A278241 -id:A278241 - cp's OEIS frontend

A278478 a(n) is the 2-adic valuation of A000041(n).

0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 1, 3, 0, 0, 0, 4, 0, 0, 0, 1, 0, 3, 1, 0, 0, 1, 2, 1, 1, 0, 2, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 1, 0, 0, 0, 0, 2, 0, 1, 2, 2, 0, 0, 2, 0, 1, 1, 6, 0, 0, 0, 5, 0, 0, 0, 2, 3, 0, 0, 2, 1, 2, 0, 0, 0, 2, 0, 1, 0, 0, 0, 0, 0, 0, 0, 3, 0, 11, 1, 3, 0, 2, 1, 0, 1, 0, 0, 4, 0, 2, 7, 1, 0, 2, 2, 0, 0, 3, 2, 0

Offset: 0

Joerg Arndt, Nov 23 2016

nonn

Write A000041(n) = 2^k * s where s is odd, then a(n) = k.

Joerg Arndt, Table of n, a(n) for n = 0..10000

Cf. A000041, A007814, A069935, A278479.
Cf. A052002, A237280, A278779, A278780, A278781, A278782, A278783, A278784 (positions of terms 0, 1, 2, ..., 7 in this sequence).
Cf. also A278241.

a:= n-> padic[ordp](combinat[numbpart](n), 2):
seq(a(n), n=0..120);  # Alois P. Heinz, Nov 23 2016

a[n_] := IntegerExponent[PartitionsP[n], 2]; Array[a, 100, 0] (* Amiram Eldar, May 25 2024 *)

{ my( x='x+O('x^100), v=Vec(1/eta(x)) ); vector(#v,n,valuation(v[n],2)) }

From Amiram Eldar, May 25 2024: (Start)
a(n) = A007814(A000041(n)).
a(n) = log_2(A069935(n)). (End)

A278245 Least number with the same prime signature as the n-th Fibonacci number: a(n) = A046523(A000045(n)).

Original entry on oeis.org

1, 1, 2, 2, 2, 8, 2, 6, 6, 6, 2, 144, 2, 6, 30, 30, 2, 120, 6, 210, 30, 6, 2, 10080, 12, 6, 210, 210, 2, 9240, 6, 210, 30, 6, 30, 166320, 30, 30, 30, 30030, 6, 9240, 2, 2310, 2310, 30, 2, 2882880, 30, 4620, 30, 210, 6, 120120, 210, 60060, 2310, 30, 6, 232792560, 6, 30, 2310, 30030, 30, 9240, 30, 2310, 2310, 510510, 6, 1396755360, 6, 210, 4620, 2310, 210, 120120, 6

Offset: 1

Antti Karttunen, Nov 16 2016

nonn

This sequence can be used as a filter for certain sequences involving Fibonacci numbers as it matches to any sequence that is obtained as f(A000045(n)), where f(n) is any function that depends only on the prime signature of n (see the index entry for "sequences computed from exponents in ...").

Matching in this context means that the sequence a matches with the sequence b iff for all i, j: a(i) = a(j) => b(i) = b(j). In other words, iff the sequence b partitions the natural numbers to the same or coarser equivalence classes (as/than the sequence a) by the distinct values it obtains.

			From _Michael De Vlieger_, May 18 2017: (Start)
a(6) = 8 because Fibonacci(6) = 8, the multiplicity of the prime factor of 8 is 3; the smallest p^3 = 2^3 = 8.
a(7) = 2 because Fibonacci(7) = 13, the multiplicity of the prime factor of 13 is 1; the smallest p^1 = 2^1 = 2.
a(15) = 30 because Fibonacci(15) = 610. The multiplicities of the prime factors of 610, in order from greatest to least, are {1, 1, 1}, the smallest prime power product p^1 * q^1 * r^1 = 2 * 3 * 5 = 30.
a(18) = 120 because Fibonacci(18) = 2584 = 2^3 * 17 * 19 -> 2^3 * 3 * 5 = 120. (End)

Antti Karttunen (terms 1..374) & Hans Havermann, Table of n, a(n) for n = 1..1300
Index entries for sequences computed from exponents in factorization of n

Cf. A000045, A046523, A278241, A278248.
Cf. A286545 (rgs-version of this sequence), A286467.
Cf. A001605 (positions of 2's), A072381 (of 6's).
Sequences with matching equivalence classes: A063375, A105307, A152774.

Table[If[# == 1, 1, Times @@ MapIndexed[Prime[First[#2]]^#1 &,
Sort[FactorInteger[#][[All, -1]], Greater]]] &@ Fibonacci@ n, {n, 79}] (* Michael De Vlieger, May 18 2017 *)

A046523(n) = my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]) \\ From Charles R Greathouse IV, Aug 17 2011
f0 = 0; f1 = 1; for(n=1, 10000, write("b278245.txt", n, " ", A046523(f1)); old_f0 = f0; f0 = f1; f1 = f1 + old_f0; );

(define (A278245 n) (A046523 (A000045 n)))

a(n) = A046523(A000045(n)).

A278248 Least number with the same prime signature as the n-th number in Perrin sequence: a(n) = A046523(A001608(n)), a(1) = 0.

