A278245 -id:A278245 - cp's OEIS frontend

A286545 Restricted growth sequence of A278245 (prime signature of Fibonacci numbers).

1, 1, 2, 2, 2, 3, 2, 4, 4, 4, 2, 5, 2, 4, 6, 6, 2, 7, 4, 8, 6, 4, 2, 9, 10, 4, 8, 8, 2, 11, 4, 8, 6, 4, 6, 12, 6, 6, 6, 13, 4, 11, 2, 14, 14, 6, 2, 15, 6, 16, 6, 8, 4, 17, 8, 18, 14, 6, 4, 19, 4, 6, 14, 13, 6, 11, 6, 14, 14, 20, 4, 21, 4, 8, 16, 14, 8, 17, 4, 22, 20, 6, 2, 23, 8, 6, 8, 22, 4, 24, 25, 13, 8, 4, 13, 26, 8, 14, 13, 27, 4, 17, 6, 20, 20, 6, 4, 28

Offset: 1

Antti Karttunen, May 17 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..1300 (computed from the b-file of A278245 provided by Hans Havermann)

Cf. A278245, A286467.

Cf. A001605 (positions of 2's), A072381 (of 4's).

A278241 Least number with the same prime signature as the n-th partition number: a(n) = A046523(A000041(n)).

Original entry on oeis.org

1, 1, 2, 2, 2, 2, 2, 6, 6, 30, 30, 24, 6, 2, 24, 48, 30, 24, 30, 60, 30, 360, 30, 6, 180, 30, 420, 210, 60, 30, 60, 30, 60, 180, 30, 60, 2, 30, 60, 1680, 420, 210, 30, 240, 60, 30, 210, 420, 30, 60, 30, 60, 2310, 60, 2310, 420, 30, 30, 420, 4620, 30, 2310, 420, 30, 2310, 6, 6720, 6, 420, 30, 3360, 30, 30, 30, 2520, 120120, 6, 2, 420, 420, 1260, 6, 840, 30, 4620, 12

Offset: 0

Antti Karttunen, Nov 16 2016

nonn

This sequence works as a "sentinel" for partition numbers by matching to any sequence that is obtained as f(A000041(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 ..."). The last line in Crossrefs section lists such sequences that were present in the database as of Nov 11 2016.

Amiram Eldar, Table of n, a(n) for n = 0..10000 (terms 0..2527 from Antti Karttunen)
Index entries for sequences computed from exponents in factorization of n

Cf. A000041, A046523, A278245, A278248.

Sequences that partition N into same or coarser equivalence classes: A085543, A085561, A087175.

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
A278241(n) = A046523(numbpart(n));
for(n=0, 2310, write("b278241.txt", n, " ", A278241(n)));

(define (A278241 n) (A046523 (A000041 n)))

a(n) = A046523(A000041(n)).

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

Original entry on oeis.org

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)).

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)).

A286467 Compound filter (prime signature of n & prime signature of the n-th Fibonacci number): a(n) = P(A101296(n), A286545(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, 33, 18, 25, 5, 51, 5, 25, 40, 73, 5, 72, 12, 84, 40, 25, 5, 128, 69, 25, 71, 84, 5, 180, 12, 146, 40, 25, 40, 242, 23, 40, 40, 198, 12, 180, 5, 177, 177, 40, 5, 337, 31, 216, 40, 84, 12, 284, 59, 308, 140, 40, 12, 478, 12, 40, 177, 339, 40, 180, 23, 177, 140, 387, 12, 610, 12, 59, 216, 177, 59, 309, 12, 540, 332, 40, 5, 608, 59, 40, 59

Offset: 1

Antti Karttunen, May 17 2017

nonn

Nonsquare semiprimes pq for which F(pq) is also a semiprime is given by the positions where 25's occur in this sequence: 10, 14, 22, 26, 34, 94, (any more terms?). This is a subsequence of A072381.

Antti Karttunen, Table of n, a(n) for n = 1..1300 (based also on the b-file of A278245 provided by Hans Havermann)

Cf. A000027, A046523, A072381, A101296, A278245, A286545.

Cf. A083668 (positions of 5's).

(define (A286467 n) (* (/ 1 2) (+ (expt (+ (A101296 n) (A286545 n)) 2) (- (A101296 n)) (- (* 3 (A286545 n))) 2)))

a(n) = (1/2)*(2 + ((A101296(n) + A286545(n))^2) - A101296(n) - 3*A286545(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)).

A346491 Number of factorizations of the n-th Fibonacci number.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 2, 2, 2, 1, 29, 1, 2, 5, 5, 1, 21, 2, 15, 5, 2, 1, 719, 4, 2, 15, 15, 1, 296, 2, 15, 5, 2, 5, 4323, 5, 5, 5, 203, 2, 296, 1, 52, 52, 5, 1, 32653, 5, 135, 5, 15, 2, 1315, 15, 566, 52, 5, 2, 270920, 2, 5, 52, 203, 5, 296, 5, 52, 52, 877, 2

Offset: 1

Alois P. Heinz, Jul 19 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..119

Cf. A000045, A001055, A001605, A046523, A072381, A278245.

b:= proc(n, k) option remember; `if`(n>k, 0, 1)+`if`(isprime(n), 0,
      add(`if`(d>k, 0, b(n/d, d)), d=numtheory[divisors](n) minus {1, n}))
    end:
a:= proc(n) option remember; b((l-> mul(ithprime(i)^l[i], i=1..nops(l)))(
      sort(map(i-> i[2], ifactors(combinat[fibonacci](n))[2]), `>`))$2)
    end:
seq(a(n), n=1..80);

T[, 1] = T[1, ] = 1;
T[n_, m_] := T[n, m] = DivisorSum[n, If[1 < # <= m, T[n/#, #], 0]&];
f[n_] := T[n, n];
a[n_] := f[Fibonacci[n]];
Table[Print[n, " ", a[n]]; a[n], {n, 1, 119}] (* Jean-François Alcover, Sep 08 2022 *)

a(n) = A001055(A000045).
a(n) = A001055(A046523(A000045(n))).
a(n) = A001055(A278245(n)).
a(n) = 1 <=> n in { A001605 } union {1,2}.
a(n) = 2 <=> n in { A072381 }.

cp's OEIS Frontend

A286545 Restricted growth sequence of A278245 (prime signature of Fibonacci numbers).

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A278241 Least number with the same prime signature as the n-th partition number: a(n) = A046523(A000041(n)).

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

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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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A286467 Compound filter (prime signature of n & prime signature of the n-th Fibonacci number): a(n) = P(A101296(n), A286545(n)), where P(n,k) is sequence A000027 used as a pairing function.

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

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A346491 Number of factorizations of the n-th Fibonacci number.

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