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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. 1
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 (list; graph; refs; listen; history; text; internal format)
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;

....................................................

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;

....................................................

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 A221732

Adjacent sequences:  A249478 A249479 A249480 * A249482 A249483 A249484

KEYWORD

nonn

AUTHOR

Michel Lagneau, Jan 13 2015

STATUS

approved

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Last modified January 28 09:52 EST 2020. Contains 331319 sequences. (Running on oeis4.)