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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. 1
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified August 3 21:13 EDT 2021. Contains 346441 sequences. (Running on oeis4.)