A244417 -id:A244417 - cp's OEIS frontend

A322316 Lexicographically earliest such sequence a that a(i) = a(j) => A122841(i) = A122841(j) and A244417(i) = A244417(j), for all i, j.

1, 2, 2, 3, 1, 4, 1, 5, 3, 2, 1, 6, 1, 2, 2, 7, 1, 6, 1, 3, 2, 2, 1, 8, 1, 2, 5, 3, 1, 4, 1, 9, 2, 2, 1, 10, 1, 2, 2, 5, 1, 4, 1, 3, 3, 2, 1, 11, 1, 2, 2, 3, 1, 8, 1, 5, 2, 2, 1, 6, 1, 2, 3, 12, 1, 4, 1, 3, 2, 2, 1, 13, 1, 2, 2, 3, 1, 4, 1, 7, 7, 2, 1, 6, 1, 2, 2, 5, 1, 6, 1, 3, 2, 2, 1, 14, 1, 2, 3, 3, 1, 4, 1, 5, 2

Offset: 1

Antti Karttunen, Dec 04 2018

nonn

Restricted growth sequence transform of the ordered pair [A122841(n), A244417(n)].
Essentially also the restricted growth sequence transform of the unordered pair {A007814(n), A007949(n)}.
For all i, j: a(i) = a(j) => A072078(i) = A072078(j).

Antti Karttunen, Table of n, a(n) for n = 1..19683
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A007814, A007949, A122841, A244417, A322026, A322317 (ordinal transform).

up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A007814(n) = valuation(n,2);
A007949(n) = valuation(n,3);
A122841(n) = min(A007814(n), A007949(n));
A244417(n) = max(valuation(n,2), valuation(n,3));
v322316 = rgs_transform(vector(up_to, n, [A122841(n), A244417(n)]));
\\ The following is equivalent:
\\ v322316 = rgs_transform(vector(up_to, n, Set([A007814(n), A007949(n)])));
A322316(n) = v322316[n];

A054707 Number of powers of 6 modulo n.

Original entry on oeis.org

1, 2, 2, 3, 1, 2, 2, 4, 3, 2, 10, 3, 12, 3, 2, 5, 16, 3, 9, 3, 3, 11, 11, 4, 5, 13, 4, 4, 14, 2, 6, 6, 11, 17, 2, 3, 4, 10, 13, 4, 40, 3, 3, 12, 3, 12, 23, 5, 14, 6, 17, 14, 26, 4, 10, 5, 10, 15, 58, 3, 60, 7, 4, 7, 12, 11, 33, 18, 12, 3, 35, 4, 36, 5, 6, 11, 10, 13, 78, 5, 5, 41, 82, 4, 16

Offset: 1

Henry Bottomley, Apr 20 2000

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from David W. Wilson)

Cf. A007737, A244417.

Cf. A054703 (base 2), A054704 (3), A054705 (4), A054706 (5), A054708 (7), A054709 (8), A054717 (9), A054710 (10), A351524 (11), A054712 (12), A054713 (13), A054714 (14), A054715 (15), A054716 (16).

a[n_] := Module[{e = IntegerExponent[n, {2, 3}]}, Max[e] + MultiplicativeOrder[6, n/Times @@ ({2, 3}^e)]]; Array[a, 100] (* Amiram Eldar, Aug 25 2024 *)

a(n) = A007737(n) + A244417(n). - Amiram Eldar, Aug 25 2024

A322026 Lexicographically earliest infinite sequence such that a(i) = a(j) => A007814(i) = A007814(j) and A007949(i) = A007949(j), for all i, j, where A007814 and A007949 give the 2- and 3-adic valuations of n.

Original entry on oeis.org

1, 2, 3, 4, 1, 5, 1, 6, 7, 2, 1, 8, 1, 2, 3, 9, 1, 10, 1, 4, 3, 2, 1, 11, 1, 2, 12, 4, 1, 5, 1, 13, 3, 2, 1, 14, 1, 2, 3, 6, 1, 5, 1, 4, 7, 2, 1, 15, 1, 2, 3, 4, 1, 16, 1, 6, 3, 2, 1, 8, 1, 2, 7, 17, 1, 5, 1, 4, 3, 2, 1, 18, 1, 2, 3, 4, 1, 5, 1, 9, 19, 2, 1, 8, 1, 2, 3, 6, 1, 10, 1, 4, 3, 2, 1, 20, 1, 2, 7, 4, 1, 5, 1, 6, 3

