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This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A331285 a(n) is the index of the first occurrence of n in A331284.

Original entry on oeis.org

1, 8, 27, 108, 180, 396, 672, 1056, 1372, 1760, 2352, 3087, 3696, 3744, 4896, 5733, 7497, 9724, 11907, 13600, 15200, 18513, 19773, 23940, 24752, 28917, 32319, 33534, 42282, 45472, 47500, 52668, 55890, 59976, 66048, 74240, 77792, 81144, 86944, 100035, 105248, 109368, 122825, 127908, 134368, 144648, 156325, 168948, 175770
Offset: 1

Views

Author

Antti Karttunen, Jan 14 2020

Keywords

Comments

Also positions of records in A331284.
a(n) is the least k such that in range 1 .. k of A329605 (equally: in A329606[1..k]) there can be found exactly n occurrences of some term. In A329605 these "champion terms" are A329605(a(n)): 1, 4, 16, 24, 48, 192, 96, 192, 384, 288, 288, 576, 576, 576, 1152, 2304, 4608, 9216, 576, 2304, ..., that appear all to be 3-smooth numbers (in A003586).
For example, a(5)=180 and 48 is the first term in A329605 to occur for five times, as A329605(22) = A329605(42) = A329605(75) = A329605(112) = A329605(180) = 48.

Crossrefs

Cf. A331284 (a left inverse).
Cf. A003586, A329605, A329606.

Formula

A331284(a(n)) = n for all n >= 1.

A329605 Number of divisors of A108951(n), where A108951 is fully multiplicative with a(prime(i)) = prime(i)# = Product_{i=1..i} A000040(i).

Original entry on oeis.org

1, 2, 4, 3, 8, 6, 16, 4, 9, 12, 32, 8, 64, 24, 18, 5, 128, 12, 256, 16, 36, 48, 512, 10, 27, 96, 16, 32, 1024, 24, 2048, 6, 72, 192, 54, 15, 4096, 384, 144, 20, 8192, 48, 16384, 64, 32, 768, 32768, 12, 81, 36, 288, 128, 65536, 20, 108, 40, 576, 1536, 131072, 30, 262144, 3072, 64, 7, 216, 96, 524288, 256, 1152, 72, 1048576, 18, 2097152, 6144, 48
Offset: 1

Views

Author

Antti Karttunen, Nov 18 2019

Keywords

Links

Crossrefs

Cf. A000005, A000040, A000142, A002110, A010052, A034386, A052126, A056239, A108951, A329902, A329378, A329382, A329614, A329617, A331283, A331286 (odd part).
Cf. A329606 (rgs-transform), A329608, A331284 (ordinal transform).
Cf. A331285 (the position where for the first time some term has occurred n times in this sequence).

Programs

  • Mathematica
    Block[{a}, a[n_] := a[n] = Module[{f = FactorInteger[n], p, e}, If[Length[f] > 1, Times @@ a /@ Power @@@ f, {{p, e}} = f; Times @@ (Prime[Range[PrimePi[p]]]^e)]]; a[1] = 1; Array[DivisorSigma[0, a@ #] &, 75]] (* Michael De Vlieger, Jan 24 2020, after Jean-François Alcover at A108951 *)
  • PARI
    A034386(n) = prod(i=1, primepi(n), prime(i));
A108951(n) = { my(f=factor(n)); prod(i=1, #f~, A034386(f[i, 1])^f[i, 2]) };  \\ From A108951
A329605(n) = numdiv(A108951(n));
  • PARI
    A329605(n) = if(1==n,1,my(f=factor(n),e=1,m=1); forstep(i=#f~,1,-1, e += f[i,2]; m *= e^(primepi(f[i,1])-if(1==i,0,primepi(f[i-1,1])))); (m)); \\ Antti Karttunen, Jan 14 2020
  • PARI
    A329605(n) = if(1==n,1,my(f=factor(n),e=0,d); forstep(i=#f~,1,-1, e += f[i,2]; d = (primepi(f[i,1])-if(1==i,0,primepi(f[i-1,1]))); f[i,1] = (e+1); f[i,2] = d); factorback(f)); \\ Antti Karttunen, Jan 14 2020

Formula

a(n) = A000005(A108951(n)).
a(n) >= A329382(n) >= A329617(n) >= A329378(n).
A020639(a(n)) = A329614(n).
From Antti Karttunen, Jan 14 2020: (Start)
a(A052126(n)) = A329382(n).
a(A002110(n)) = A000142(1+n), for all n >= 0.
a(n) > A056239(n).
a(A329902(n)) = A002183(n).
A000265(a(n)) = A331286(n).
gcd(n,a(n)) = A331283(n).
If n = p(k1)^e(k1) * p(k2)^e(k2) * p(k3)^e(k3) * ... * p(kx)^e(kx), with p(n) = A000040(n) and k1 > k2 > ... > kx, then a(n) = (1+e(k1))^(k1-k2) * (1+e(k1)+e(k2))^(k2-k3) * ... * (1+e(k1)+e(k2)+...+e(kx))^kx.
A000035(a(n)) = A000035(A000005(n)) = A010052(n).
(End)
Showing 1-2 of 2 results.

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