A144695 Numbers n such that sigma_1(n)/sigma_0(n) = c^2, c an integer.

1, 7, 17, 22, 30, 31, 71, 94, 97, 115, 119, 127, 138, 154, 164, 165, 199, 210, 214, 217, 241, 260, 265, 318, 337, 343, 374, 382, 449, 497, 510, 513, 517, 527, 577, 647, 658, 668, 679, 682, 705, 745, 759, 805, 862, 881, 889, 894, 930, 966, 967, 996, 1102, 1125

Offset: 1

Author

Ctibor O. Zizka, Sep 19 2008

Keywords

• easy
• nonn

Comments

A000203(n)/A000005(n) = c^2. Generalized sigma-sequences are sequences of numbers n for which sigma_r(n)/sigma_s(n) = c^t . Sigma_i(n) denotes sum of i-th powers of divisors of n; c,r,s,t positive integers. This sequence has r=1,s=0,t=2, sequence A003601 has r=1,s=0,t=1, sequence {1,21,53,85,102,110,127,217,431,....} has r=1,s=0,t=3, sequence A020487 has r=2,s=1,t=1, sequence A020486 has r=2,s=0,t=1, sequence A140480 has r=2,s=0,t=2.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Eric Weisstein's World of Mathematics, Divisor function

Crossrefs

Cf. A000005, A000203, A140480, A003601, A020487, A020486.

Programs

Maple

A000005 := proc(n) numtheory[tau](n) ; end: A000203 := proc(n) numtheory[sigma](n) ; end: isA144695 := proc(n) local s ; s := A000005(n) ; if s <> 0 then issqr(A000203(n)/s) ; else false ; fi; end: for n from 1 to 5000 do if isA144695(n) then printf("%d,",n) ; fi; od: # R. J. Mathar, Sep 20 2008

Mathematica

Select[Range[1125], IntegerQ @ Sqrt[DivisorSigma[1, #]/DivisorSigma[0, #]] &] (* Amiram Eldar, Nov 20 2019 *)

PARI

isok(m) = my(f=factor(m), q=sigma(f)/numdiv(f)); issquare(q) && !frac(q); \\ Michel Marcus, Mar 15 2022

Extensions

More terms from R. J. Mathar, Sep 20 2008