Offset: 1

Antti Karttunen, Dec 03 2018

nonn

Restricted growth sequence transform of the ordered pair [A007814(n), A007949(n)].
For all i, j:
  A305900(i) = A305900(j) => a(i) = a(j),
  a(i) = a(j) => A122841(i) = A122841(j),
  a(i) = a(j) => A244417(i) = A244417(j),
  a(i) = a(j) => A322316(i) = A322316(j) => A072078(i) = A072078(j).
If and only if a(k) > a(i) for all k > i then k is in A003586, - David A. Corneth, Dec 03 2018
That is, A003586 gives the positions of records (1, 2, 3, 4, 5, ...) in this sequence.
Sequence A126760 (without its initial zero) and this sequence are ordinal transforms of each other.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Jon Maiga, Computer-generated formulas for A322026, Sequence Machine.

Cf. A003586 (positions of records, the first occurrence of n), A007814, A007949, A065331, A071521, A072078, A087465, A122841, A126760 (ordinal transform), A322316, A323883, A323884.

Cf. also A247714 and A255975.

up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A007814(n) = valuation(n,2);
A007949(n) = valuation(n,3);
v322026 = rgs_transform(vector(up_to, n, [A007814(n), A007949(n)]));
A322026(n) = v322026[n];

A065331(n) = (3^valuation(n, 3)<A065331
A071521(n) = { my(t=1/3); sum(k=0, logint(n, 3), t*=3; logint(n\t, 2)+1); }; \\ From A071521.
A322026(n) = A071521(A065331(n)); \\ Antti Karttunen, Sep 08 2024

For s = A003586(n), a(s) = n = a((6k+1)*s) = a((6k-1)*s), where s is the n-th 3-smooth number and k > 0. - David A. Corneth, Dec 03 2018
A065331(n) = A003586(a(n)). - David A. Corneth, Dec 04 2018
From Antti Karttunen, Sep 08 2024: (Start)
a(n) = Sum{k=1..n} [A126760(k)==A126760(n)], where [ ] is the Iverson bracket.
a(n) = A071521(A065331(n)). [Found by Sequence Machine and also by LODA miner]
a(n) = A323884(25*n). [Conjectured by Sequence Machine]
(End)

A375538 Numerator of the asymptotic mean over the positive integers of the maximum exponent in the prime factorization of the largest prime(n)-smooth divisor function.

Original entry on oeis.org

1, 13, 51227, 926908275845, 548123689541583443758024333411, 629375533747930240763697631488051776709110194920714685268467462860005271344878614119

Offset: 1

Amiram Eldar, Aug 19 2024

The numbers of digits of the terms are 1, 2, 5, 12, 30, 84, 215, 537, 1237, 2930, 6775, 15484, 35185, ... .

			Fractions begins: 1, 13/10, 51227/36540, 926908275845/636617813832, 548123689541583443758024333411/369693143251781030056182487680, ...
For n = 1, prime(1) = 2, the "2-smooth numbers" are the powers of 2 (A000079), and the sequence that gives the exponent of the largest power of 2 that divides n is A007814, whose asymptotic mean is 1.
For n = 2, prime(2) = 3, the 3-smooth numbers are in A003586, and the sequence that gives the maximum exponent in the prime factorization of the largest 3-smooth divisor of n is A244417, whose asymptotic mean is 13/10.

Cf. A033150, A375537, A375539 (denominators).
Cf. A375538 (numerators).
Cf. A000079, A003586, A007814, A244417, A375536.

d[k_, n_] := Product[1 - 1/Prime[i]^k, {i, 1, n}]; f[n_] := Sum[k * (d[k+1, n] - d[k, n]), {k, 1, Infinity}]; Numerator[Array[f, 6]]

Let f(n) = a(n)/A375539(n). Then:
f(n) = lim_{m->oo} (1/m) * Sum_{i=1..m} A375537(n, i).
f(n) = Sum_{k>=1} k * (d(k+1, prime(n)) - d(k, prime(n))), where d(k, p) = Product_{q prime <= p} (1 - 1/q^k).
Limit_{n->oo} f(n) = A033150.

A322317 Ordinal transform of A322316.

