A346096 -id:A346096 - cp's OEIS frontend

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

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).

A346098 a(n) = A064989(A346096(n)) = A064989(A319626(A324886(n))).

Original entry on oeis.org

1, 2, 3, 4, 5, 9, 7, 3, 5, 25, 11, 81, 13, 49, 15625, 9, 17, 25, 19, 625, 117649, 121, 23, 45, 2401, 169, 21, 2401, 29, 4375, 31, 10, 1771561, 289, 14235529, 625, 37, 361, 4826809, 7, 41, 11, 43, 14641, 16807, 529, 47, 125, 2093663, 5764801, 24137569, 28561, 53, 441, 20449, 5764801, 47045881, 841, 59, 343, 61, 961, 1331, 100, 396067447082177

Offset: 1

Antti Karttunen, Jul 07 2021

nonn

Cf. A064989, A319626, A324886, A346095, A346096, A346097, A346099 [= gcd(n, a(n))].

A064989(n) = { my(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)));
A346098(n) = A064989(A319626(A324886(n))); \\ Rest of program given in A324886.

a(n) = A064989(A346096(n)) = A064989(A319626(A324886(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).

A346097 Denominator of the primorial deflation of A276086(A108951(n)): a(n) = A319627(A324886(n)).

Original entry on oeis.org

1, 2, 3, 4, 5, 9, 7, 2, 3, 25, 11, 81, 13, 49, 15625, 4, 17, 9, 19, 625, 117649, 121, 23, 27, 1225, 169, 21, 2401, 29, 3125, 31, 10, 1771561, 289, 5764801, 81, 37, 361, 4826809, 5, 41, 7, 43, 14641, 12005, 529, 47, 75, 1127357, 1500625, 24137569, 28561, 53, 441, 14641, 5764801, 47045881, 841, 59, 125, 61, 961, 343, 100, 302875106592253

Offset: 1

Antti Karttunen, Jul 07 2021

Denominator of ratio A324886(n) / A329044(n).

Cf. A346096 (numerators).
Cf. A006530, A020639, A108951, A276086, A319627, A324886, A329044, A345941 [= gcd(n, a(n))], A346095, A346098, A346107, A346109.
Cf. also A337377.

A064989(n) = { my(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); };
A319627(n) = (A064989(n) / gcd(n, A064989(n)));
A346097(n) = A319627(A324886(n)); \\ Rest of program given in A324886.

a(n) = A319627(A324886(n)).
a(n) = A329044(n) / A346095(n) = A329044(n) / gcd(A324886(n), A329044(n)).
A020639(a(n)) = A006530(n).
A108951(a(n)) = A346107(n).
A346105(a(n)) = A346109(n).

A337376 Primorial deflation (numerator) of Doudna-tree.

Original entry on oeis.org

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)).

A346095 a(n) = gcd(A324886(n), A064989(A324886(n))).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 3, 5, 1, 1, 1, 1, 1, 1, 9, 1, 25, 1, 1, 1, 1, 1, 5, 49, 1, 1, 1, 1, 7, 1, 1, 1, 1, 121, 625, 1, 1, 1, 7, 1, 11, 1, 1, 7, 1, 1, 5, 143, 2401, 1, 1, 1, 1, 169, 1, 1, 1, 1, 343, 1, 1, 1331, 1, 17, 1, 1, 1, 1, 161051, 1, 175, 1, 1, 41503, 1, 169, 1, 1, 49, 35, 1, 1, 121, 19, 1, 1, 1, 1, 49, 24137569

Offset: 1

Antti Karttunen, Jul 07 2021

nonn

Cf. A006530, A324886, A329044, A330749, A346096, A346097.

A330749(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)); gcd(n, factorback(f)); };
A346095(n) = A330749(A324886(n)); \\ Rest of program given in A324886.

a(n) = A330749(A324886(n)) = gcd(A324886(n), A329044(n)) = gcd(A324886(n), A064989(A324886(n))).
a(n) = A324886(n) / A346096(n).
a(n) = A329044(n) / A346097(n).
a(n) mod A006530(n) > 0, for all n > 1.

A346099 a(n) = gcd(n, A346098(n)).

Original entry on oeis.org

1, 2, 3, 4, 5, 3, 7, 1, 1, 5, 11, 3, 13, 7, 5, 1, 17, 1, 19, 5, 7, 11, 23, 3, 1, 13, 3, 7, 29, 5, 31, 2, 11, 17, 7, 1, 37, 19, 13, 1, 41, 1, 43, 11, 1, 23, 47, 1, 1, 1, 17, 13, 53, 9, 11, 7, 19, 29, 59, 1, 61, 31, 1, 4, 13, 11, 67, 17, 23, 1, 71, 3, 73, 37, 25, 19, 1, 13, 79, 1, 1, 41, 83, 1, 17, 43, 29, 11, 89

Offset: 1

Antti Karttunen, Jul 07 2021

nonn

Only powers of primes (A000961) occur as terms. A346100 lists the exponents.

Cf. A000961, A064989, A319626, A324886, A346095, A346096, A346097, A346098, A346100.

Cf. A346090 (positions of ones).

A064989(n) = { my(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)));
A346099(n) = gcd(n, A064989(A319626(A324886(n)))); \\ Rest of program in A324886.

a(n) = gcd(n, A346098(n)) = gcd(n, A064989(A319626(A324886(n)))).

A346100 a(n) = A100995(gcd(n, A064989(A319626(A324886(n))))).

Original entry on oeis.org

0, 1, 1, 2, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 0, 0, 1, 1, 1, 2, 1, 1, 1, 1, 1, 0, 1, 1, 0, 2, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 2, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 3

Offset: 1

Antti Karttunen, Jul 07 2021

nonn

Cf. A064989, A100995, A319626, A324886, A346087, A346095, A346096, A346098, A346099.

Cf. A346090 (positions of 0's).

A064989(n) = { my(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)));
A346100(n) = isprimepower(gcd(n, A064989(A319626(A324886(n))))); \\ Rest of program given in A324886.

a(n) = A100995(A346099(n)) = A100995(gcd(n, A064989(A319626(A324886(n))))).

cp's OEIS Frontend

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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A346098 a(n) = A064989(A346096(n)) = A064989(A319626(A324886(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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A346097 Denominator of the primorial deflation of A276086(A108951(n)): a(n) = A319627(A324886(n)).

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A337376 Primorial deflation (numerator) of Doudna-tree.

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Mathematica

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A346095 a(n) = gcd(A324886(n), A064989(A324886(n))).

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A346099 a(n) = gcd(n, A346098(n)).

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A346100 a(n) = A100995(gcd(n, A064989(A319626(A324886(n))))).

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