A244417 -id:A244417

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A322316

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

+20
4

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

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).

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..19683

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

PROG

(PARI)

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];

CROSSREFS

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

KEYWORD

nonn

AUTHOR

Antti Karttunen, Dec 04 2018

STATUS

approved

A054707

Number of powers of 6 modulo n.

+10
15

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

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

FORMULA

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

MATHEMATICA

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 *)

CROSSREFS

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).

KEYWORD

easy,nonn

AUTHOR

Henry Bottomley, Apr 20 2000

STATUS

approved

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.

+10
13

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

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.

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..65537

Jon Maiga, Computer-generated formulas for A322026, Sequence Machine.

FORMULA

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)

PROG

(PARI)

up_to = 65537;

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];

(PARI)

A065331(n) = (3^valuation(n, 3)<<valuation(n, 2)); \\ From 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

CROSSREFS

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.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Dec 03 2018

STATUS

approved

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.

+10
4

1, 13, 51227, 926908275845, 548123689541583443758024333411, 629375533747930240763697631488051776709110194920714685268467462860005271344878614119

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

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

LINKS

Amiram Eldar, Table of n, a(n) for n = 1..8

Index entries for sequences computed from exponents in factorization of n.

FORMULA

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.

EXAMPLE

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.

MATHEMATICA

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]]

CROSSREFS

Cf. A033150, A375537, A375539 (denominators).

Cf. A375538 (numerators).

Cf. A000079, A003586, A007814, A244417, A375536.

KEYWORD

nonn,frac

AUTHOR

Amiram Eldar, Aug 19 2024

STATUS

approved

A322317

Ordinal transform of A322316.

+10
3

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..19683

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

PROG

(PARI)

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; };

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];

CROSSREFS

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

Cf. also A126760.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Dec 04 2018

STATUS

approved

A375536

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

+10
3

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,4

LINKS

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

Index entries for sequences computed from exponents in factorization of n.

FORMULA

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...

MATHEMATICA

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

PROG

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

CROSSREFS

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

Cf. A007814, A007949, A112765.

KEYWORD

nonn,easy

AUTHOR

Amiram Eldar, Aug 19 2024

STATUS

approved

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.

+10
3

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

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

1,10

COMMENTS

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.

LINKS

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.

FORMULA

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).

EXAMPLE

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

MATHEMATICA

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

PROG

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

CROSSREFS

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

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

KEYWORD

nonn,easy,tabl

AUTHOR

Amiram Eldar, Aug 19 2024

STATUS

approved

A244416

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

+10
1

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

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.

REFERENCES

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

LINKS

Table of n, a(n) for n=1..86.

FORMULA

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

EXAMPLE

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, ...

MATHEMATICA

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

PROG

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

CROSSREFS

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

KEYWORD

nonn,easy

AUTHOR

Wolfdieter Lang, Jun 30 2014

STATUS

approved

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