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
Keywords
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
Programs
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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))));
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