A076274 -id:A076274 - cp's OEIS frontend

A038547 Least number with exactly n odd divisors.

1, 3, 9, 15, 81, 45, 729, 105, 225, 405, 59049, 315, 531441, 3645, 2025, 945, 43046721, 1575, 387420489, 2835, 18225, 295245, 31381059609, 3465, 50625, 2657205, 11025, 25515, 22876792454961, 14175, 205891132094649, 10395, 1476225, 215233605

Offset: 1

Tom Verhoeff

Also least odd number with exactly n divisors. - Lekraj Beedassy, Aug 30 2006
a(2n-1) = {1, 9, 81, 729, 225, 59049, ...} are the squares. A122842(n) = sqrt(a(2n-1)) = {1, 3, 9, 27, 15, 243, 729, 45, 6561, 19683, 135, 177147, 225, 105, 4782969, 14348907, 1215, ...}. - Alexander Adamchuk, Sep 13 2006
Also the least number k such that there are n partitions of k whose elements are consecutive integers. I.e., 1=1, 3=1+2=3, 9=2+3+4=4+5=9, 15=1+2+3+4+5=4+5+6=7+8=15, etc. - Robert G. Wilson v, Jun 02 2007
The politeness of an integer, A069283(n), is defined to be the number of its nontrivial runsum representations, and the sequence 3, 9, 15, 81, 45, 729, 105, ... represents the least integers to have a politeness of 1, 2, 3, 4, ... This is also the sequence of smallest integers with n+1 odd divisors and so apart from the leading 1, is precisely this sequence. - Ant King, Sep 23 2009
a(n) is also the least number k with the property that the symmetric representation of sigma(k) has n subparts. - Omar E. Pol, Dec 31 2016

			a(2^3) = 105 = 3*5 while a(2^4) = 945 = 3^3 * 5 * 7. There are 5 partition lists for the exponents of numbers with 16 odd divisors; they are {1, 1, 1, 1}, {3, 1, 1}, {3, 3}, {7, 1}, and {15} that result in the 5 numbers 1155, 945, 3375, 10935, and 14348907. Number a(3^8) = a(6561) = 3^2 * 5^2 * ... * 19^2 * 23^2 = 12442607161209225 while a(3^9) = a(19683) = 3^8 * 5^2 * ... * 19^2 * 23^2 = 9070660620521525025. The numbers a(5^52) = 3^4 * 5^4 * 7^4 * ... and a(5^53) = 3^24 * 5^4 * 7^4 * ... have 393 and 402 digits, respectively.  - _Hartmut F. W. Hoft_, Nov 03 2022

Don Reble, Table of n, a(n) for n = 1..2000
T. Verhoeff, Rectangular and Trapezoidal Arrangements, J. Integer Sequences, Vol. 2, 1999, #99.1.6.

A122842 = sqrt( a(2n-1) ).
Cf. A001227, A005179, A002110, A070826, A000005, A053640, A053624, A237593, A279387.
Row 1 of A266531. - Omar E. Pol, Dec 31 2016
Cf. A006093, A076274, A080577.

import Data.List  (find)
import Data.Maybe (fromJust)
a038547 n = fromJust $ find ((== n) . length . divisors) [1,3..]
   where divisors m = filter ((== 0) . mod m) [1..m]
-- Reinhard Zumkeller, Feb 24 2011

Table[Select[Range[1,532000,2],DivisorSigma[0,#]==k+1 &,1],{k,0,15}]//Flatten (* Ant King, Nov 28 2010 *)
2#-1&/@With[{ds=DivisorSigma[0,Range[1,600000,2]]},Table[Position[ds,n,1,1],{n,16}]]//Flatten (* The program is not suitable for generating terms beyond a(16) *) (* Harvey P. Dale, Jun 06 2017 *)
(* direct computation of A038547(n) *)
(* Function by _Vaclav Kotesovec_in A005179, Apr 04 2021, modified for odd divisors *)
mp[1, m_] := {{}}; mp[n_, 1] := {{}}; mp[n_?PrimeQ, m_] := If[mHartmut F. W. Hoft, Mar 05 2023 *)

