A246371 -id:A246371 - cp's OEIS frontend

A064989 Multiplicative with a(2^e) = 1 and a(p^e) = prevprime(p)^e for odd primes p.

1, 1, 2, 1, 3, 2, 5, 1, 4, 3, 7, 2, 11, 5, 6, 1, 13, 4, 17, 3, 10, 7, 19, 2, 9, 11, 8, 5, 23, 6, 29, 1, 14, 13, 15, 4, 31, 17, 22, 3, 37, 10, 41, 7, 12, 19, 43, 2, 25, 9, 26, 11, 47, 8, 21, 5, 34, 23, 53, 6, 59, 29, 20, 1, 33, 14, 61, 13, 38, 15, 67, 4, 71, 31, 18, 17, 35, 22, 73, 3, 16

Offset: 1

Vladeta Jovovic, Oct 30 2001

From Antti Karttunen, May 12 2014: (Start)
a(A003961(n)) = n for all n. [This is a left inverse function for the injection A003961.]
Bisections are A064216 (the terms at odd indices) and A064989 itself (the terms at even indices), i.e., a(2n) = a(n) for all n.
(End)
From Antti Karttunen, Dec 18-21 2014: (Start)
When n represents an unordered integer partition via the indices of primes present in its prime factorization (for n >= 2, n corresponds to the partition given as the n-th row of A112798) this operation subtracts one from each part. If n is of the form 2^k (a partition having just k 1's as its parts) the result is an empty partition (which is encoded by 1, having an "empty" factorization).
For all odd numbers n >= 3, a(n) tells which number is located immediately above n in square array A246278. Cf. also A246277.
(End)
Alternatively, if numbers are represented as the multiset of indices of prime factors with multiplicity, this operation subtracts 1 from each element and discards the 0's. - M. F. Hasler, Dec 29 2014

			a(20) = a(2^2*5) = a(2^2)*a(5) = prevprime(5) = 3.

Harry J. Smith (first 1000 terms) & Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A064216 (odd bisection), A003961 (inverse), A151799.
Cf. A000040, A008578, A027746, A032742, A049084, A052126, A055396, A061395, A112798, A249817, A249818.
Other sequences whose definition involve or are some other way related with this sequence: A105560, A108951, A118306, A122111, A156552, A163511, A200746, A241909, A243070, A243071, A243072, A243073, A244319, A245605, A245607, A246165, A246266, A246268, A246277, A246278, A246361, A246362, A246371, A246372, A246373, A246374, A246376, A246380, A246675, A246682, A249745, A250470.
Similar prime-shifts towards smaller numbers: A252461, A252462, A252463.

a064989 1 = 1
a064989 n = product $ map (a008578 . a049084) $ a027746_row n
-- Reinhard Zumkeller, Apr 09 2012
(MIT/GNU Scheme, with Aubrey Jaffer's SLIB Scheme library)
(require 'factor)
(define (A064989 n) (if (= 1 n) n (apply * (map (lambda (k) (if (zero? k) 1 (A000040 k))) (map -1+ (map A049084 (factor n)))))))
;; Antti Karttunen, May 12 2014
(definec (A064989 n) (if (= 1 n) n (* (A008578 (A055396 n)) (A064989 (A032742 n))))) ;; One based on given recurrence and utilizing memoizing definec-macro.
(definec (A064989 n) (cond ((= 1 n) n) ((even? n) (A064989 (/ n 2))) (else (A163511 (/ (- (A243071 n) 1) 2))))) ;; Corresponds to one of the alternative formulas, but is very unpractical way to compute this sequence. - Antti Karttunen, Dec 18 2014

q:= proc(p) prevprime(p) end proc: q(2):= 1:
[seq(mul(q(f[1])^f[2], f = ifactors(n)[2]), n = 1 .. 1000)]; # Robert Israel, Dec 21 2014

Table[Times @@ Power[Which[# == 1, 1, # == 2, 1, True, NextPrime[#, -1]] & /@ First@ #, Last@ #] &@ Transpose@ FactorInteger@ n, {n, 81}] (* Michael De Vlieger, Jan 04 2016 *)

{ for (n=1, 1000, f=factor(n)~; a=1; j=1; if (n>1 && f[1, 1]==2, j=2); for (i=j, length(f), a*=precprime(f[1, i] - 1)^f[2, i]); write("b064989.txt", n, " ", a) ) } \\ Harry J. Smith, Oct 02 2009

a(n) = {my(f = factor(n)); for (i=1, #f~, if ((p=f[i,1]) % 2, f[i,1] = precprime(p-1), f[i,1] = 1);); factorback(f);} \\ Michel Marcus, Dec 18 2014

