A216023 - OEIS

%I #13 Oct 17 2019 21:09:53

%S 645,1105,1905,2465,10585,11305,16705,18705,34945,39865,41665,55245,

%T 62745,72885,74665,83665,107185,121465,208465,215265,223345,266305,

%U 278545,449065,451905,464185,493885,588745,743665,757945,800605,825265,831405,898705,1050985

%N Fermat pseudoprimes to base 2 divisible by 5.

%C Many Fermat pseudoprimes to base 2 divisible by 5 have one of the following four properties:

%C (1) the sum of their prime factors is divisible by the sum of their digits:

%C for 1105 = 3*5*17 we have 35 divisible by 15;

%C for 1905 = 3*5*127 we have 135 divisible by 15;

%C for 2465 = 5*17*29 we have 51 divisible by 17;

%C for 34945 = 5*29*241 we have 275 divisible by 25;

%C for 62745 = 3*5*47*89 we have 144 divisible by 24;

%C for 107185 = 3*5*47*89 we have 132 divisible by 22;

%C for 223345 = 5*19*2351 we have 2375 divisible by 19;

%C for 451905 = 3*5*47*641 we have 696 divisible by 24.

%C (2) the sum of their prime factors is divisible by 5 (1105, 1905, 16705, 18705, 34945, 223345, 757945, 800605).

%C (3) the sum of their digits is divisible by 5 (645, 1905, 11305, 34945, 72885, 208465, 72885);

%C (4) they are Harshad numbers (645, 1905, 2465, 223345, 757945).

%C 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.

%C 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).

%H Charles R Greathouse IV, <a href="/A216023/b216023.txt">Table of n, a(n) for n = 1..10000</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PouletNumber.html">Poulet Number</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HarshadNumber.html">Harshad Number</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SumofPrimeFactors.html">Sum of Prime Factors</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/DigitSum.html">Digit Sum</a>

%t Select[5*Range[2, 210200], PowerMod[2, # - 1, #] == 1 &] (* _T. D. Noe_, Aug 31 2012 *)

%o (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

%o 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

%Y Cf. A001567.

%K nonn

%O 1,1

%A _Marius Coman_, Aug 30 2012