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

1,1

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, ...).

Links

Table of n, a(n) for n=1..45.

Formula

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

Example

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, ...

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 *)

Prog

(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))));

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.

Sequence in context: A147983 A307933 A235239 * A235232 A304284 A167562

Adjacent sequences: A337466 A337467 A337468 * A337470 A337471 A337472

Keyword

nonn

Author

Antti Karttunen and Peter Munn, Sep 07 2020

Status

approved