Original entry on oeis.org

2, 0, 2, 2, 2, 2, 2, 2, 6, 12, 2, 6, 2, 6, 6, 12, 60, 6, 6, 6, 2, 2, 96, 60, 2, 30, 6, 6, 6, 840, 30, 6, 30, 6, 2, 6, 6, 60, 2, 420, 1260, 30, 30, 420, 210, 30, 30, 210, 6, 30, 30, 12, 6, 2310, 30, 840, 6, 240, 6, 30, 6, 420, 6, 6, 30, 420, 6, 210, 6, 6, 6, 4620, 60, 210, 30030, 2, 6, 30, 2310, 13860, 60, 210, 6, 6, 6, 120, 6, 2310, 210, 210, 6, 210, 30, 60, 4620

Offset: 0

Antti Karttunen, Nov 16 2016

nonn

This sequence works as a "sentinel" for Perrin sequence by matching to any other sequence that is obtained as f(A001608(n)), where f(n) is any function that depends only on the prime signature of n (see the index entry for "sequences computed from exponents in ..."). As of Nov 11 2016 no such sequences were present in the database.

Antti Karttunen, Table of n, a(n) for n = 0..555
Index entries for sequences computed from exponents in factorization of n

Cf. A001608, A046523, A278241, A278245.

A046523(n) = my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]) \\ From Charles R Greathouse IV, Aug 17 2011
p0 = 3; p1 = 0; p2 = 2; for(n=0, 555, write("b278248.txt", n, " ", if(!p0,p0,A046523(p0))); old_p0 = p0; old_p1 = p1; p0 = p1; p1 = p2; p2 = old_p1 + old_p0; );

(define (A278248 n) (if (= 1 n) 0 (A046523 (A001608 n))))

a(1) = 0; for any other n, a(n) = A046523(A001608(n)).

A278258 Least number with the prime signature of the n-th Catalan number.

Original entry on oeis.org

1, 1, 2, 2, 6, 30, 60, 30, 210, 210, 420, 2310, 4620, 13860, 360360, 60060, 1021020, 9699690, 58198140, 223092870, 446185740, 446185740, 892371480, 1338557220, 1338557220, 6692786100, 2677114440, 12939386460, 802241960520, 802241960520, 1604483921040, 200560490130, 14841476269620, 608500527054420, 608500527054420, 304250263527210, 608500527054420, 608500527054420

Offset: 0

Antti Karttunen, Nov 19 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 0..150

Cf. A000108, A046523.

Cf. also A278241, A278245, A278248, A278250, A278165.

Table[Times @@ MapIndexed[(Prime@ First@ #2)^#1 &, #] &@ If[Length@ # == 1 && #[[1, 1]] == 1, {0}, Reverse@ Sort@ #[[All, -1]]] &@ FactorInteger[CatalanNumber@ n], {n, 0, 37}] (* Michael De Vlieger, Nov 21 2016 *)

A000108(n) = binomial(2*n, n)/(n+1);
A046523(n) = my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]) \\ From Charles R Greathouse IV, Aug 17 2011
A278258(n) = A046523(A000108(n));
for(n=0, 150, write("b278258.txt", n, " ", A278258(n)));

(define (A278258 n) (A046523 (A000108 n)))

a(n) = A046523(A000108(n)).

A278249 Least number with the prime signature of A000123(n), the number of partitions of 2n into powers of 2.

Original entry on oeis.org

1, 2, 4, 6, 6, 6, 12, 6, 36, 6, 60, 6, 6, 30, 60, 6, 6, 30, 12, 210, 210, 180, 12, 30, 12, 30, 900, 60, 6, 6, 60, 30, 12, 210, 720, 30, 420, 30, 60, 6, 12, 30, 60, 6, 6, 60, 60, 30, 60, 210, 2520, 210, 210, 30, 180, 210, 60, 120, 60, 210, 6, 30, 60, 30, 6, 30, 60, 30, 6, 30, 12, 30, 60, 30, 420, 210, 60, 30, 420, 60, 6, 30, 2520, 30, 30, 210, 12, 210, 60, 210

Offset: 0

Antti Karttunen, Nov 22 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 0..321

Cf. A000123, A046523.

Cf. also A278241.

A000123(n) = if(n<1, n==0, A000123(n\2)+A000123(n-1)); \\ From M. F. Hasler, Apr 30 2009 (?)
A046523(n) = my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]) \\ From Charles R Greathouse IV, Aug 17 2011
A278249(n) = A046523(A000123(n));
for(n=0, 321, write("b278249.txt", n, " ", A278249(n)));

(define (A278249 n) (A046523 (A000123 n)))

a(n) = A046523(A000123(n)).

cp's OEIS Frontend

A278478 a(n) is the 2-adic valuation of A000041(n).

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A278245 Least number with the same prime signature as the n-th Fibonacci number: a(n) = A046523(A000045(n)).

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A278248 Least number with the same prime signature as the n-th number in Perrin sequence: a(n) = A046523(A001608(n)), a(1) = 0.

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A278258 Least number with the prime signature of the n-th Catalan number.

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A278249 Least number with the prime signature of A000123(n), the number of partitions of 2n into powers of 2.

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PARI

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