Original entry on oeis.org

1, 1, 2, 1, 2, 1, 3, 1, 2, 3, 4, 1, 5, 4, 5, 1, 6, 2, 7, 3, 6, 7, 8, 1, 9, 8, 2, 4, 10, 2, 11, 1, 9, 10, 12, 1, 13, 11, 12, 3, 14, 3, 15, 5, 6, 13, 16, 1, 17, 14, 15, 7, 18, 2, 19, 4, 16, 17, 20, 3, 21, 18, 8, 1, 22, 4, 23, 9, 19, 20, 24, 1, 25, 21, 22, 10, 26, 5, 27, 2, 3, 23, 28, 4, 29, 24, 25, 5, 30, 5, 31, 11, 26, 27, 32, 1

Offset: 1

Antti Karttunen, Dec 04 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..19683
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A007814, A007949, A122841, A244417, A322316.

Cf. also A126760.

up_to = 65537;
ordinal_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), pt); for(i=1, length(invec), if(mapisdefined(om,invec[i]), pt = mapget(om, invec[i]), pt = 0); outvec[i] = (1+pt); mapput(om,invec[i],(1+pt))); outvec; };
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A007814(n) = valuation(n,2);
A007949(n) = valuation(n,3);
A122841(n) = min(A007814(n), A007949(n));
A244417(n) = max(valuation(n,2), valuation(n,3));
v322316 = rgs_transform(vector(up_to, n, [A122841(n), A244417(n)]));
v322317 = ordinal_transform(v322316);
A322317(n) = v322317[n];

A375536 The maximum exponent in the prime factorization of the largest 5-smooth divisor of n.

Original entry on oeis.org

0, 1, 1, 2, 1, 1, 0, 3, 2, 1, 0, 2, 0, 1, 1, 4, 0, 2, 0, 2, 1, 1, 0, 3, 2, 1, 3, 2, 0, 1, 0, 5, 1, 1, 1, 2, 0, 1, 1, 3, 0, 1, 0, 2, 2, 1, 0, 4, 0, 2, 1, 2, 0, 3, 1, 3, 1, 1, 0, 2, 0, 1, 2, 6, 1, 1, 0, 2, 1, 1, 0, 3, 0, 1, 2, 2, 0, 1, 0, 4, 4, 1, 0, 2, 1, 1, 1, 3, 0, 2, 0, 2, 1, 1, 1, 5, 0, 1, 2, 2, 0, 1, 0, 3, 1

Offset: 1

Amiram Eldar, Aug 19 2024

Cf. A007775, A051903, A244417 (3-smooth analog), A355582, A375537, A375538, A375539.

Cf. A007814, A007949, A112765.

a[n_] := Max[IntegerExponent[n, {2, 3, 5}]]; Array[a, 100]

a(n) = max(max(valuation(n, 2), valuation(n, 3)), valuation(n, 5));

a(n) = A051903(A355582(n)).
a(n) = max(A007814(n), A007949(n), A112765(n)).
a(n) = 0 if and only if n is a 7-rough number (A007775).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = A375538(3)/A375539(3) = 51227/36540 = 1.401943076...

A375537 Square array A(n, k) (n, k >= 1) read by antidiagonals in ascending order: A(n, k) = Max_{i = 1..n} v_prime(i)(k), where v_p(k) is the p-adic valuation of k.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Aug 19 2024

For a given n, A(n, k) is the sequence that gives the maximum exponent in the prime factorization of the largest prime(n)-smooth divisor of k.

			Array begins:
   n | n-th row
  ---+-----------------------------
   1 | 0, 1, 0, 2, 0, 1, 0, 3, 0, 1
   2 | 0, 1, 1, 2, 0, 1, 0, 3, 2, 1
   3 | 0, 1, 1, 2, 1, 1, 0, 3, 2, 1
   4 | 0, 1, 1, 2, 1, 1, 1, 3, 2, 1
   5 | 0, 1, 1, 2, 1, 1, 1, 3, 2, 1
   6 | 0, 1, 1, 2, 1, 1, 1, 3, 2, 1
   7 | 0, 1, 1, 2, 1, 1, 1, 3, 2, 1
   8 | 0, 1, 1, 2, 1, 1, 1, 3, 2, 1
   9 | 0, 1, 1, 2, 1, 1, 1, 3, 2, 1
  10 | 0, 1, 1, 2, 1, 1, 1, 3, 2, 1

Amiram Eldar, Table of n, a(n) for n = 1..10011 (first 141 antidiagonals, flattened)
Index entries for sequences computed from exponents in factorization of n.

