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A216023
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Fermat pseudoprimes to base 2 divisible by 5.
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3
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645, 1105, 1905, 2465, 10585, 11305, 16705, 18705, 34945, 39865, 41665, 55245, 62745, 72885, 74665, 83665, 107185, 121465, 208465, 215265, 223345, 266305, 278545, 449065, 451905, 464185, 493885, 588745, 743665, 757945, 800605, 825265, 831405, 898705, 1050985
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OFFSET
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1,1
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COMMENTS
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Many Fermat pseudoprimes to base 2 divisible by 5 have one of the following four properties:
(1) the sum of their prime factors is divisible by the sum of their digits:
for 1105 = 3*5*17 we have 35 divisible by 15;
for 1905 = 3*5*127 we have 135 divisible by 15;
for 2465 = 5*17*29 we have 51 divisible by 17;
for 34945 = 5*29*241 we have 275 divisible by 25;
for 62745 = 3*5*47*89 we have 144 divisible by 24;
for 107185 = 3*5*47*89 we have 132 divisible by 22;
for 223345 = 5*19*2351 we have 2375 divisible by 19;
for 451905 = 3*5*47*641 we have 696 divisible by 24.
(2) the sum of their prime factors is divisible by 5 (1105, 1905, 16705, 18705, 34945, 223345, 757945, 800605).
(3) the sum of their digits is divisible by 5 (645, 1905, 11305, 34945, 72885, 208465, 72885);
(4) they are Harshad numbers (645, 1905, 2465, 223345, 757945).
Interesting is that the first property is found to other squarefree numbers, not Fermat pseudoprimes, divisible by 5 (e.g., for 1505 = 5*7*43 we have 55 divisible by 11, for 2555 = 5*7*73 we have 85 divisible by 17). It looks like it's a property which deserves further study.
Note: the four properties from above are also found to other Fermat pseudoprimes to base 2, but not in this high density (taking, for the second and third properties, a prime factor beside 5 and not considering for the third property the prime factor 3, because would be obviously satisfied).
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LINKS
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Eric Weisstein's World of Mathematics, Digit Sum
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MATHEMATICA
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Select[5*Range[2, 210200], PowerMod[2, # - 1, #] == 1 &] (* T. D. Noe, Aug 31 2012 *)
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PROG
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(PARI) Korselt(n)=my(f=factor(n)); for(i=1, #f[, 1], if(f[i, 2]>1||(n-1)%(f[i, 1]-1), return(0))); 1
list(lim)=my(v=List()); forstep(n=645, lim, 20, if(Korselt(n), listput(v, n))); Vec(v) \\ Charles R Greathouse IV, Jun 30 2017
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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