A329900 -id:A329900 - cp's OEIS frontend

A330687 Positions of records in A050377, number of ways to factor n into "Fermi-Dirac primes" (A050376).

1, 4, 16, 64, 144, 256, 576, 1024, 1296, 2304, 5184, 9216, 20736, 82944, 186624, 331776, 746496, 1327104, 2073600, 2985984, 5308416, 8294400, 18662400, 21233664, 26873856, 33177600, 47775744, 51840000, 74649600, 107495424, 132710400, 207360000, 429981696, 530841600, 671846400, 829440000, 1194393600, 1719926784, 1866240000, 2687385600

Offset: 1

Antti Karttunen, Dec 28 2019

nonn

From David A. Corneth, Dec 29 2019: (Start)
Each term is a perfect square. Proof: A050377(n) is multiplicative with a(p^e) = A018819(e) and A018819(2k) = A018819(2k+1) and this sequence considers just records so we only need exponents of the form 2k; i.e., terms are squares.
Furthermore, the exponent 2 occurs at most once in the prime factorization of a(n) as A018819(2)^2 = A018819(4) = 4. So if the last two exponents in the prime factorization of m are 2's then setting the first of those two exponents to 4 and the other to 0 gives the same A050377(m).
Example of an application of this proof: we have 3600 = 2^4 * 3^2 * 5^2. We see the last two exponents are 2's so we can set the first of those two to 4 and the second to 0. This gives 2^4 * 3^4 = 1296 and, indeed, A050377(1296) = A050377(3600) = 16.
It seems that most exponents of a(n) are divisible by 4.
More specifically: Let S(n) be the list, possibly with duplicates, of exponents occurring in the prime factorizations of terms with the sum of exponents in the prime factorization <= n.
Let R(n) = |{x : x==4, S(n)}| / |S(n)|.
For example, S(8) is found from the following terms: 4, 16, 64, 144, 256, 576 and 1296 as the exponents in the prime factorization are (2), (4), (6), (4, 2), (8), (6, 2), (4, 4). The sums of each of these exponents per term is <= 8. There are 10 exponents listed. Of these 10 there are 5 that are divisible by 4. Therefore R(8) = 5/10.
Then it seems that R(n) tends to some value > 0.8 as n grows. (End)

David A. Corneth, Table of n, a(n) for n = 1..1004 (terms <= 10^100; first 294 terms from Antti Karttunen)

Cf. A018819, A050376, A050377, A329900, A330684 (square roots), A330688 (the record values), A330689 (primorial deflation).

Subsequence of A025487.

Block[{s = Rest@ Nest[Function[{a, n, b}, Append[a, {Times @@ Map[a[[# + 1, -1]] &, FactorInteger[n][[All, -1]] ], b}]] @@ {#1, #2, #1[[-1, -1]] + If[EvenQ@ #2, #1[[#2/2 + 1, -1]], 0 ]} & @@ {#, Length@ #} &, {{0, 1}, {1, 1}}, 10^5][[All, 1]], t}, t = Union@ FoldList[Max, s]; Map[FirstPosition[s, #][[1]] &, t]] (* Michael De Vlieger, Dec 29 2019 *)

upto_e = 101; \\ 101 --> 211 terms.
A018819(n) = if( n<1, n==0, if( n%2, A018819(n-1), A018819(n/2)+A018819(n-1))); \\ From A018819
v018819 = vector(upto_e,n,A018819(n)); \\ Precompute.
A050377(n) = factorback(apply(e -> v018819[e], factor(n)[, 2]));
A283980(n) = {my(f=factor(n)); prod(i=1, #f~, my(p=f[i, 1], e=f[i, 2]); if(p==2, 6, nextprime(p+1))^e)}; \\ From A283980
A330687list(e) = { my(lista = List([1, 2]), i=2, u = 2^e, t, m=0, v025487); while(lista[i] != u, if(2*lista[i] <= u, listput(lista,2*lista[i]); t = A283980(lista[i]); if(t <= u, listput(lista,t))); i++); v025487 = vecsort(Vec(lista)); lista = List([]); for(i=1,#v025487,if((t=A050377(v025487[i]))>m, listput(lista,v025487[i]); m=t)); Vec(lista); };
v330687 = A330687list(upto_e);
A330687(n) = v330687[n];
for(n=1,#v330687,print1(A330687(n),", "));

A050377(a(n)) = A330688(n).
A329900(a(n)) = A330689(n).
a(n) = A330684(n)^2.