A064989(n)=factorback(Mat(apply(t->[max(precprime(t[1]-1),1),t[2]],Vec(factor(n)~))~)) \\ M. F. Hasler, Dec 29 2014

from sympy import factorint, prevprime
from operator import mul
from functools import reduce
def a(n):
    f=factorint(n)
    return 1 if n==1 else reduce(mul, [1 if i==2 else prevprime(i)**f[i] for i in f])
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Jun 15 2017

from math import prod
from sympy import prevprime, factorint
def A064989(n): return prod(prevprime(p)**e for p, e in  factorint(n>>(~n&n-1).bit_length()).items()) # Chai Wah Wu, Jan 05 2023

From Antti Karttunen, Dec 18 2014: (Start)
If n = product A000040(k)^e(k) then a(n) = product A008578(k)^e(k) [where A000040(n) gives the n-th prime, and A008578(n) gives 1 for 1 and otherwise the (n-1)-th prime].
a(1) = 1; for n > 1, a(n) = A008578(A055396(n)) * a(A032742(n)). [Above formula represented as a recurrence. Cf. A252461.]
a(1) = 1; for n > 1, a(n) = A008578(A061395(n)) * a(A052126(n)). [Compare to the formula of A252462.]
This prime-shift operation is used in the definitions of many other sequences, thus it can be expressed in many alternative ways:
a(n) = A200746(n) / n.
a(n) = A242424(n) / A105560(n).
a(n) = A122111(A122111(n)/A105560(n)) = A122111(A052126(A122111(n))). [In A112798-partition context: conjugate, remove the largest part (the largest prime factor), and conjugate again.]
a(1) = 1; for n > 1, a(2n) = a(n), a(2n+1) = A163511((A243071(2n+1)-1) / 2).
a(n) = A249818(A250470(A249817(n))). [A250470 is an analogous operation for "going one step up" in the square array A083221 (A083140).]
(End)
Product_{k=1..n} a(k) = n! / A307035(n). - Vaclav Kotesovec, Mar 21 2019
Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * Product_{p prime} ((p^2-p)/(p^2-q(p))) = 0.220703928... , where q(p) = prevprime(p) (A151799) if p > 2 and q(2) = 1. - Amiram Eldar, Nov 18 2022

A064216 Replace each p^e with prevprime(p)^e in the prime factorization of odd numbers; inverse of sequence A048673 considered as a permutation of the natural numbers.

Original entry on oeis.org

1, 2, 3, 5, 4, 7, 11, 6, 13, 17, 10, 19, 9, 8, 23, 29, 14, 15, 31, 22, 37, 41, 12, 43, 25, 26, 47, 21, 34, 53, 59, 20, 33, 61, 38, 67, 71, 18, 35, 73, 16, 79, 39, 46, 83, 55, 58, 51, 89, 28, 97, 101, 30, 103, 107, 62, 109, 57, 44, 65, 49, 74, 27, 113, 82, 127, 85, 24, 131

Offset: 1

Howard A. Landman, Sep 21 2001

a((A003961(n) + 1) / 2) = n and A003961(a(n)) = 2*n - 1 for all n. If the sequence is indexed by odd numbers only, it becomes multiplicative. In this variant sequence, denoted b, even indices don't exist, and we get b(1) = a(1) = 1, b(3) = a(2) = 2, b(5) = 3, b(7) = 5, b(9) = 4 = b(3) * b(3), ... , b(15) = 6 = b(3) * b(5), and so on. This property can also be stated as: a(x) * a(y) = a(((2x - 1) * (2y - 1) + 1) / 2) for x, y > 0. - Reinhard Zumkeller [re-expressed by Peter Munn, May 23 2020]
Not multiplicative in usual sense - but letting m=2n-1=product_j (p_j)^(e_j) then a(n)=a((m+1)/2)=product_j (p_(j-1))^(e_j). - Henry Bottomley, Apr 15 2005
From Antti Karttunen, Jul 25 2016: (Start)
Several permutations that use prime shift operation A064989 in their definition yield a permutation obtained from their odd bisection when composed with this permutation from the right. For example, we have:
A243505(n) = A122111(a(n)).
A243065(n) = A241909(a(n)).
A244153(n) = A156552(a(n)).
A245611(n) = A243071(a(n)).
(End)

			For n=11, the 11th odd number is 2*11 - 1 = 21 = 3^1 * 7^1. Replacing the primes 3 and 7 with the previous primes 2 and 5 gives 2^1 * 5^1 = 10, so a(11) = 10. - _Michael B. Porter_, Jul 25 2016