for(nd=1,15,forstep(k=1,10^66,2,if(nd==numdiv(k),print1(k,", ");break())))

from math import prod
from sympy import isprime, divisors, prime
def A038547(n):
    def mult_factors(n):
        if isprime(n):
            return [(n,)]
        c = []
        for d in divisors(n,generator=True):
            if 1Chai Wah Wu, Aug 17 2024

a(p) = 3^(p-1) for primes p. - Zak Seidov, Apr 18 2006
a(n) = A119265(n,n). - Reinhard Zumkeller, May 11 2006
It was suggested by Alexander Adamchuk that for all n >= 1, we have a(3^(n-1)) = (p(n)#/2)^2 = (A002110(n)/2)^2 = A070826(n)^2. But this is false! E.g., (p(n)#/2)^2 = 3^2 * 5^2 * 7^2 * ... * 23^2 * 29^2 does indeed have 3^9 odd factors, but it is greater than 3^8 * 5^2 * 7^2 * ... * 23^2 which has 9*3*3*3*3*3*3*3 = 9*3^7 = 3^9 odd factors. - Richard Sabey, Oct 06 2007
a(A053640(m)) = a(A000005(A053624(m))) = A053624(m). - Rick L. Shepherd, Apr 20 2008
a(p^k) = Product_{i=1..k} prime(i+1)^(p-1), p prime and k >= 0, only when p_(k+1) < 3^p. - Hartmut F. W. Hoft, Nov 03 2022

Corrected by Ron Knott, Feb 22 2001
a(30) from Zak Seidov, Apr 18 2006
a(32)-a(34) from Lekraj Beedassy, Aug 30 2006

A076271 a(1) = 1, a(2) = 2, and for n > 2, a(n) = a(n-1) + gpf(a(n-1)), where gpf = greatest prime factor = A006530.

Original entry on oeis.org

1, 2, 4, 6, 9, 12, 15, 20, 25, 30, 35, 42, 49, 56, 63, 70, 77, 88, 99, 110, 121, 132, 143, 156, 169, 182, 195, 208, 221, 238, 255, 272, 289, 306, 323, 342, 361, 380, 399, 418, 437, 460, 483, 506, 529, 552, 575, 598, 621, 644, 667, 696, 725, 754, 783, 812, 841

Offset: 1

Reinhard Zumkeller, Oct 04 2002

nonn

a(n+1) is the smallest number such that the largest prime divisor of a(n) is the highest common factor of a(n) and a(n+1). - Amarnath Murthy, Oct 17 2002
Essentially the same as A036441(n) = a(n+1) and A180107(n) = a(n-1) (n > 1).
The equivalent sequence with A020639 = spf instead of A006530 = gpf begins a(1) = 1, a(2) = 2, and from then on we get all even numbers: a(n) = a(2) + 2*(n-2), n > 1. - M. F. Hasler, Apr 08 2015
From David James Sycamore, Apr 27 2017: (Start)
The sequence contains only one prime; a(2)=2, all other terms (excluding a(1)=1) being composite, since if a(n) for some n > 2 is assumed to be the first prime after 2, then a(n) = a(n-1) + gpf(a(n-1))= m*q+q = q*(m+1) for some integer m > 1 and some prime q. This number is composite; contradiction. Terms after a(3)=4 alternate between even and odd values since each is created by addition of a prime (odd term).
All terms a(n) arise as consecutive multiples of consecutive primes occurring in their natural ascending order, 2,3,5,7.... (A000040). The number of (consecutive) terms which arise as multiples of p(n)= A000040(n) is 1 + p(n+1)- p(n-1), namely n-th term of the sequence: 2,4,5,7,7,7,7,7,11, etc. Example: Number of multiples of 17, the 7th prime, is 1+p(8)-p(6) = 1+19-13 = 7.
For any pair of consecutive primes, p,q (p < q) a(p+q-1) = p*q, the (semiprime) term where multiples of p end and multiples of q start. Example a(7+11-1) = a(17) = 77 = 11*7, the last multiple of 7 and first multiple of 11. Every string of multiples of prime p contains the term p^2, located at a(2*p-1). E.g.: a(3)=4, a(5)=9, a(9)=25. (End)

