A337469 a(n) is the least k that is a multiple of A071395(n) (the n-th primitive abundant number) for which A003961(k) is abundant.

120, 420, 1320, 1560, 4080, 4560, 5520, 6960, 1650, 3432, 3900, 4488, 7524, 1890, 17760, 19680, 20640, 4290, 22560, 3150, 25440, 5610, 28320, 29280, 12012, 6270, 4410, 6630, 7410, 7590, 23256, 8970, 28152, 9570, 9690, 10230, 6930, 52440, 22620, 59160, 24180, 12210, 8190, 63240, 64320

Offset: 1

Author

Antti Karttunen and Peter Munn, Sep 07 2020

Keywords

• nonn

Comments

A003961(k) replaces each prime factor of k with the next larger prime. Thus for all terms a(n), A003961(a(n)) is an odd abundant number (some of which are also primitive abundant numbers, starting with n = 1, 2, 9, 10, 12, ...).

Examples

			The table below shows a(n), for n less than 16, alongside A071395(n) and its prime factors, and the additional prime factors that are needed to produce a(n).
   n   a(n)               A071395(n)
   1    120 / (2 * 3)  =    20  =  2^2 * 5,
   2    420 / (2 * 3)  =    70  =  2 * 5 * 7,
   3   1320 / (3 * 5)  =    88  =  2^3 * 11,
   4   1560 / (3 * 5)  =   104  =  2^3 * 13,
   5   4080 / (3 * 5)  =   272  =  2^4 * 17,
   6   4560 / (3 * 5)  =   304  =  2^4 * 19,
   7   5520 / (3 * 5)  =   368  =  2^4 * 23,
   8   6960 / (3 * 5)  =   464  =  2^4 * 29,
   9   1650 / (3)      =   550  =  2 * 5^2 * 11,
  10   3432 / (2 * 3)  =   572  =  2^2 * 11 * 13,
  11   3900 / (2 * 3)  =   650  =  2 * 5^2 * 13,
  12   4488 / (2 * 3)  =   748  =  2^2 * 11 * 17,
  13   7524 / (3 * 3)  =   836  =  2^2 * 11 * 19,
  14   1890 / (2)      =   945  =  3^3 * 5 * 7,
  15  17760 / (3 * 5)  =  1184  =  2^5 * 37, ...
		

Crossrefs

See A000203 and A005101 for the definition of abundant.

A003961 and A071395 are used to define the sequence.

Sequences with related definitions: A337386, A337479, A337538.

Cf. A003973.

Programs

Mathematica

Map[Block[{k = 1}, While[DivisorSigma[1, #] <= 2 # &[Times @@ Map[#1^#2 & @@ # &, FactorInteger[k #] /. {p_, e_} /; e > 0 :> {Prime[PrimePi@ p + 1], e}]], k++]; # k] &, Select[Range[5*10^3], DivisorSigma[1, #] > 2 # && Times @@ Boole@ Map[DivisorSigma[1, #] < 2 # &, Most@ Divisors@ #] == 1 &]] (* Michael De Vlieger, Oct 05 2020 *)

PARI

isA071395(n) = if(sigma(n) <= 2*n, 0, fordiv(n, d, if((d != n)&&(sigma(d) >= 2*d), return(0))); (1)); \\ After code in A071395
A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
isA337386(n) = { my(x=A003961(n)); (sigma(x)>=2*x); };
for(n=1,2^13,if(isA071395(n), i=0; for(k=1,oo,if(isA337386(k*n),i++; print1(k*n,", "); break))));

Formula

a(n) = A071395(n) * A337538(n).