A307035 a(n) is the unique integer k such that A108951(k) = n!.

Original entry on oeis.org

1, 1, 2, 3, 12, 20, 60, 84, 672, 1512, 5040, 7920, 47520, 56160, 157248, 393120, 6289920, 8225280, 37013760, 41368320, 275788800, 579156480, 1820206080, 2203407360, 26440888320, 73446912000, 173601792000, 585906048000, 3281073868800, 4137006182400, 20685030912000

Offset: 0

Allan C. Wechsler, Mar 20 2019

For all n, n! = A108951(k) for some unique k. This sequence gives that k for each n. In some sense this sequence tells how to factor factorials into primorials.

Represent n! as a product of primorials p#. Then replace each primorial with its base prime to calculate a(n).

			Represent 7! as a product of primorials:
7! = 2^4 * 3^2 * 5 * 7 = (2#)^2 * 3# * 7#
Replace primorials by primes:
2^2 * 3 * 7 = 84.
So a(7) = 84.

Alois P. Heinz, Table of n, a(n) for n = 0..1911 (terms n = 1..1000 from Vaclav Kotesovec)

Cf. A000142, A108951, A025487, A122111, A319626, A319627, A325508, A329900.

f:= proc(n) option remember; `if`(n<2, 0, f(n-1)+add(
      i[2]*x^numtheory[pi](i[1]), i=ifactors(n)[2]))
    end:
a:= proc(n) local d, p, r; p, r:= f(n), 1;
      do d:= degree(p); if d<1 then break fi;
         p, r:= p-add(x^i, i=1..d), ithprime(d)*r
      od: r
    end:
seq(a(n), n=0..35);  # Alois P. Heinz, Mar 21 2019

q[n_] := Apply[Times, Table[Prime[i], {i, 1, PrimePi[n]}]]; Flatten[{1, 1, Table[val = 1; fak = n!; Do[If[PrimeQ[k], Do[If[Divisible[fak, q[k]], val = val*k; fak = fak/q[k]], {j, 1, n}]], {k, n, 2, -1}]; val, {n, 2, 30}]}] (* Vaclav Kotesovec, Mar 21 2019 *)

g(n) = my(f=factor(n)); prod(k=1, #f~, my(p=f[k, 1]); (p/if(p>2, precprime(p-1), 1))^f[k, 2]); \\ A319626/A319627
a(n) = prod(k=1, n, g(k)); \\ Daniel Suteu, Mar 21 2019

A307035(n) = { my(m=1, pp=1); n=n!; while(1, forprime(p=2, ,if(n%p, if(2==p, return(m), break), n /= p; pp = p)); m *= pp); }; \\ Antti Karttunen, Dec 29 2019

a(0) = 1, a(n) = a(n-1) * (A319626(n) / A319627(n)), for n > 0. - Daniel Suteu, Mar 21 2019
a(n) = n! / Product_{k=1..n} A064989(k). - Vaclav Kotesovec, Mar 21 2019
a(n) = A122111(A325508(n)) = A319626(A000142(n)) = A329900(A000142(n)). - Antti Karttunen, Nov 19 & Dec 29 2019

a(12)-a(13) from Michel Marcus, Mar 21 2019
a(14)-a(15) from Vaclav Kotesovec, Mar 21 2019
a(0), a(16)-a(30) from Alois P. Heinz, Mar 21 2019

A342012 Primorial deflation of the n-th colossally abundant number: the unique integer k such that A108951(k) = A004490(n).

