A098012 -id:A098012 - cp's OEIS frontend

A104887 Triangle T(n,k) = (n-k+1)-th prime, read by rows.

2, 3, 2, 5, 3, 2, 7, 5, 3, 2, 11, 7, 5, 3, 2, 13, 11, 7, 5, 3, 2, 17, 13, 11, 7, 5, 3, 2, 19, 17, 13, 11, 7, 5, 3, 2, 23, 19, 17, 13, 11, 7, 5, 3, 2, 29, 23, 19, 17, 13, 11, 7, 5, 3, 2, 31, 29, 23, 19, 17, 13, 11, 7, 5, 3, 2, 37, 31, 29, 23, 19, 17, 13, 11, 7, 5, 3, 2, 41, 37, 31, 29

Offset: 1

Gary W. Adamson, Mar 29 2005

Repeatedly writing the prime sequence backwards.

Sequence B is called a reverse reluctant sequence of sequence A, if B is triangle array read by rows: row number k lists first k elements of the sequence A in reverse order. Sequence A104887 is the reverse reluctant sequence of sequence the prime numbers (A000040). - Boris Putievskiy, Dec 13 2012

			Triangle begins:
   2;
   3,  2;
   5,  3,  2;
   7,  5,  3,  2;
  11,  7,  5,  3,  2;
  13, 11,  7,  5,  3,  2;
  17, 13, 11,  7,  5,  3,  2;

Reinhard Zumkeller, Rows n = 1..125 of triangle, flattened
Boris Putievskiy, Transformations Integer Sequences And Pairing Functions, arXiv:1212.2732 [math.CO], 2012.

Reflected triangle of A037126.

Cf. A098012 (partial products per row).

P:=Filtered([1..200],IsPrime);;
T:=Flat(List([1..13],n->List([1..n],k->P[n-k+1]))); # Muniru A Asiru, Mar 16 2019

import Data.List (inits)
a104887 n k = a104887_tabl !! (n-1) !! (k-1)
a104887_row n = a104887_tabl !! (n-1)
a104887_tabl = map reverse $ tail $ inits a000040_list
-- Reinhard Zumkeller, Oct 02 2014

T:=(n,k)->ithprime(n-k+1): seq(seq(T(n,k),k=1..n),n=1..13); # Muniru A Asiru, Mar 16 2019

Module[{nn=15,prms},prms=Prime[Range[nn]];Table[Reverse[Take[prms,n]],{n,nn}]]//Flatten (* Harvey P. Dale, Aug 10 2021 *)

T(n,k) = A000040(n-k+1); a(n) = A000040(A004736(n)).

a(n) = A000040(m), where m=(t*t+3*t+4)/2-n, t=floor((-1+sqrt(8*n-7))/2). - Boris Putievskiy, Dec 13 2012

Edited by Ralf Stephan, Apr 05 2009

A060381 a(n) = prime(n)prime(n+1)...prime(2n-1), where prime(i) is the i-th prime.

Original entry on oeis.org

1, 2, 15, 385, 17017, 1062347, 86822723, 10131543907, 1204461778591, 198229051666003, 35224440615606707, 6295457783127226289, 1331590860773071702483, 310692537866322378582047, 78832548083496383033878901, 21381953681344611984282084241

Offset: 0

Jason Earls, Apr 03 2001

Central terms of triangle A098012. - Reinhard Zumkeller, Oct 02 2014

For n >= 0, a(n+1) is the n-th power of 15 in the monoid defined by A306697. - Peter Munn, Feb 18 2020

			a(1)=2; a(2) = 3*5 = 15; a(3) = 5*7*11 = 385.

Reinhard Zumkeller, Table of n, a(n) for n = 0..250

Cf. A002110, A098012.
Related to A006516 via A019565.
A003961, A059896, A306697 are used to express relationship between terms of this sequence.

