A286567 -id:A286567 - 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)).

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

Original entry on oeis.org

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.

A363837 Numbers k such that k-th Jacobsthal number A001045(k) is a semiprime.

Original entry on oeis.org

6, 8, 10, 14, 26, 29, 34, 37, 38, 41, 47, 49, 53, 62, 67, 71, 73, 103, 107, 109, 122, 139, 151, 179, 223, 229, 251, 254, 269, 277, 311, 349, 353, 433, 457, 487, 503, 599, 601, 613, 619, 643, 739, 757, 827, 839, 1031, 1061, 1117

Offset: 1

Sean A. Irvine, Oct 19 2023

			10 is a term because Jacobsthal(10) = A001045(10) = 341 = 11*31 is a semiprime.

Eric Weisstein's World of Mathematics, Jacobsthal Number.
Eric Weisstein's World of Mathematics, Semiprime.

Cf. A001045, A001358, A277356 (the actual semiprimes), A250292, A085726, A072381, A101757, A286567, A271314.

isok(k) = bigomega((2^k - (-1)^k)/3) == 2; \\ Michel Marcus, Oct 19 2023

a(47)-a(49) from Amiram Eldar, Feb 25 2024

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

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

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A363837 Numbers k such that k-th Jacobsthal number A001045(k) is a semiprime.

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