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.

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.
		

Crossrefs

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.
		

Crossrefs

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.
		

Crossrefs

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.
		

Crossrefs

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

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

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
Showing 1-5 of 5 results.