Original entry on oeis.org

2, 3, 6, 10, 20, 30, 42, 84, 132, 156, 312, 468, 780, 1020, 1140, 1380, 2760, 3480, 3720, 5208, 7812, 9324, 10332, 10836, 21672, 23688, 26712, 29736, 49560, 51240, 56280, 59640, 61320, 96360, 104280, 208560, 219120, 328680, 352440, 384120, 453960, 472680, 482040, 500760, 510120, 528840, 594360, 613080, 641160, 650520, 1301040

Offset: 1

Antti Karttunen, Mar 08 2021

nonn

In contrast to A329902, this sequence is monotonic, because each term is obtained from the previous, either by multiplying it by 2, or by "bumping" one [or hypothetically: two] of its prime factors one step up (i.e., replacing it with the next larger prime), and both operations are guaranteed to make the number larger.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A004490, A005940, A073751, A108951, A319626, A329900, A342010 [= A001222(a(n))], A342011, A342013.

Cf. also A217867, A329902.

v073751 = readvec("b073751_to.txt");
A073751(n) = v073751[n];
A004490list(v073751) = { my(v=vector(#v073751)); v[1] = 2; for(n=2,#v,v[n] = v073751[n]*v[n-1]); (v); };
v004490 = A004490list(v073751);
A004490(n) = v004490[n];
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A319626(n) = (n / gcd(n, A064989(n)));
A342012(n) = A319626(A004490(n));

a(n) = A319626(A004490(n)) = A329900(A004490(n)).

a(n) = A005940(1+A342013(n)).

A329899 If A181815(n) is odd, a(n) = A064989(A025487(n)), otherwise a(n) = A025487(n)/2.

Original entry on oeis.org

1, 1, 2, 2, 4, 6, 8, 12, 6, 16, 4, 24, 30, 32, 36, 48, 60, 64, 72, 12, 96, 30, 8, 120, 128, 144, 180, 192, 210, 216, 240, 256, 288, 360, 384, 420, 432, 36, 480, 512, 24, 576, 60, 16, 720, 768, 840, 864, 900, 960, 1024, 1080, 1152, 210, 1260, 1296, 1440, 1536, 1680, 1728, 1800, 1920, 2048, 2160, 2304, 2310, 2520, 2592, 72

Offset: 1

Antti Karttunen, Dec 24 2019

nonn

If 2-adic and 3-adic valuations of A025487(n) are equal, then a(n) = A064989(A025487(n)), otherwise a(n) = A025487(n)/2.

Only terms of A025487 occur, and each one of them occurs exactly twice.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A025487, A064989, A181815, A329897, A329898, A329904.

A181815(n) = A329900(A025487(n));
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A329899(n) = if(A181815(n)%2,A064989(A025487(n)),A025487(n)/2);

If A181815(n) is odd, a(n) = A064989(A025487(n)), otherwise a(n) = A025487(n)/2.

a(n) = A025487(A329904(n)).

A330686 Primorial deflation of (nonzero) K-champion numbers: a(n) is the unique integer x such that A108951(x) = A307866(1+n).

Original entry on oeis.org

1, 4, 3, 8, 6, 12, 9, 24, 18, 48, 20, 36, 96, 40, 72, 30, 54, 80, 144, 60, 108, 160, 288, 120, 216, 320, 90, 576, 240, 432, 180, 480, 864, 360, 960, 720, 1920, 540, 252, 1440, 3840, 1080, 504, 2880, 1200, 2160, 1008, 5760, 2688, 2400, 4320, 2016, 11520, 3240, 4800, 1512, 8640, 4032, 23040, 1680, 6480, 9600, 3024, 17280, 8064, 7200

Offset: 1

Antti Karttunen, Dec 28 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..761 (computed from the b-file of A307866)

Cf. A074206, A181815, A307866, A329900, A353568.

a(n) = A329900(A307866(1+n)).

a(n) = A122111(A353568(n)). - Antti Karttunen, May 20 2022

A330689 Primorial deflation of A330687 (record positions in A050377): a(n) is the unique integer x such that A108951(x) = A330687(n).

Original entry on oeis.org

1, 4, 16, 64, 36, 256, 144, 1024, 81, 576, 324, 2304, 1296, 5184, 2916, 20736, 11664, 82944, 14400, 46656, 331776, 57600, 32400, 1327104, 104976, 230400, 746496, 40000, 129600, 419904, 921600, 160000, 1679616, 3686400, 291600, 640000, 2073600, 6718464, 360000, 1166400, 2560000, 26873856, 1440000, 313600, 4665600, 10240000, 810000

Offset: 1

Antti Karttunen, Dec 28 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..294

Cf. A050376, A050377, A108951, A329900, A330687.

