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%I #15 Jan 15 2015 13:09:43
%S 8,152,224,279,423,440,584,711,855,872,1016,1143,1287,1304,1448,1575,
%T 1719,1736,1824,1880,2007,2151,2168,2312,2439,2583,2600,2744,2871,
%U 2975,3015,3032,3176,3303,3424,3447,3464,3608,3735,3879,3896,3968,4040,4167,4311
%N 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.
%C 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:
%C The subsequence where (p,q) = (3,5) is {26000, 34749, 56375, 65124, 117125, 125874, 208250, 216999, 238625, 247374, 329750, 338499,...}
%C The subsequence where (p,q) = (3,7) is {494262, 1244403, 1756160, 2744685, 3256442, 3494826, 4006583, 4995108, 5506865, 5745249, ...}.
%C Is the number of distinct pairs (p,q) infinite?
%C We observe interesting properties with periodicities from the values a(n) - a(n-1). For example
%C a(2) - a(1) = a(5) - a(4) = a(7) - a(6) = ... = 144;
%C a(6) - a(5) = a(10) - a(9) = a(14) - a(13) = ... = 17;
%C ....................................................
%C We observe similar periodicities if we consider the subsequence b(n) when (p,q) = (3,5). For example
%C b(2)-b(1) = 34749 - 26000 = 8749;
%C b(4)-b(3) = 65124 - 56375 = 8749;
%C b(6)-b(5) = 125874 - 117125 = 8749;
%C ....................................................
%C We observe also the same behavior when (p,q) = (3,7).
%H Michel Lagneau, <a href="/A249481/b249481.txt">Table of n, a(n) for n = 1..10000</a>
%e 152 is in the sequence because 152 = 19*2^3 and 153 = 3^2*17 => (p,q) = (2,3);
%e 26000 is in the sequence because 26000 = 2^4*5^3*13 and 26001 = 3^5*107 => (p,q) = (3,5).
%p with(numtheory):nn:=10000:
%p for n from 2 to nn do:
%p x0:=ifactors(n):x1:=x0[2]:nx0:=nops(x1):
%p y0:=ifactors(n+1):y1:=y0[2]:ny0:=nops(y1):
%p for i from 1 to nx0 do:
%p xx1:=x1[i]:p1:=xx1[1]:q1:=xx1[2]:
%p for j from 1 to ny0 do:
%p yy1:=y1[j]:p2:=yy1[1]:q2:=yy1[2]:
%p if p1=q2 and p2=q1
%p then
%p printf(`%d, `,n):
%p else
%p fi:
%p od:
%p od:
%p od:
%K nonn
%O 1,1
%A _Michel Lagneau_, Jan 13 2015
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