P:=Filtered([1..200],IsPrime);;
a:=List([1..15],n->Product([0..n-1],k->P[n+k])); # Muniru A Asiru, Mar 16 2019

a060381 n = a098012 (2 * n - 1) n  -- Reinhard Zumkeller, Oct 02 2014

seq(mul(ithprime(n+k),k=0..n-1),n=0..15); # Muniru A Asiru, Mar 16 2019

Table[Times@@Prime[Range[n,2n-1]],{n,20}] (* Harvey P. Dale, Jul 19 2018 *)

a(n) = prod(k=n, 2*n-1, prime(k)); \\ Michel Marcus, Mar 16 2019

a(n) = A002110(2*n-1)/A002110(n-1). - Michel Marcus, Mar 16 2019
From Peter Munn, Feb 18 2020: (Start)
a(n) = A019565(A006516(n)).
For n >= 1, a(n) = A098012(n,n), reading A098012 as a square array.
For n > 1, a(n) = A306697(a(n-1), 15) = A059896(A003961^2(a(n-1)), A003961(a(n-1))).
(End)

a(0)=1 prepended by Alois P. Heinz, Mar 16 2019

A285639 a(n) = n*A117366(n)/spf(n), where A117366(n) is the smallest prime larger than all prime factors of n, and spf is the smallest prime factor of n (or 1 if n = 1).

Original entry on oeis.org

2, 3, 5, 6, 7, 15, 11, 12, 15, 35, 13, 30, 17, 77, 35, 24, 19, 45, 23, 70, 77, 143, 29, 60, 35, 221, 45, 154, 31, 105, 37, 48, 143, 323, 77, 90, 41, 437, 221, 140, 43, 231, 47, 286, 105, 667, 53, 120, 77, 175, 323, 442, 59, 135, 143, 308, 437, 899, 61, 210, 67, 1147

Offset: 1

M. F. Hasler, Apr 30 2017

nonn

The smallest prime factor of n is removed, and a prime factor larger than all others is added. This is somewhat in the spirit of A003961 where each of the prime factors is increased to the next larger prime. Therefore a(n) = A003961(n) when n is a prime or a product of consecutive primes.

Leaves invariant A073485, i.e., for all n in A073485, a(n) is again in A073485. More precisely, a(A098012(m,n)) = A098012(m,n+1). - M. F. Hasler, May 03 2017

			a(1) = nextprime(1) = 2.
a(2) = 2 / 2 * nextprime(2) = 3.
a(3) = 3 / 3 * nextprime(3) = 5, and in the same way, a(prime(k))=prime(k+1).
a(4) = 4 / 2 * nextprime(2) = 2*3 = 6.
a(6) = 6 / 2 * nextprime(3) = 3*5 = 15.

Ivan Neretin, Table of n, a(n) for n = 1..10000

Cf. A020639, A003961, A006530, A117366, A151800.

Table[d = FactorInteger[n]; n*NextPrime[d[[-1, 1]]]/d[[1, 1]], {n, 62}] (* Ivan Neretin, Jan 23 2018 *)

a(n,f=factor(n)[,1])={f||f=[1];n\f[1]*nextprime(f[#f]+1)}

A381500 a(n) = A019565(A187769(n)).

Original entry on oeis.org

1, 2, 3, 6, 5, 10, 15, 30, 7, 14, 21, 35, 42, 70, 105, 210, 11, 22, 33, 55, 77, 66, 110, 165, 154, 231, 385, 330, 462, 770, 1155, 2310, 13, 26, 39, 65, 91, 143, 78, 130, 195, 182, 273, 455, 286, 429, 715, 1001, 390, 546, 910, 1365, 858, 1430, 2145, 2002, 3003

Offset: 0

Michael De Vlieger, Peter Munn, and Antti Karttunen, Feb 27 2025

The squarefree numbers, ordered first by largest prime factor (dividing the sequence into rows), then by number of prime factors, then lexicographically by their prime factors (written in descending order).

We index (a(n)) from offset 0, matching the choice for A019565 and similar sequences.