PARI
```
A330689(n) = A329900(A330687(n));
```

a(n) = A329900(A330687(n)).

A337478 Primorial deflation of A336389.

Original entry on oeis.org

1, 3, 20, 38, 159, 749, 1337

Offset: 0

Antti Karttunen, Aug 29 2020

Cf. A319626, A329900, A336389, A337474, A337476.

Cf. also A337477.

a(n) = A319626(A336389(n)) = A329900(A336389(n)).

For all n >= 0, A337474(a(n)) >= n and a(n) >= A337476(n).

A342013 Position of the n-th colossally abundant number in A329886, the primorial inflation of Doudna-tree.

Original entry on oeis.org

1, 2, 5, 9, 19, 21, 37, 75, 139, 267, 535, 539, 555, 1067, 2091, 4139, 8279, 16471, 32855, 32919, 32923, 65691, 131227, 262299, 524599, 1048887, 2097463, 4194615, 4194647, 8388951, 16777559, 33554775, 67109207, 67109463, 134218327, 268436655, 536872111, 536872119, 1073743031, 2147484855, 2147485879, 4294969527, 8589936823

Offset: 1

Antti Karttunen and Michael De Vlieger, Mar 08 2021

nonn

a(n) is the unique integer k such that A329886(k) = A004490(n).
Like A342012, also this sequence is monotonic. Proof: the doubling step corresponds here to step *2 + 1, and "bumping up" some of the prime factors likewise results a larger A156552-code, thus both steps keep the result growing.
The binary length of these numbers (A070939, = 1+A000523) grows by 0 or 1 at each step, thus the next colossally abundant number is always found on either on the same row (right of the current CA-number), or the next row of A329886, the row immediately below. The next CA-number will be on the same row only when its factorization contains neither a new prime nor yet another instance of prime 2.

Cf. A004490, A005940, A073751, A108951, A156552, A319626, A329886, A329900, A342000, A342010 (the binary weight), A342011, A342012.

A156552(n) = {my(f = factor(n), p, p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res};
A342013(n) = A156552(A342012(n)); \\ Uses also code from A342012.

a(n) = A156552(A342012(n)) = A156552(A319626(A004490(n))).

A329887 a(0) = 1, a(1) = 2; for n > 1, if n is even, then a(n) = 2*a(n/2), and if n is odd, a(n) = A283980(a((n-1)/2)).

Original entry on oeis.org

1, 2, 4, 6, 8, 36, 12, 30, 16, 216, 72, 900, 24, 180, 60, 210, 32, 1296, 432, 27000, 144, 5400, 1800, 44100, 48, 1080, 360, 6300, 120, 1260, 420, 2310, 64, 7776, 2592, 810000, 864, 162000, 54000, 9261000, 288, 32400, 10800, 1323000, 3600, 264600, 88200, 5336100, 96, 6480, 2160, 189000, 720, 37800, 12600, 485100, 240

Offset: 0

Antti Karttunen, Dec 24 2019

nonn

			This irregular table can be represented as a binary tree. Each child to the left is obtained by doubling the parent, and each child to the right is obtained by applying A283980 to the parent:
                                     1
                                     |
                  ...................2...................
                 4                                       6
       8......../ \........36                 12......../ \........30
      / \                 / \                 / \                 / \
     /   \               /   \               /   \               /   \
    /     \             /     \             /     \             /     \
  16      216         72      900         24      180         60       210
etc.
A329886 is the mirror image of the same tree.

Antti Karttunen, Table of n, a(n) for n = 0..8192

Permutation of A025487.
Cf. A054429, A108951, A163511, A283980, A329900.
Cf. also A322827, A329886.