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A036441, A076272(n) = a(n+1) - a(n).
See also A180107.
Cf. A070229.

a076271 n = a076271_list !! (n-1)
a076271_list = iterate a070229 1  -- Reinhard Zumkeller, Nov 07 2015

NestList[#+FactorInteger[#][[-1,1]]&,1,60] (* Harvey P. Dale, May 11 2015 *)

print1(n=1);for(i=1,199,print1(","n+=A006530(n))) \\ M. F. Hasler, Apr 08 2015

a(A076274(n)) = A008578(n)^2 for all n.

a(n+1) = A070229(a(n)). - Reinhard Zumkeller, Nov 07 2015

Edited by M. F. Hasler, Apr 08 2015

A327263 Array T(n,k) in which the i-th row consists of numbers > 1 not in array U(i;n,k) = (ink - (i-2)*A319929(n,k))/2 where i >= 1, n >= 1 and k >= 1, read by antidiagonals.

Original entry on oeis.org

3, 5, 2, 9, 3, 2, 13, 5, 3, 2, 21, 7, 4, 3, 2, 25, 11, 5, 4, 3, 2, 33, 13, 7, 5, 4, 3, 2, 37, 17, 9, 6, 5, 4, 3, 2, 45, 19, 10, 7, 6, 5, 4, 3, 2, 57, 23, 13, 9, 7, 6, 5, 4, 3, 2, 61, 29, 15, 11, 8, 7, 6, 5, 4, 3, 2, 73, 31, 17, 12, 9, 8, 7, 6, 5, 4, 3, 2

Offset: 1

David Lovler, Oct 15 2019

All the U(i;n,k) mimic the ordinary multiplication table in that they are commutative, associative, have identity element 1 and have 0. However (except when i=2) they are partially distributive, meaning that distributivity works except if an even number is partitioned into a sum of two odd numbers. Only when i=2, the odd-even-dependent A319929 term disappears and normal distributivity holds.
U(0;n,k) = A319929(n,k);
U(1;n,k) = A322630(n,k);
U(2;n,k) = n*k;
U(3;n,k) = A322744(n,k);
U(4;n,k) = A327259(n,k);
U(i;n,k) = 2i*floor(n/2)*floor(k/2) + A319929(n,k).
Row 1 is 2p-1 where p is a prime number (A076274 without 1).
Row 2 is the prime numbers.
Row 3 is A307002.
Row 4 is A327261.
The i-th row of T(n,k) consists of numbers that sieve out of the array U(i;n,k) = (i*n*k - (i-2)*A319929(n,k))/2, in numerical order.
From David Lovler, Sep 02 2020: (Start)
Row 1 has no even numbers. Row 2 has one even number. Generally, the even numbers of the i-th row start with i-1 consecutive even numbers (from 2). This is because U(i;2,2) = 2*i gives the first even number not in row i.
Row 3 seems to have even numbers that, after 2, coincide with A112774 which has an infinite number of terms. For i > 3, as i increases, row i has a denser presence of even numbers, thus each row has an infinite number of even terms.
Generalization of the twin prime conjecture: Since row 2 is the prime numbers, we can observe the twin prime conjecture that after the first three odd primes, the sprinkling of pairs of consecutive prime numbers never ends. Concerning just odd terms, a similar conjecture can be stated for rows i >= 3. Row 3 starts with four odd numbers then the sprinkling of three consecutive odd number never ends. Row 4 starts with five odd numbers then the sprinkling of four consecutive odd numbers never ends. The pattern continues as row i starts with i+1 odd numbers then the sprinkling of i consecutive odd numbers never ends. We can take this back to row 1 which starts with two odd numbers then continues with isolated odd numbers.
Studying the even terms, there is an analog to the above generalization of the twin prime conjecture. Row 3 starts with two even numbers then continues with isolated even numbers. Row 4 starts with three even numbers then the sprinkling of pairs of consecutive even numbers never ends. Row 5 starts with four even numbers then the sprinkling of three consecutive even numbers never ends. The pattern continues as row i starts with i-1 even numbers then the sprinkling of i-2 consecutive even numbers never ends.
(End)

