A319626 -id:A319626 - cp's OEIS frontend

A337376 Primorial deflation (numerator) of Doudna-tree.

1, 2, 3, 4, 5, 3, 9, 8, 7, 10, 5, 6, 25, 9, 27, 16, 11, 14, 21, 20, 7, 5, 15, 12, 49, 50, 25, 9, 125, 27, 81, 32, 13, 22, 33, 28, 55, 21, 63, 40, 11, 14, 7, 10, 35, 15, 45, 24, 121, 98, 147, 100, 49, 25, 25, 18, 343, 250, 125, 27, 625, 81, 243, 64, 17, 26, 39, 44, 65, 33, 99, 56, 91, 110, 55, 42, 275, 63, 189, 80, 13, 22

Offset: 0

Antti Karttunen, Michael De Vlieger and Peter Munn, Aug 25 2020

Tree with both numerators (this sequence) and denominators (A337377) shown starts as:
                                     1/1
                                      |
                                      2
                                      -
                                      1
                  3                  / \                  4
                  - .................   ................. -
                  2                                       1
        5        / \        3                   9        / \        8
        - .......   ....... -                   - .......   ....... -
        3                   1                   4                   1
       / \                 / \                 / \                 / \
      /   \               /   \               /   \               /   \
     /     \             /     \             /     \             /     \
    7       10          5       6          25       9          27       16
    -       --          -       -          --       -          --       --
    5        3          2       1           9       2           8        1
   / \      / \        / \     / \         / \     / \         / \      / \
  11 14   21  20      7   5   15 12      49  50   25  9     125  27   81  32
  -- --   --  --      -   -   -- --      --  --   --  -     ---  --   --  --
   7  5   10   3      3   1    4  1      25  9     6  1      27   4   16   1
etc.

Antti Karttunen, Table of n, a(n) for n = 0..8191
Antti Karttunen, Data supplement: n, a(n) computed for n = 0..65537
Index entries for fraction trees

A005940, A319626, A337375 are used in a formula defining this sequence.
Cf. A064989.
Cf. A337377 (denominators).
A000265, A001222, A003961, A007814, A337821 are used to express relationship between terms of this sequence.
Cf. also A329886, A346096.

Array[#1/GCD[#1, #2] & @@ {#, Apply[Times, Map[If[#1 <= 2, 1, NextPrime[#1, -1]]^#2 & @@ # &, FactorInteger[#]]]} &@ Function[p, Times @@ Flatten@ Table[Prime[Count[Flatten[#], 0] + 1]^#[[1, 1]] &@ Take[p, -i], {i, Length[p]}]]@ Partition[Split[Join[IntegerDigits[# - 1, 2], {2}]], 2] &, 82] (* Michael De Vlieger, Aug 27 2020 *)

A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); (t); };
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)));
A337376(n) = A319626(A005940(1+n));

a(n) = A319626(A005940(1+n)).
a(n) = A005940(1+n) / A337375(n).
a(2*n) = A003961(a(n)).
If A007814(n+1) < A337821(n+1) then a(2*n+1) = a(n), otherwise a(2*n+1) = 2 * a(n).
If A337377(n) mod 2 = 0 then a(2*n+1) = a(n), otherwise a(2*n+1) = 2 * a(n).
A000265(a(2*n+1)) = A000265(a(n)).
A001222(a(2*n)) = A001222(A337377(2*n)) = A001222(a(n)).
A001222(a(2*n+1)) - A001222(A337377(2*n+1)) = 1 + A001222(a(n)) - A001222(A337377(n)).

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)).

A346108 a(n) = A276085(A108951(A346096(n))), where A346096(n) gives the numerator of the primorial deflation of A276086(A108951(n)).