{1}~Join~Nest[Append[#1, If[EvenQ@ #2, 2 #1[[#2/2]], (Times @@ Map[#1^#2 & @@ # &, FactorInteger[#] /. {p_, e_} /; e > 0 :> {Prime[PrimePi@ p + 1], e}] - Boole[# == 1])*2^IntegerExponent[#, 2] &[#1[[(#2 - 1)/2]] ]]] & @@ {#, Length@ # + 1} &, {2}, 55] (* Michael De Vlieger, Dec 29 2019 *)

A283980(n) = {my(f=factor(n)); prod(i=1, #f~, my(p=f[i, 1], e=f[i, 2]); if(p==2, 6, nextprime(p+1))^e)}; \\ From A283980
A329887(n) = if(n<2,1+n,if(n%2,A283980(A329887(n\2)),2*A329887(n/2)));

a(0) = 1, a(1) = 2; for n > 1, if n is odd, a(n) = A283980(a((n-1)/2)), and if n is even, then a(n) = 2*a(n/2).
a(n) = A108951(A163511(n)).
a(2^n) = 2^(1+n). [And all the terms following after a(2^n) are > 2^(1+n).]
For n >= 1, a(n) = A329886(A054429(n)).

A330743 a(n) is the first term k of A329902 for which A056239(k) = n.

Original entry on oeis.org

1, 2, 4, 6, 12, 24, 40, 60, 84, 168, 336, 528, 792, 936, 1872, 2448, 3060, 4560, 4788, 8280, 15456, 23184, 29232, 31248, 62496, 74592, 124320, 137760, 144480, 157920, 315840, 356160, 559680, 623040, 644160, 966240, 1061280, 1124640, 1686960, 1734480, 2049840, 2218320, 2330640, 2499120, 4165200, 4539600, 4726800, 4820400

Offset: 0

Antti Karttunen, Jan 13 2020

nonn

Note that in contrast to A330744 this is not monotonic. The first point where a(n) > a(n+1) occurs is at a(120) = 5481774144 > a(121) = 5452302240. See also comment in A328521, whose primorial deflation this sequence is.

a(n-1) differs from A330744(n) at n = 17, 19, 21, 51, 52, 55, 56, 57, 58, 59, 60, 61, ...

Antti Karttunen, Table of n, a(n) for n = 0..4257

Primorial deflation of A328521.
Cf. A056239, A319626, A329900, A329902, A330748.
Cf. also A330744.

A330743(n) = { for(k=1,oo,if(A056239(A329902(k))==n,return(A329902(k)))); };

v329902 = readvec("a329902.txt"); \\ File for the first 779674 terms of A329902 as prepared by Michael De Vlieger.
A056239(n) = if(1==n,0,my(f=factor(n)); sum(i=1, #f~, f[i,2] * primepi(f[i,1])));
A330743list() = { my(m=Map(), lista=List([]), t); for(i=1, #v329902, t = A056239(v329902[i]); if(!mapisdefined(m,t), mapput(m,t,v329902[i]))); for(n=0,oo,if(mapisdefined(m,n,&t), listput(lista,t), return(Vec(lista)))); };
v330743 = A330743list();
A330743(n) = v330743[1+n];
for(n=0,#v330743-1,write("b330743.txt", n, " ", A330743(n)));

a(n) = A329902(min{i: A056239(A329902(i))==n}).
a(n) = A329902(A330748(n)).
a(n) = A329900(A328521(n)) = A319626(A328521(n)).

cp's OEIS Frontend

A330687 Positions of records in A050377, number of ways to factor n into "Fermi-Dirac primes" (A050376).

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A307035 a(n) is the unique integer k such that A108951(k) = n!.

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A342012 Primorial deflation of the n-th colossally abundant number: the unique integer k such that A108951(k) = A004490(n).

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A329899 If A181815(n) is odd, a(n) = A064989(A025487(n)), otherwise a(n) = A025487(n)/2.

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A330686 Primorial deflation of (nonzero) K-champion numbers: a(n) is the unique integer x such that A108951(x) = A307866(1+n).

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A330689 Primorial deflation of A330687 (record positions in A050377): a(n) is the unique integer x such that A108951(x) = A330687(n).

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A337478 Primorial deflation of A336389.

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A342013 Position of the n-th colossally abundant number in A329886, the primorial inflation of Doudna-tree.

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