A054994 Numbers of the form q1^b1 * q2^b2 * q3^b3 * q4^b4 * q5^b5 * ... where q1=5, q2=13, q3=17, q4=29, q5=37, ... (A002144) and b1 >= b2 >= b3 >= b4 >= b5 >= ....

1, 5, 25, 65, 125, 325, 625, 1105, 1625, 3125, 4225, 5525, 8125, 15625, 21125, 27625, 32045, 40625, 71825, 78125, 105625, 138125, 160225, 203125, 274625, 359125, 390625, 528125, 690625, 801125, 1015625, 1185665, 1221025, 1373125, 1795625

Offset

1,2

Comments

This sequence is related to Pythagorean triples regarding the number of hypotenuses which are in a particular number of total Pythagorean triples and a particular number of primitive Pythagorean triples.

Least integer "mod 4 prime signature" values that are the hypotenuse of at least one primitive Pythagorean triple. - Ray Chandler, Aug 26 2004

See A097751 for definition of "mod 4 prime signature"; terms of A097752 with all prime factors of form 4*k+1.

Sequence A006339 (Least hypotenuse of n distinct Pythagorean triangles) is a subset of this sequence. - Ruediger Jehn, Jan 13 2022

Links

Ray Chandler, Table of n, a(n) for n = 1..10000

Eric Weisstein's World of Mathematics, Pythagorean Triple

Formula

Sum_{n>=1} 1/a(n) = Product_{n>=1} 1/(1 - 1/A006278(n)) = 1.2707219403... - Amiram Eldar, Oct 20 2020

Example

1=5^0, 5=5^1, 25=5^2, 65=5*13, 125=5^3, 325=5^2*13, 625=5^4, etc.

Mathematica

maxTerm = 10^15; (* this limit gives ~ 500 terms *) maxNumberOfExponents = 9; (* this limit has to be increased until the number of reaped terms no longer changes *) bmax = Ceiling[ Log[ maxTerm]/Log[q]]; q = Reap[For[k = 0 ; cnt = 0, cnt <= maxNumberOfExponents, k++, If[PrimeQ[4*k + 1], Sow[4*k + 1]; cnt++]]][[2, 1]]; Clear[b]; b[maxNumberOfExponents + 1] = 0; iter = Sequence @@ Table[{b[k], b[k + 1], bmax[[k]]}, {k, maxNumberOfExponents, 1, -1}]; Reap[ Do[an = Product[q[[k]]^b[k], {k, 1, maxNumberOfExponents}]; If[an <= maxTerm, Print[an]; Sow[an]], Evaluate[iter]]][[2, 1]] // Flatten // Union (* Jean-François Alcover, Jan 18 2013 *)

Prog

(PARI) list(lim)=
{
  my(u=[1], v=List(), w=v, pr, t=1);
  forprime(p=5, ,
    if(p%4>1, next);
    t*=p;
    if(t>lim, break);
    listput(w, t)
  );
  for(i=1, #w,
    pr=1;
    for(e=1, logint(lim\=1, w[i]),
      pr*=w[i];
      for(j=1, #u,
        t=pr*u[j];
        if(t>lim, break);
        listput(v, t)
      )
    );
    if(w[i]^2<lim, u=Set(concat(Vec(v), u)); v=List());
  );
  Set(concat(Vec(v), u));
} \\ Charles R Greathouse IV, Dec 11 2016

Crossrefs

Subsequence of A097752.

Cf. A002144, A006278, A006339, A071383, A097751, A097752, A097753, A097754, A097755, A097756.

Sequence in context: A146404 A323187 A179131 * A108403 A007058 A071383

Adjacent sequences: A054991 A054992 A054993 * A054995 A054996 A054997

Keyword

easy,nonn

Author

Bernard Altschuler (Altschuler_B(AT)bls.gov), May 30 2000

Extensions

More terms from Henry Bottomley, Mar 14 2001

Status

approved