Original entry on oeis.org

1, 3, 9, 6, 39, 18, 249, 9, 39, 78, 2559, 36, 32589, 498, 234, 18, 543099, 78, 10242789, 156, 1494, 5118, 233335659, 57, 996, 65178, 258, 996, 6703028889, 405, 207263519019, 42, 15354, 1086198, 6612, 156, 7628001653829, 20485578, 195534, 249, 311878265181039, 2559, 13394639596851069, 10236, 1245, 466671318, 628284422185342479

Offset: 1

Antti Karttunen, Jul 08 2021

nonn

Cf. A002110, A064989, A108951, A276085, A276086, A324886, A319626, A346096, A346105, A346106, A346109.

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)));
A346096(n) = A319626(A324886(n));
A346106(n) = A108951(A346096(n));
A002110(n) = prod(i=1,n,prime(i));
A276085(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*A002110(primepi(f[k, 1])-1)); };
A346108(n) = A276085(A346106(n)); \\ Rest of program given in A324886.

a(n) = A346105(A346096(n)) = A276085(A346106(n)) = A276085(A108951(A346096(n))).

a(n) = A108951(n) + A346109(n).

A346106 a(n) = A108951(A346096(n)), where A346096(n) gives the numerator of the primorial deflation of A276086(A108951(n)).

Original entry on oeis.org

2, 6, 30, 36, 210, 900, 2310, 30, 210, 44100, 30030, 810000, 510510, 5336100, 85766121000000, 900, 9699690, 44100, 223092870, 1944810000, 151939915084881000000, 901800900, 6469693230, 189000, 28473963210000, 260620460100, 69300, 28473963210000, 200560490130, 4492511100000, 7420738134810, 1260, 733384949590939374729000000

Offset: 1

Antti Karttunen, Jul 08 2021

nonn

Cf. A064989, A108951, A319626, A324886, A346096, A346107, A346108.

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)));
A346096(n) = A319626(A324886(n));
A346106(n) = A108951(A346096(n)); \\ Rest of program in A324886.

a(n) = A108951(A346096(n)) = A108951(A319626(A324886(n))).

a(n) = A324886(n) * A346107(n).

A348994 a(n) = A003961(n) / gcd(n, A003961(n)), where A003961(n) is fully multiplicative with a(prime(k)) = prime(k+1).

Original entry on oeis.org

1, 3, 5, 9, 7, 5, 11, 27, 25, 21, 13, 15, 17, 33, 7, 81, 19, 25, 23, 63, 55, 39, 29, 45, 49, 51, 125, 99, 31, 7, 37, 243, 65, 57, 11, 25, 41, 69, 85, 189, 43, 55, 47, 117, 35, 87, 53, 135, 121, 147, 95, 153, 59, 125, 91, 297, 115, 93, 61, 21, 67, 111, 275, 729, 119, 65, 71, 171, 145, 33, 73, 75, 79, 123, 49, 207

Offset: 1

Antti Karttunen, Nov 10 2021

Numerator of ratio A003961(n) / n. This ratio is fully multiplicative, and a(n) / A348990(n) = A319626(A003961(n)) / A319627(A003961(n)) gives it in its lowest terms.

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences computed from indices in prime factorization

Cf. A003961, A319626, A319627, A348990 (denominators).

Array[#2/GCD[##] & @@ {#, If[# == 1, 1, Times @@ Map[NextPrime[#1]^#2 & @@ # &, FactorInteger[#]]]} &, 76] (* Michael De Vlieger, Nov 11 2021 *)

A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A348994(n) = (A003961(n) / gcd(n, A003961(n)));

a(n) = A003961(n) / gcd(n, A003961(n)).

a(n) = A319626(A003961(n)).

A349176 Odd numbers k for which gcd(k, A003961(k)) = gcd(sigma(k), A003961(k)) > 1, where A003961(n) is fully multiplicative with a(prime(k)) = prime(k+1), and sigma is the sum of divisors function.