			Table begins:
  Row 0:  1;
  Row 1:  2;
  Row 2:  3,  6;
  Row 3:  5, 10, 15, 30;
  Row 4:  7, 14, 21, 35, 42, 70, 105, 210;
  Row 5: 11, 22, 33, 55, 77, 66, 110, 165, 154, 231, 385, 330, 462, 770, 1155, 2310;
  ...
Table of a(n) for n = 0..31, demonstrating relationship of this sequence with s = A187769:
          <-factors                    <-factors
   n  a(n)  2 3 5 7  s(n)  |   n   a(n)  2 3 5 7 11 s(n)
  -------------------------|----------------------------
   0    1   .          0   |  16    11   . . . . x   16
   1    2   x          1   |  17    22   x . . . x   17
   2    3   . x        2   |  18    33   . x . . x   18
   3    6   x x        3   |  19    55   . . x . x   20
   4    5   . . x      4   |  20    77   . . . x x   24
   5   10   x . x      5   |  21    66   x x . . x   19
   6   15   . x x      6   |  22   110   x . x . x   21
   7   30   x x x      7   |  23   165   . x x . x   22
   8    7   . . . x    8   |  24   154   x . . x x   25
   9   14   x . . x    9   |  25   231   . x . x x   26
  10   21   . x . x   10   |  26   385   . . x x x   28
  11   35   . . x x   12   |  27   330   x x x . x   23
  12   42   x x . x   11   |  28   462   x x . x x   27
  13   70   x . x x   13   |  29   770   x . x x x   29
  14  105   . x x x   14   |  30  1155   . x x x x   30
  15  210   x x x x   15   |  31  2310   x x x x x   31
  -------------------------|----------------------------
            1 2 4 8  s(n)  |             1 2 4 8 16 s(n)
             bits->                         bits->

Michael De Vlieger, Table of n, a(n) for n = 0..16383
Michael De Vlieger, Log log scatterplot of a(n), n = 0..2^16-1.
Michael De Vlieger, Plot p | a(n) at (x,y) = (n, pi(p)), n = 0..2047, 12X vertical exaggeration.
Michael De Vlieger, Fan style binary tree showing a(n), n = 0..2048, with a color function showing smallest and greatest terms in each row in green and red, respectively.

Cf. A019565, A119416, A163866, A187769, A253550, A294648, A344085.

A002110, A098012 are subsequences.

a187769 = {{0}}~Join~Table[SortBy[Range[2^n, 2^(n + 1) - 1], DigitCount[#, 2, 1] &], {n, 0, 8}] // Flatten; a019565[x_] := Times @@ Prime@ Flatten@ Position[#, 1] &@ Reverse@ IntegerDigits[x, 2]; Map[a019565, a187769]

a(n) = A019565(A187769(n)).
As an irregular triangle T(n,k), where row 0 = {1}:
For n > 1, omega(T(n,1)) = 1, omega(T(n, 2^(n-1))) = n, thus row n is divided into n segments S such that with S, omega(T(n,k)) = m, where m = 1..n. (See A187769 for the lengths of segments associated with Pascal's triangle A007318.)
S(-1,-1) = (1).
For n >= 0:
S(n-1, n) = (); S(n, -1) = ();
for 0 <= m <= n, S(n,m) = ( A253550'(S(n-1, m)), A119416'(S(n-1, m-1)) ), where Axxx'((i_1, i_2, ..., i_j)) denotes Axxx(i_1), Axxx(i_2), ..., Axxx(i_j).
a(A163866(n)) = A098012(n).

cp's OEIS Frontend

A104887 Triangle T(n,k) = (n-k+1)-th prime, read by rows.

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A060381 a(n) = prime(n)*prime(n+1)*...*prime(2*n-1), where prime(i) is the i-th prime.

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A285639 a(n) = n*A117366(n)/spf(n), where A117366(n) is the smallest prime larger than all prime factors of n, and spf is the smallest prime factor of n (or 1 if n = 1).

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A381500 a(n) = A019565(A187769(n)).

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A060381 a(n) = prime(n)prime(n+1)...prime(2n-1), where prime(i) is the i-th prime.