A257042 a(n) = (3*n+7)*n^2.

0, 10, 52, 144, 304, 550, 900, 1372, 1984, 2754, 3700, 4840, 6192, 7774, 9604, 11700, 14080, 16762, 19764, 23104, 26800, 30870, 35332, 40204, 45504, 51250, 57460, 64152, 71344, 79054, 87300, 96100, 105472, 115434, 126004, 137200, 149040, 161542, 174724

Offset

0,2

Comments

Consider a natural number r such that r has 15 proper divisors and 5 prime factors (note that these prime factors do not have to be distinct). The difference between these two values, say d(r), is in this case 10. Where n is a positive integer, d(r^n)=(3*n+7)*n^2.

Links

Table of n, a(n) for n=0..38.

Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Formula

G.f.: x*(10+12*x-4*x^2)/(1-x)^4. - Vincenzo Librandi, Apr 15 2015

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n>3. - Vincenzo Librandi, Apr 15 2015

Example

The smallest integer that satisfies this is 120: it has 15 proper divisors (1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60) and 5 prime factors (2, 2, 2, 3, 5), so d(120)=10. The square of 120, 14400, we would expect to have a difference of 52 between the number of its proper divisors and prime factors, and with respectively 62 and 10, d(120)=52 indeed. Checking this with further integer powers of 120 will continue to generate terms in this sequence.
The integers which satisfy the proper-divisor-prime-factor requirement are those of A189975.

Maple

A257042:=n->(3*n+7)*n^2: seq(A257042(n), n=0..50); # Wesley Ivan Hurt, Apr 16 2015

Mathematica

Table[(3 n + 7) n^2, {n, 40}] (* or *) CoefficientList[Series[(10 + 12 x - 4 x^2) / (1 - x)^4, {x, 0, 40}], x] (* Vincenzo Librandi, Apr 15 2015 *)

Prog

(Magma) [(3*n+7)*n^2: n in [0..65]]  // Vincenzo Librandi, Apr 15 2015
(PARI) lista(nn) = {v = 1; while(!((numdiv(v)-1 == 15) && (bigomega(v) == 5)), v++); for (n=0, nn, vn = v^n; nb = numdiv(vn)-1-bigomega(vn); print1(nb, ", "); ); } \\ Michel Marcus, Apr 16 2015

Crossrefs

Cf. A189975.

Sequence in context: A058827 A232909 A028994 * A092966 A281401 A050494

Adjacent sequences: A257039 A257040 A257041 * A257043 A257044 A257045

Keyword

nonn,easy

Author

Garrett Frandson, Apr 14 2015

Extensions

More terms from Vincenzo Librandi, Apr 15 2015

Status

approved