			3  5  9  13  21  25  33  37  45  57  61  73  81  85  93 105 117 121 133 141 145 ...
2  3  5   7  11  13  17  19  23  29  31  37  41  43  47  53  59  61  67  71  73 ...
2  3  4   5   7   9  10  13  15  17  21  22  23  25  29  31  34  37  39  41  45 ...
2  3  4   5   6   7   9  11  12  14  15  17  19  21  22  25  27  28  29  31  35 ...
2  3  4   5   6   7   8   9  11  13  14  16  17  18  19  21  23  25  26  28  29 ...
2  3  4   5   6   7   8   9  10  11  13  15  16  18  19  20  21  22  23  25  27 ...
2  3  4   5   6   7   8   9  10  11  12  13  15  17  18  20  21  22  23  24  25 ...
2  3  4   5   6   7   8   9  10  11  12  13  14  15  17  19  20  22  23  24  25 ...
2  3  4   5   6   7   8   9  10  11  12  13  14  15  16  17  19  21  22  24  25 ...
2  3  4   5   6   7   8   9  10  11  12  13  14  15  16  17  18  19  21  23  24 ...
...

Cf. A000040, A003991, A076274, A112774.
Cf. A319929, A322630, A322744, A327259, A345474.
Cf. A354596.

row=12;max=200;U[i_,n_,k_]:=(i*n*k-(i-2)If[OddQ@n,If[OddQ@k,n+k-1,k],If[OddQ@k,n,0]])/2;t=Table[c=Union@Flatten@Table[U[i,n,k],{n,2,max},{k,2,max}];Complement[Range[2,max],c][[;;row]],{i,row}];Flatten@Table[t[[m,k-m+1]],{k,row},{m,k}] (* Giorgos Kalogeropoulos, Jun 08 2021 *)

With one exception there are likely no formulas for the rows of T(n,k) since their creation is based on a sieving process like the familiar prime number sieve. The exception is T(1,k) = 2*T(2,k)-1.

A139286 a(n) = 2^(2*prime(n) - 1).

Original entry on oeis.org

8, 32, 512, 8192, 2097152, 33554432, 8589934592, 137438953472, 35184372088832, 144115188075855872, 2305843009213693952, 9444732965739290427392, 2417851639229258349412352, 38685626227668133590597632, 9903520314283042199192993792

Offset: 1

Omar E. Pol, Apr 13 2008

Cf. A000040, A000079, A076274, A139287, A139288, A139289, A139290, A139291, A139292, A139293, A139294, A139306.

[2^(2*p-1): p in PrimesUpTo(200)]; // Vincenzo Librandi, Dec 15 2010

2^(2Prime[Range[20]]-1) (* Harvey P. Dale, Dec 01 2017 *)

a(n)=2^(2*prime(n)-1) \\ Charles R Greathouse IV, Jul 02 2013

Corrected and extended by Harvey P. Dale, Dec 01 2017

A139287 2^(2p - 1) - 1, where p is prime.

Original entry on oeis.org

7, 31, 511, 8191, 2097151, 33554431, 8589934591, 137438953471, 35184372088831, 144115188075855871, 2305843009213693951, 9444732965739290427391, 2417851639229258349412351, 38685626227668133590597631

Offset: 1

Omar E. Pol, Apr 13 2008

Vincenzo Librandi, Table of n, a(n) for n = 1..200

Cf. A000040, A000079, A076274, A139286, A139288, A139289, A139290, A139291, A139292, A139293, A139294, A139306.

[2^(2*p-1)-1: p in PrimesUpTo(200)]; // Vincenzo Librandi, Dec 15 2010

Table[(2^(2 Prime[n] - 1) - 1), {n, 1, 20}] (* Vincenzo Librandi, May 24 2014 *)

a(9) corrected by Vincenzo Librandi, May 24 2014

A322630 Arithmetic table similar to multiplication with different rules for odd and even products, read by antidiagonals. T(n,k) = (n*k + A319929(n,k))/2.