Original entry on oeis.org

135, 285, 435, 455, 855, 885, 1185, 1287, 1305, 1335, 1425, 1435, 1485, 1635, 2235, 2275, 2295, 2655, 2685, 2905, 2985, 3105, 3135, 3185, 3311, 3395, 3435, 3555, 3585, 4005, 4035, 4185, 4425, 4785, 4865, 4905, 4995, 5385, 5685, 5805, 5835, 5845, 5925, 6135, 6237, 6335, 6345, 6585, 6675, 6735, 7125, 7155, 7175, 7185

Offset: 1

Antti Karttunen, Nov 11 2021

nonn

			For n = 135 = 3^3 * 5, sigma(135) = 240 = 2^4 * 3 * 5, A003961(135) = 5^3 * 7 = 875, and gcd(135,875) = gcd(240,875) = 5, which is larger than 1, therefore 135 is included in the sequence.

Intersection of A104210 and A349174, or equally, intersection of A349166 and A349174.
Subsequence of A372567.
Cf. A000203, A003961, A319626, A348994, A349161, A349164, A349169.

Select[Range[1, 7200, 2], And[#1/#2 == #1/#3, #2 > 1] & @@ {#3, GCD[#1, #3], GCD[#2, #3]} & @@ {#, DivisorSigma[1, #], Times @@ Map[NextPrime[#1]^#2 & @@ # &, FactorInteger[#]]} &] (* Michael De Vlieger, Nov 11 2021 *)

A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
isA349176(n) = if(!(n%2),0,my(u=A003961(n),t=gcd(u,n)); (t>1)&&(gcd(u,sigma(n))==t));

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))).

A348990 a(n) = n / gcd(n, A003961(n)), where A003961(n) is fully multiplicative function with a(prime(k)) = prime(k+1).

Original entry on oeis.org

1, 2, 3, 4, 5, 2, 7, 8, 9, 10, 11, 4, 13, 14, 3, 16, 17, 6, 19, 20, 21, 22, 23, 8, 25, 26, 27, 28, 29, 2, 31, 32, 33, 34, 5, 4, 37, 38, 39, 40, 41, 14, 43, 44, 9, 46, 47, 16, 49, 50, 51, 52, 53, 18, 55, 56, 57, 58, 59, 4, 61, 62, 63, 64, 65, 22, 67, 68, 69, 10, 71, 8, 73, 74, 15, 76, 7, 26, 79, 80, 81, 82, 83, 28

Offset: 1

Antti Karttunen, Nov 10 2021

Denominator of ratio A003961(n) / n. This ratio is fully multiplicative, and A348994(n) / a(n) = A319626(A003961(n)) / A319627(A003961(n)) gives it in its lowest terms.

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences computed from indices in prime factorization

Cf. A000035, A000961, A002110, A003961, A319626, A319627, A319630 (fixed points), A322361, A349169 (where equal to A348992).

Cf. A348994 (numerators).

Array[#1/GCD[##] & @@ {#, Times @@ Map[NextPrime[#1]^#2 & @@ # &, FactorInteger[#]]} &, 84] (* Michael De Vlieger, Nov 11 2021 *)

A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A348990(n) = (n/gcd(n, A003961(n)));

a(n) = n / A322361(n) = n / gcd(n, A003961(n)).
a(n) = A319627(A003961(n)).
For all odd numbers n, a(n) = A003961(A319627(n)).
For all n >= 1, A000035(A348990(n)) = A000035(n). [Preserves the parity]

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

A337376 Primorial deflation (numerator) of Doudna-tree.

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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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A346108 a(n) = A276085(A108951(A346096(n))), where A346096(n) gives the numerator of the primorial deflation of A276086(A108951(n)).

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A346106 a(n) = A108951(A346096(n)), where A346096(n) gives the numerator of the primorial deflation of A276086(A108951(n)).

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A348994 a(n) = A003961(n) / gcd(n, A003961(n)), where A003961(n) is fully multiplicative with a(prime(k)) = prime(k+1).

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A349176 Odd numbers k for which gcd(k, A003961(k)) = gcd(sigma(k), A003961(k)) > 1, where A003961(n) is fully multiplicative with a(prime(k)) = prime(k+1), and sigma is the sum of divisors function.

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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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A348990 a(n) = n / gcd(n, A003961(n)), where A003961(n) is fully multiplicative function with a(prime(k)) = prime(k+1).

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