A278165 -id:A278165 - cp's OEIS frontend

A286565 Restricted growth sequence of A278165 (= A046523(A001045(n))).

1, 1, 2, 2, 2, 3, 2, 3, 4, 3, 2, 5, 2, 3, 6, 6, 2, 7, 2, 7, 6, 6, 2, 8, 6, 3, 9, 10, 3, 8, 2, 5, 5, 3, 5, 11, 3, 3, 6, 12, 3, 12, 2, 8, 13, 6, 3, 14, 3, 8, 10, 8, 3, 15, 7, 16, 5, 10, 6, 17, 2, 3, 13, 8, 10, 18, 3, 8, 5, 18, 3, 19, 3, 5, 8, 8, 10, 16, 2, 11, 20, 5, 10, 19, 6, 5, 5, 14, 6, 17, 10, 18, 5, 10, 5, 21, 5, 5, 22, 23, 2, 24, 3, 14, 24, 10, 3, 25, 3

Offset: 1

Antti Karttunen, May 21 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..1032 (computed from b-file of A278165 provided by Hans Havermann)

Cf. A001045, A046523, A278165.

Cf. A107036 (positions of 2's), A286566, A286567.

A286567 Smallest prime factor of the n-th Jacobsthal number: a(n) = A020639(A001045(n)), with a(1)=a(2)=1.

Original entry on oeis.org

1, 1, 3, 5, 11, 3, 43, 5, 3, 11, 683, 3, 2731, 43, 3, 5, 43691, 3, 174763, 5, 3, 23, 2796203, 3, 11, 2731, 3, 5, 59, 3, 715827883, 5, 3, 43691, 11, 3, 1777, 174763, 3, 5, 83, 3, 2932031007403, 5, 3, 47, 283, 3, 43, 11, 3, 5, 107, 3, 11, 5, 3, 59, 2833, 3, 768614336404564651, 715827883, 3, 5, 11, 3, 7327657, 5, 3, 11, 56409643, 3, 1753, 223, 3, 5, 43, 3

Offset: 1

Antti Karttunen & Hans Havermann, May 21 2017

nonn

Hans Havermann, Table of n, a(n) for n = 1..1122

Cf. A001045, A020639, A278165.

Rest[FactorInteger[#][[1,1]]&/@LinearRecurrence[{1,2},{0,1},100]] (* Harvey P. Dale, Sep 23 2018 *)

(define (A286567 n) (A020639 (A001045 n)))

a(n) = A020639(A001045(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)).

A286566 Compound filter (prime signature of n & prime signature of the n-th Jacobsthal number): a(n) = P(A101296(n), A286566(n)), where P(n,k) is sequence A000027 used as a pairing function.

Original entry on oeis.org

1, 3, 5, 9, 5, 19, 5, 26, 18, 19, 5, 51, 5, 19, 40, 73, 5, 72, 5, 72, 40, 40, 5, 113, 31, 19, 83, 111, 8, 129, 5, 101, 32, 19, 32, 221, 8, 19, 40, 179, 8, 199, 5, 84, 159, 40, 8, 312, 13, 84, 82, 84, 8, 239, 49, 261, 32, 82, 23, 419, 5, 19, 159, 224, 82, 334, 8, 84, 32, 334, 8, 543, 8, 32, 84, 84, 82, 285, 5, 243, 332, 32, 57, 478, 40, 32, 32, 218, 23, 419, 82

Offset: 1

Antti Karttunen, May 21 2017

nonn

Here, instead of A046523 and A278165 we use as the components of a(n) their rgs-versions A101296 and A286565 because of the latter sequences' more moderate growth rates.

Antti Karttunen, Table of n, a(n) for n = 1..1032 (computed using the b-file of A278165 provided by Hans Havermann)

Cf. A001045, A046523, A101296, A278165, A286565.
Cf. A000978 (positions of 5's).
Cf. A286467 (similar filter).

(define (A286566 n) (* (/ 1 2) (+ (expt (+ (A101296 n) (A286565 n)) 2) (- (A101296 n)) (- (* 3 (A286565 n))) 2)))

a(n) = (1/2)*(2 + ((A101296(n)+A286565(n))^2) - A101296(n) - 3*A286565(n)).

cp's OEIS Frontend

A286565 Restricted growth sequence of A278165 (= A046523(A001045(n))).

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A286567 Smallest prime factor of the n-th Jacobsthal number: a(n) = A020639(A001045(n)), with a(1)=a(2)=1.

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

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A286566 Compound filter (prime signature of n & prime signature of the n-th Jacobsthal number): a(n) = P(A101296(n), A286566(n)), where P(n,k) is sequence A000027 used as a pairing function.

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