A048765 -id:A048765 - cp's OEIS frontend

A278225 Filter-sequence for factorial base (cycles in A060117/A060118-permutations): Least number with the same prime signature as A275725.

2, 4, 12, 8, 12, 8, 60, 36, 24, 16, 24, 16, 60, 24, 24, 16, 36, 16, 60, 24, 36, 16, 24, 16, 420, 180, 180, 72, 180, 72, 120, 72, 48, 32, 48, 32, 120, 48, 48, 32, 72, 32, 120, 48, 72, 32, 48, 32, 420, 180, 120, 48, 120, 48, 120, 72, 48, 32, 48, 32, 180, 72, 48, 32, 72, 32, 180, 72, 72, 32, 48, 32, 420, 120, 120, 48, 180, 48, 180, 72, 48, 32, 72, 32, 120, 48, 48

Offset: 0

Antti Karttunen, Nov 16 2016

nonn

This sequence can be used for filtering certain sequences related to cycle-structures in finite permutations as ordered by lists A060117 / A060118 (and thus also related to factorial base representation, A007623) because it matches only with any such sequence b that can be computed as b(n) = f(A275725(n)), where f(n) is any function that depends only on the prime signature of n (some of these are listed under 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.

Cf. A007623, A060117, A060118, A046523, A275725.
Other filter-sequences related to factorial base: A278234, A278235, A278236.
Sequences that partition N into same or coarser equivalence classes: A048764, A048765, A060129, A060130, A060131, A084558, A275803, A275851, A257510.

(define (A278225 n) (A046523 (A275725 n)))

a(n) = A046523(A275725(n)).

A151918 a(n) = k! - prime(n) where k is the smallest number for which prime(n) <= k!.

Original entry on oeis.org

0, 3, 1, 17, 13, 11, 7, 5, 1, 91, 89, 83, 79, 77, 73, 67, 61, 59, 53, 49, 47, 41, 37, 31, 23, 19, 17, 13, 11, 7, 593, 589, 583, 581, 571, 569, 563, 557, 553, 547, 541, 539, 529, 527, 523, 521, 509, 497, 493, 491, 487, 481, 479, 469, 463, 457, 451, 449, 443, 439, 437

Offset: 0

Ctibor O. Zizka, Apr 06 2008

How many times does each prime number appear in this sequence?
Are there infinitely many solutions of the form
(k!-p(n)) = p(r_1)*...*p(r_i); r_i < n for all i?

			a(1)  = 2! - p(1)  =   2 -  2 =  0;
a(2)  = 3! - p(2)  =   6 -  3 =  3;
a(3)  = 3! - p(3)  =   6 -  5 =  1;
a(4)  = 4! - p(4)  =  24 -  7 = 17;
a(5)  = 4! - p(5)  =  24 - 11 = 13;
a(6)  = 4! - p(6)  =  24 - 13 = 11;
a(7)  = 4! - p(7)  =  24 - 17 =  7;
a(8)  = 4! - p(8)  =  24 - 19 =  5;
a(9)  = 4! - p(9)  =  24 - 23 =  1;
a(10) = 5! - p(10) = 120 - 29 = 91;
etc.

Cf. A000040, A000142.

A048765 := proc(n) for i from 1 do if i! >= n then return i! ; end if; end do: end proc:
A151918 := proc(n) p := ithprime(n) ; A048765(p)-p ; end proc:
seq(A151918(n),n=1..80) ; # R. J. Mathar, Aug 25 2010

Module[{fs=Range[10]!,p},Join[{0},Flatten[Table[p=Prime[n];Select[ fs,#>p&,1]-p,{n,2,70}]]]] (* Harvey P. Dale, Oct 04 2013 *)

a(n) = A048765(prime(n)) - prime(n). - R. J. Mathar, Aug 25 2010

More terms from R. J. Mathar, Aug 25 2010

cp's OEIS Frontend

A278225 Filter-sequence for factorial base (cycles in A060117/A060118-permutations): Least number with the same prime signature as A275725.

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A151918 a(n) = k! - prime(n) where k is the smallest number for which prime(n) <= k!.

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