A249481 Numbers n such that p|n and q|n+1 => p^q|n and q^p|n+1, where p is prime with multiplicity q and q prime with multiplicity p.

8, 152, 224, 279, 423, 440, 584, 711, 855, 872, 1016, 1143, 1287, 1304, 1448, 1575, 1719, 1736, 1824, 1880, 2007, 2151, 2168, 2312, 2439, 2583, 2600, 2744, 2871, 2975, 3015, 3032, 3176, 3303, 3424, 3447, 3464, 3608, 3735, 3879, 3896, 3968, 4040, 4167, 4311

Offset

1,1

Comments

The majority of numbers generate pairs (p,q) = (2,3), but there exist subsequences of numbers such that (p,q) is different from (2,3). Examples:

The subsequence where (p,q) = (3,5) is {26000, 34749, 56375, 65124, 117125, 125874, 208250, 216999, 238625, 247374, 329750, 338499,...}

The subsequence where (p,q) = (3,7) is {494262, 1244403, 1756160, 2744685, 3256442, 3494826, 4006583, 4995108, 5506865, 5745249, ...}.

Is the number of distinct pairs (p,q) infinite?

We observe interesting properties with periodicities from the values a(n) - a(n-1). For example

a(2) - a(1) = a(5) - a(4) = a(7) - a(6) = ... = 144;

a(6) - a(5) = a(10) - a(9) = a(14) - a(13) = ... = 17;

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We observe similar periodicities if we consider the subsequence b(n) when (p,q) = (3,5). For example

b(2)-b(1) = 34749 - 26000 = 8749;

b(4)-b(3) = 65124 - 56375 = 8749;

b(6)-b(5) = 125874 - 117125 = 8749;

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We observe also the same behavior when (p,q) = (3,7).

Links

Michel Lagneau, Table of n, a(n) for n = 1..10000

Example

152 is in the sequence because 152 = 19*2^3 and 153 = 3^2*17 => (p,q) = (2,3);
26000 is in the sequence because 26000 = 2^4*5^3*13 and 26001 = 3^5*107 => (p,q) = (3,5).

Maple

with(numtheory):nn:=10000:
  for n from 2 to nn do:
    x0:=ifactors(n):x1:=x0[2]:nx0:=nops(x1):
    y0:=ifactors(n+1):y1:=y0[2]:ny0:=nops(y1):
      for i from 1 to nx0 do:
        xx1:=x1[i]:p1:=xx1[1]:q1:=xx1[2]:
         for j from 1 to ny0 do:
          yy1:=y1[j]:p2:=yy1[1]:q2:=yy1[2]:
          if p1=q2 and p2=q1
          then
          printf(`%d, `, n):
          else
         fi:
       od:
      od:
    od:

Crossrefs

Sequence in context: A300872 A217502 A229955 * A003491 A053606 A367701

Adjacent sequences: A249478 A249479 A249480 * A249482 A249483 A249484

Keyword

nonn

Author

Michel Lagneau, Jan 13 2015

Status

approved