - 1/2)/(p - 1/2) are integers for each prime p dividing k.

Original entry on oeis.org

260054438, 597892523, 1200695738, 3287998643, 3423456563, 10524308498, 13292859563, 15646705718, 19441707170, 33309521438, 38848586123, 43312628678, 61899936935, 72422400713, 75439031063, 85338414662, 112419230963, 132624705038, 136084511063, 141236121758
Offset: 1

Views

Author

Paolo P. Lava and Giovanni Resta, Jun 07 2013

Keywords

Comments

Also composite squarefree numbers k such that (2p - 1) | (2k - 1).

Examples

			3287998643 is a term since it is equal to 743*787*5623 and 3287998643-1/2 divided by 743-1/2, 787-1/2 and 5623-1/2 gives 3 integers, namely 4428281, 4180545 and 584793.

Links

Crossrefs

Cf. A208728, A225702-A225720, A226020, A226111-A226114, A226364.

Programs

  • Maple
    with(numtheory); ListA226448:=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: ListA226448(10^9, 1/2); # Paolo P. Lava, Oct 06 2013
  • PARI
    is(n, P)=n=2*n-1; for(i=1, #P, if(n%(2*P[i]-1), return(0))); 1
list(lim, P=[], n=1, mx=lim\2)=my(v=[], t); if(#P>1&&is(n, P), v=[n]); P=concat(P, 0); forprime(p=2, min(lim, mx), P[#P]=p; t=list(lim\p, P, n*p, p-1); if(#t, v=concat(v, t))); v \\ Charles R Greathouse IV, Jun 07 2013

A274443 Least composite squarefree number k such that (p-n) | (k-1) for all primes p dividing n.

Original entry on oeis.org

561, 21, 85, 15, 21, 35, 33, 21, 65, 91, 57, 91, 133, 55, 161, 91, 57, 133, 33, 253, 65, 91, 145, 115, 217, 451, 161, 703, 253, 551, 561, 253, 481, 217, 129, 451, 301, 1081, 161, 1189, 145, 989, 217, 235, 481, 703, 649, 329, 265, 1081, 1121, 1219, 145, 1037, 721
Offset: 1

Views

Author

Paolo P. Lava, Jun 23 2016

Keywords

Examples

			Prime factors of 561 are 3, 11 and 17: (561 - 1) / (3 - 1) = 560 / 2 = 280, (561 - 1) / (11 - 1) = 560 / 10 = 56 and (561 - 1) / (17 - 1) = 560 / 16 = 35.
Prime factors of 21 are 3 and 7: (21 - 1) / (3 - 2) = 20 / 1 = 20, (21 - 1) / (7 - 2) = 20 / 5 = 4.

Links

Crossrefs

Cf. A208728, A225702-A225720, A226020, A226111-A226114, A226364, A226448, A228299-A228302, A229273-A229276, A229321-A229324, A274444, A274445, 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 p[d][1]=k then ok:=0; break; else
if not type((n-1)/(p[d][1]-k),integer) then ok:=0; break; fi; fi; od;
if ok=1 then print(n); break; fi; fi; od; od; end: P(10^9);
  • Mathematica
    t = Select[Range@2000, SquareFreeQ@ # && CompositeQ@ # &]; Table[SelectFirst[t, Function[k, AllTrue[First /@ FactorInteger@ k, If[# == 0, False, Divisible[k - 1, #]] &[# - n] &]]], {n, 55}] (* Michael De Vlieger, Jun 24 2016, Version 10 *)

A274444 a(n) = smallest composite squarefree number k such that (p-n) | (k+1) for all primes dividing k.

Original entry on oeis.org

15, 65, 35, 15, 21, 35, 15, 35, 35, 77, 35, 55, 55, 143, 119, 51, 95, 155, 55, 323, 95, 119, 39, 391, 87, 209, 119, 299, 143, 341, 319, 629, 259, 899, 407, 185, 119, 299, 287, 1517, 203, 799, 159, 155, 407, 1189, 119, 517, 341, 1763, 1363, 629, 335, 2491, 493, 3599
Offset: 1

Views

Author

Paolo P. Lava, Jun 23 2016

Keywords

Examples

			a(1) = 15: Prime factors of 15 are 3 and 5: (15 + 1) / (3 - 1) = 16 / 2 = 8 and (15 + 1) / (5 - 1) = 16 / 4 = 4.
a(2) = 6: Prime factors of 65 are 5 and 13: (65 + 1) / (5 - 2) = 66 / 3 = 22 and (65 + 1) / (13 - 2) = 66 / 11 = 6.

Links

Crossrefs

Cf. A208728, A225702-A225720, A226020, A226111-A226114, A226364, A226448, A228299-A228302, A229273-A229276, A229321-A229324, A274443, A274445, 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 p[d][1]=k then ok:=0; break; else
if not type((n+1)/(p[d][1]-k),integer) then ok:=0; break; fi; fi; od;
if ok=1 then print(n); break; fi; fi; od; od; end: P(10^9);
  • Mathematica
    t = Select[Range[10^4], SquareFreeQ@ # && CompositeQ@ # &]; Table[SelectFirst[t, Function[k, AllTrue[First /@ FactorInteger@ k,
If[# == 0, False, Divisible[k + 1, #]] &[# - n] &]]], {n, 56}] (* Michael De Vlieger, Jun 24 2016, Version 10 *)

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

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

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

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

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 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.

Links

Crossrefs

Cf. A208728, A225702-A225720, A226020, A226111, A226112, A226114.

Programs

  • Maple
    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);

Extensions

a(5)-a(22) from Giovanni Resta, Jun 02 2013
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