A286565 -id:A286565 - cp's OEIS frontend

A278165 Least number with the prime signature of the n-th Jacobsthal number.

1, 1, 2, 2, 2, 6, 2, 6, 12, 6, 2, 210, 2, 6, 30, 30, 2, 420, 2, 420, 30, 30, 2, 30030, 30, 6, 120, 2310, 6, 30030, 2, 210, 210, 6, 210, 19399380, 6, 6, 30, 60060, 6, 60060, 2, 30030, 4620, 30, 6, 223092870, 6, 30030, 2310, 30030, 6, 120120, 420, 510510, 210, 2310, 30, 401120980260, 2, 6, 4620, 30030, 2310, 9699690, 6, 30030, 210, 9699690, 6, 14841476269620, 6

Offset: 1

Antti Karttunen, Nov 19 2016

nonn

Antti Karttunen (terms 1..257) & Hans Havermann, Table of n, a(n) for n = 1..1032

Cf. A001045, A046523.
Cf. A107036 (positions of 2's), A286565 (rgs-version of this sequence).
Cf. also A278245, A278248, A278258, A286567.

Table[Times @@ MapIndexed[(Prime@ First@ #2)^#1 &, #] &@ If[Length@# == 1 && #[[1, 1]] == 1, {0}, Reverse@ Sort@ #[[All, -1]]] &@ FactorInteger[ (2^n - (-1)^n)/3], {n, 120}] (* Michael De Vlieger, Nov 21 2016 *)

A001045(n) = (2^n - (-1)^n) / 3;
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
A278165(n) = A046523(A001045(n));
for(n=1, 257, write("b278165.txt", n, " ", A278165(n)));

(define (A278165 n) (A046523 (A001045 n)))

a(n) = A046523(A001045(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

A278165 Least number with the prime signature of the n-th Jacobsthal number.

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