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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.

%I A337469 #23 Oct 06 2020 02:36:56
%S A337469 120,420,1320,1560,4080,4560,5520,6960,1650,3432,3900,4488,7524,1890,
%T A337469 17760,19680,20640,4290,22560,3150,25440,5610,28320,29280,12012,6270,
%U A337469 4410,6630,7410,7590,23256,8970,28152,9570,9690,10230,6930,52440,22620,59160,24180,12210,8190,63240,64320
%N 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.
%C A337469 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, ...).
%F A337469 a(n) = A071395(n) * A337538(n).
%e A337469 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).
%e A337469    n   a(n)               A071395(n)
%e A337469    1    120 / (2 * 3)  =    20  =  2^2 * 5,
%e A337469    2    420 / (2 * 3)  =    70  =  2 * 5 * 7,
%e A337469    3   1320 / (3 * 5)  =    88  =  2^3 * 11,
%e A337469    4   1560 / (3 * 5)  =   104  =  2^3 * 13,
%e A337469    5   4080 / (3 * 5)  =   272  =  2^4 * 17,
%e A337469    6   4560 / (3 * 5)  =   304  =  2^4 * 19,
%e A337469    7   5520 / (3 * 5)  =   368  =  2^4 * 23,
%e A337469    8   6960 / (3 * 5)  =   464  =  2^4 * 29,
%e A337469    9   1650 / (3)      =   550  =  2 * 5^2 * 11,
%e A337469   10   3432 / (2 * 3)  =   572  =  2^2 * 11 * 13,
%e A337469   11   3900 / (2 * 3)  =   650  =  2 * 5^2 * 13,
%e A337469   12   4488 / (2 * 3)  =   748  =  2^2 * 11 * 17,
%e A337469   13   7524 / (3 * 3)  =   836  =  2^2 * 11 * 19,
%e A337469   14   1890 / (2)      =   945  =  3^3 * 5 * 7,
%e A337469   15  17760 / (3 * 5)  =  1184  =  2^5 * 37, ...
%t A337469 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 *)
%o A337469 (PARI)
%o A337469 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
%o A337469 A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
%o A337469 isA337386(n) = { my(x=A003961(n)); (sigma(x)>=2*x); };
%o A337469 for(n=1,2^13,if(isA071395(n), i=0; for(k=1,oo,if(isA337386(k*n),i++; print1(k*n,", "); break))));
%Y A337469 See A000203 and A005101 for the definition of abundant.
%Y A337469 A003961 and A071395 are used to define the sequence.
%Y A337469 Sequences with related definitions: A337386, A337479, A337538.
%Y A337469 Cf. A003973.
%K A337469 nonn
%O A337469 1,1
%A A337469 _Antti Karttunen_ and _Peter Munn_, Sep 07 2020