Cf. A000720, A006530, A051903, A249344, A375538, A375539.

Rows k = 1..3: A007814, A244417, A375536.

A[n_, k_] := Max[IntegerExponent[k, Prime[Range[n]]]]; Table[A[n - k + 1, k], {n, 1, 14}, {k, 1 n}] // Flatten

A(n, k) = vecmax(apply(x -> valuation(k, x), primes(n)));

A(n, k) = Max_{i=1..n} A249344(i, k).
A(n, k) = A051903(k) for n >= A000720(A006530(k)).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{i=1..m} A(n, i) = A375538(n)/A375539(n).

A244416 6-adic value of 1/n for n >= 1.

Original entry on oeis.org

1, 6, 6, 36, 1, 6, 1, 216, 36, 6, 1, 36, 1, 6, 6, 1296, 1, 36, 1, 36, 6, 6, 1, 216, 1, 6, 216, 36, 1, 6, 1, 7776, 6, 6, 1, 36, 1, 6, 6, 216, 1, 6, 1, 36, 36, 6, 1, 1296, 1, 6, 6, 36, 1, 216, 1, 216, 6, 6, 1, 36, 1, 6, 36, 46656, 1, 6, 1, 36, 6, 6, 1, 216, 1, 6, 6, 36, 1, 6, 1, 1296, 1296, 6, 1, 36, 1, 6

Offset: 1

Wolfdieter Lang, Jun 30 2014

For the definition of 'g-adic value of x', called |x|_g with g an integer >= 2, see the Mahler reference, p. 7. Sometimes also called g-adic absolute value of x. If g is not a prime then this is called a non-archimeden pseudo-valuation. See Mahler, p. 10.

			a(6) = 6^max(1,1) = 6^1 = 6. a(12) = 6^max(2,1) = 6^2 = 36,
a(18) = 6^max(1,2) = 36, a(24) = 6^max(3,1) = 6^3 = 216, ...
a(2) = 6^1 = 6, a(8) = 6^3 = 216, a(14) = 6^1 = 6, ...
a(3) = 6^1 = 6, a(9) = 6^2 = 36, a(15) = 6^1 = 6, ...
a(4) = 6^2 = 36, a(10) = 6^1 = 6, a(16) = 6^4 = 1296, ...

Kurt Mahler, p-adic numbers and their functions, second ed., Cambridge University Press, 1981.

Cf. A244417, A006519 (g=2), A038500 (g=3), A240226 (g=4), A060904 (g=5).

a[n_] := 6^Max[IntegerExponent[n, {2, 3}]]; Array[a, 100] (* Amiram Eldar, Aug 19 2024 *)

a(n) = 6^max(valuation(n, 2), valuation(n, 3)); \\ Amiram Eldar, Aug 19 2024

a(n) = 1 if n == 1 or 5 (mod 6). a(n) = 6^max(A007814(n), A007949(n)) if n == 0 (mod 6), a(n) = 6^A007814(n) if n == 2 or 4 (mod 6), a(n) = 6^A007949(n) if n == 3 (mod 6). The exponents, called f(1/n) in the Mahler reference, are given in A244417(n).

a(n) = 6^A244417(n). - Amiram Eldar, Aug 19 2024

cp's OEIS Frontend

A322316 Lexicographically earliest such sequence a that a(i) = a(j) => A122841(i) = A122841(j) and A244417(i) = A244417(j), for all i, j.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A054707 Number of powers of 6 modulo n.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A322026 Lexicographically earliest infinite sequence such that a(i) = a(j) => A007814(i) = A007814(j) and A007949(i) = A007949(j), for all i, j, where A007814 and A007949 give the 2- and 3-adic valuations of n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

PARI

Formula

A375538 Numerator of the asymptotic mean over the positive integers of the maximum exponent in the prime factorization of the largest prime(n)-smooth divisor function.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A322317 Ordinal transform of A322316.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

A375536 The maximum exponent in the prime factorization of the largest 5-smooth divisor of n.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A375537 Square array A(n, k) (n, k >= 1) read by antidiagonals in ascending order: A(n, k) = Max_{i = 1..n} v_prime(i)(k), where v_p(k) is the p-adic valuation of k.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A244416 6-adic value of 1/n for n >= 1.

Views

Author

Keywords

Comments

Examples

References

Crossrefs

Programs