Original entry on oeis.org

1, 2, 2, 3, 2, 3, 4, 4, 4, 4, 5, 4, 7, 4, 5, 6, 6, 8, 8, 6, 6, 7, 6, 11, 8, 11, 6, 7, 8, 8, 12, 12, 12, 12, 8, 8, 9, 8, 15, 12, 17, 12, 15, 8, 9, 10, 10, 16, 16, 18, 18, 16, 16, 10, 10, 11, 10, 19, 16, 23, 18, 23, 16, 19, 10, 11

Offset: 1

David Lovler, Dec 20 2018

This table is akin to multiplication in that it is associative, 1 is the identity and 0 takes any number to 0. Associativity is proved by checking eight cases of three ordered odd and even numbers. Distributivity works except if an even number is partitioned into a sum of two odd numbers.

Excluding the first row and the first column, every number in the table is of the form 2i*j or 2i*j - 1 where i and j > 0. Every positive even number appears in the table. Odd numbers that do not appear are of the form 2p - 1 where p is a prime number.

			Array T(n,k) begins:
   1   2   3   4   5   6   7   8   9  10
   2   2   4   4   6   6   8   8  10  10
   3   4   7   8  11  12  15  16  19  20
   4   4   8   8  12  12  16  16  20  20
   5   6  11  12  17  18  23  24  29  30
   6   6  12  12  18  18  24  24  30  30
   7   8  15  16  23  24  31  32  39  40
   8   8  16  16  24  24  32  32  40  40
   9  10  19  20  29  30  39  40  49  50
  10  10  20  20  30  30  40  40  50  50

Michael De Vlieger, Table of n, a(n) for n = 1..11325 (rows 1..150, flattened).

Cf. A076274, A327263.

0 and diagonal is A213037.

Table[Function[n, Switch[FromDigits[Mod[{n, k}, 2], 2], 0, n k/2, 1, (n k + n)/2, 2, (n k + k)/2, , (n k + n + k - 1)/2]][m - k + 1], {m, 11}, {k, m}] // Flatten (* _Michael De Vlieger, Jan 14 2022 *)

T319929(n,k) = if (n%2, if (k%2, n+k-1, k), if (k%2, n, 0));
T(n,k) = (T319929(n,k) + n*k)/2;
matrix(6, 6, n, k, T(n,k)) \\ Michel Marcus, Dec 22 2018

T(n,k) = (n*k + n + k - 1)/2 if n is odd and k is odd;
T(n,k) = (n*k + n)/2 if n is even and k is odd;
T(n,k) = (n*k + k)/2 if n is odd and k is even;
T(n,k) = n*k/2 if n is even and k is even.

Name clarified by David Lovler, Jan 24 2022

A128281 a(n) is the least product of n distinct odd primes m=p_1p_2...*p_n, such that (d+m/d)/2 are all primes for each d dividing m.

Original entry on oeis.org

3, 21, 105, 1365, 884037

Offset: 1

Kok Seng Chua (chuakokseng(AT)hotmail.com), Mar 05 2007

From Iain Fox, Aug 26 2020: (Start)
a(6) > 10^9 if it exists.
All terms are members of A076274 since the definition requires that (1+m)/2 be prime.
The number of prime factors of m congruent to 3 (mod 4) must be even except for n=1.
(End)
a(6) > 2*10^11 if it exists. - David A. Corneth, Aug 27 2020
a(n) >= A070826(n+1) by definition of the sequence. - Iain Fox, Aug 28 2020

			105=3*5*7, (3*5*7+1)/2=53, (3+5*7)/2=19, (5+3*7)/2=13, (7+3*5)/2=11 are all primes and 105 is the least such number which is the product of 3 primes, so a(3)=3.

