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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 September 26 17:49 EDT 2023. Contains 365666 sequences. (Running on oeis4.)