Subsequence of A076274.
Lower bound: A070826.
Cf. A002145, A128276, A128282, A128283, A128284, A128285, A128286, A168352.

a(n)=if(n==1, return(3)); my(p=prod(k=1, n, prime(k+1))); forstep(m=p+if(p%4-1, 2), +oo, 4, if(bigomega(m)==n && omega(m)==n, fordiv(m, d, if(!isprime((d+m/d)/2), next(2))); return(m))) \\ Iain Fox, Aug 27 2020

Definition corrected by Iain Fox, Aug 25 2020

A347682 Array read by antidiagonals: T(n,k) (n>=1, k>=1) = f(prime(n),prime(k)), where f(x,y) = xred_inv(x,y) + yred_inv(y,x) if gcd(x,y)=1, or 0 if gcd(x,y)>1, and red_inv is defined in the comments.

Original entry on oeis.org

0, 5, 5, 9, 0, 9, 13, 11, 11, 13, 21, 13, 0, 13, 21, 25, 23, 29, 29, 23, 25, 33, 25, 21, 0, 21, 25, 33, 37, 35, 51, 43, 43, 51, 35, 37, 45, 37, 69, 27, 0, 27, 69, 37, 45, 57, 47, 39, 69, 131, 131, 69, 39, 47, 57, 61, 59, 91, 113, 67, 0, 67, 113, 91, 59, 61, 73, 61, 59, 139, 153, 103, 103, 153, 139, 59, 61, 73

Offset: 1

N. J. A. Sloane, Sep 18 2021

If u, v are positive integers with gcd(u,v) = 1, the "reduced inverse" red_inv(u,v) of u mod v is u^(-1) mod v if u^(-1) mod v <= v/2, otherwise it is v - u^(-1) mod v.
That is, we map u to whichever of +-u has a representative mod v in the range 0 to v/2. Stated another way, red_inv(u,v) is a number r in the range 0 to v/2 such that r*u == +-1 mod v.
For example, red_inv(3,11) = 4, since 3^(-1) mod 11 = 4. But red_inv(2,11) = 5 = 11-6, since red_inv(2,11) = 6.
Arises in the study of A344005.

			The array begins:
0, 5, 9, 13, 21, 25, 33, 37, 45, 57, 61, 73,...
5, 0, 11, 13, 23, 25, 35, 37, 47, 59, 61, 73,...
9, 11, 0, 29, 21, 51, 69, 39, 91, 59, 61, 149,...
13, 13, 29, 0, 43, 27, 69, 113, 139, 57, 125, 223,...
21, 23, 21, 43, 0, 131, 67, 153, 45, 175, 309, 221,...
25, 25, 51, 27, 131, 0, 103, 77, 183, 233, 311, 443,...
33, 35, 69, 69, 67, 103, 0, 305, 137, 407, 373, 443,...
37, 37, 39, 113, 153, 77, 305, 0, 229, 115, 495, 75,...
...
The first few antidiagonals are:
[0]
[5, 5]
[9, 0, 9]
[13, 11, 11, 13]
[21, 13, 0, 13, 21]
[25, 23, 29, 29, 23, 25]
[33, 25, 21, 0, 21, 25, 33]
[37, 35, 51, 43, 43, 51, 35, 37]
...

N. J. A. Sloane, Table of n, a(n) for n = 1..5050 [First 100 antidiagonals, flattened]

Cf. A344005, A347681, A347683, A347684.

Rows 1 and 2 are (essentially) A076274 and A208296.

myfun1 := proc(A,B) local Ar,Br;
if igcd(A,B) > 1 then return(0); fi;
  Ar:=(A)^(-1) mod B;
   if 2*Ar > B then Ar:=B-Ar; fi;
  Br:=(B)^(-1) mod A;
   if 2*Br > A then Br:=A-Br; fi;
A*Ar+B*Br;
end;
myfun2:=(i,j)->myfun1(ithprime(i),ithprime(j));
for i from 1 to 30 do lprint([seq(myfun2(i-j+1,j),j=1..i)]); od:

A120632 Number of numbers >1 up to 2*prime(n) which are divisible by primes up to prime(n).

Original entry on oeis.org

2, 4, 8, 11, 18, 22, 29, 33, 40, 51, 54, 64, 72, 76, 84, 94, 104, 109, 120, 127, 132, 142, 150, 161, 174, 181, 186, 194, 199, 207, 230, 238, 248, 252, 270, 275, 285, 297, 305, 317, 327, 331, 349, 353, 361, 365, 386, 407, 415, 419, 426, 438, 442, 460, 471, 482

Offset: 1

Lekraj Beedassy, Jun 21 2006

nonn

The first prime(n+1)-2 numbers >1 are divisible by primes up to prime(n).

Complement of A137624; A137621(a(n))=A000040(n); A137621(a(n)+1)=A100484(n). - Reinhard Zumkeller, Jan 30 2008

			a(4)=11 because exactly 11 numbers between 2 and 2*prime(4)=2*7=14, namely: 2,3,4,5,6,7,8,9,10,12,14 are divisible by the first four primes 2,3,5,7.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000040, A100484, A137621, A137624.

Cf. A120633, A040976, A076274, A070046.

f:= proc(n) local p;
   p:= ithprime(n); 2*p - numtheory:-pi(2*p)+n-1
end proc:
map(f, [$1..100]); # Robert Israel, Mar 02 2022

a(n) = {nb = 0; for (i = 2, 2*prime(n), for (ip = 1, n, if ( !(i % prime(ip)), nb++; break;););); nb;} \\ Michel Marcus, Oct 26 2013

a(n) = A120633(n) + A040976(n+1) = A076274(n) - A070046(n).

A278229 Least number with the prime signature of 2*prime(n) - 1.

Original entry on oeis.org

2, 2, 4, 2, 6, 4, 6, 2, 12, 6, 2, 2, 16, 6, 6, 30, 12, 4, 6, 6, 6, 2, 30, 6, 2, 6, 6, 6, 6, 36, 6, 12, 30, 2, 24, 6, 2, 12, 12, 30, 30, 4, 6, 30, 6, 2, 2, 6, 6, 2, 30, 12, 6, 6, 24, 60, 6, 2, 6, 30, 6, 60, 2, 24, 16, 6, 2, 2, 60, 6, 30, 6, 2, 6, 2, 60, 30, 6, 12, 6, 24, 4, 30, 6, 2, 30, 30, 6, 6, 12, 6, 30, 6, 12, 2, 30, 12, 6, 30, 6, 2, 30, 72, 6, 6, 2, 30, 30

Offset: 1

Antti Karttunen, Nov 19 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A000040, A046523, A076274.
Cf. A137288 (positions of 2's), A005382.
Cf. also A278227, A278228, A278230.

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

(define (A278229 n) (A046523 (+ -1 (* 2 (A000040 n)))))

a(n) = A046523(A076274(n)) = A046523((2*A000040(n))-1).

cp's OEIS Frontend

A038547 Least number with exactly n odd divisors.

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A076271 a(1) = 1, a(2) = 2, and for n > 2, a(n) = a(n-1) + gpf(a(n-1)), where gpf = greatest prime factor = A006530.

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A327263 Array T(n,k) in which the i-th row consists of numbers > 1 not in array U(i;n,k) = (i*n*k - (i-2)*A319929(n,k))/2 where i >= 1, n >= 1 and k >= 1, read by antidiagonals.

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A139286 a(n) = 2^(2*prime(n) - 1).

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A139287 2^(2p - 1) - 1, where p is prime.

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A322630 Arithmetic table similar to multiplication with different rules for odd and even products, read by antidiagonals. T(n,k) = (n*k + A319929(n,k))/2.

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A128281 a(n) is the least product of n distinct odd primes m=p_1*p_2*...*p_n, such that (d+m/d)/2 are all primes for each d dividing m.

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A327263 Array T(n,k) in which the i-th row consists of numbers > 1 not in array U(i;n,k) = (ink - (i-2)*A319929(n,k))/2 where i >= 1, n >= 1 and k >= 1, read by antidiagonals.

A128281 a(n) is the least product of n distinct odd primes m=p_1p_2...*p_n, such that (d+m/d)/2 are all primes for each d dividing m.

A347682 Array read by antidiagonals: T(n,k) (n>=1, k>=1) = f(prime(n),prime(k)), where f(x,y) = xred_inv(x,y) + yred_inv(y,x) if gcd(x,y)=1, or 0 if gcd(x,y)>1, and red_inv is defined in the comments.