Odd bisection of A064989 and A252463.
Row 1 of A251721, Row 2 of A249821.
Cf. A048673 (inverse permutation), A048674 (fixed points).
Cf. A246361 (numbers n such that a(n) <= n.)
Cf. A246362 (numbers n such that a(n) > n.)
Cf. A246371 (numbers n such that a(n) < n.)
Cf. A246372 (numbers n such that a(n) >= n.)
Cf. A246373 (primes p such that a(p) >= p.)
Cf. A246374 (primes p such that a(p) < p.)
Cf. A246343 (iterates starting from n=12.)
Cf. A246345 (iterates starting from n=16.)
Cf. A245448 (this permutation "squared", a(a(n)).)
Cf. A253894, A254044, A254045 (binary width, weight and the number of nonleading zeros in base-2 representation of a(n), respectively).
Cf. A285702, A285703 (phi and sigma applied to a(n).)
Here obviously the variant 2, A151799(n) = A007917(n-1), of the prevprime function is used.
Cf. also A003961, A270430, A270431.
Cf. also permutations A122111, A156552, A241909, A243071, A243065, A243505, A244153, A245611, A254116.

Table[Times @@ Power[If[# == 1, 1, NextPrime[#, -1]] & /@ First@ #, Last@ #] &@ Transpose@ FactorInteger[2 n - 1], {n, 69}] (* Michael De Vlieger, Dec 18 2014, revised Mar 17 2016 *)

a(n) = {my(f = factor(2*n-1)); for (k=1, #f~, f[k,1] = precprime(f[k,1]-1)); factorback(f);} \\ Michel Marcus, Mar 17 2016

from sympy import factorint, prevprime
from operator import mul
def a(n):
    f=factorint(2*n - 1)
    return 1 if n==1 else reduce(mul, [prevprime(i)**f[i] for i in f]) # Indranil Ghosh, May 13 2017

(define (A064216 n) (A064989 (- (+ n n) 1))) ;; Antti Karttunen, May 12 2014

a(n) = A064989(2n - 1). - Antti Karttunen, May 12 2014

Sum_{k=1..n} a(k) ~ c * n^2, where c = Product_{p prime > 2} ((p^2-p)/(p^2-q(p))) = 0.6621117868..., where q(p) = prevprime(p) (A151799). - Amiram Eldar, Jan 21 2023

More terms from Reinhard Zumkeller, Sep 26 2001

Additional description added by Antti Karttunen, May 12 2014

A048674 Fixed points of A048673 and A064216: Numbers n such that if n = product_{k >= 1} (p_k)^(c_k), then Product_{k >= 1} (p_{k+1})^(c_k) = (2*n)-1, where p_k indicates the k-th prime, A000040(k).

Original entry on oeis.org

1, 2, 3, 25, 26, 33, 93, 1034, 970225, 8550146, 325422273, 414690595, 1864797542, 2438037206

Offset: 1

Antti Karttunen, Jul 14 1999

Equally: after 1, numbers n such that, if the prime factorization of 2n-1 = Product_{k >= 1} (p_k)^(c_k) then Product_{k >= 1} (p_{k-1})^(c_k) = n.
Factorization of the initial terms: 1, 2, 3, 5^2, 2*13, 3*11, 3*31, 2*11*47, 5^2*197^2, 2*11*47*8269, 3*11*797*12373, 5*11^2*433*1583, 2*23*59*101*6803, 2*11*53*1201*1741.
The only 3-cycle of permutation A048673 in range 1 .. 402653184 is (2821 3460 5639).
For 2-cycles, take setwise difference of A245449 and this sequence.
Numbers k for which A336853(k) = k-1. - Antti Karttunen, Nov 26 2021

			25 is present, as 2*25 - 1 = 49 = p_4^2, and p_3^2 = 5*5 = 25.
26 is present, as 2*26 - 1 = 51 = 3*17 = p_2 * p_8, and p_1 * p_7 = 2*13 = 26.
Alternatively, as 26 = 2*13 = p_1 * p_7, and ((p_2 * p_8)+1)/2 = ((3*17)+1)/2 = 26 also, thus 26 is present.

Fixed points of permutation pair A048673/A064216.
Positions of zeros in A349573.
Subsequence of the following sequences: A245449, A269860, A319630, A349622, A378980 (see also A379216).
This sequence is also obtained as a setwise difference of the following pairs of sequences: A246281 \ A246351, A246352 \ A246282, A246361 \ A246371, A246372 \ A246362.
Cf. A000040, A003961 (A045965), A064989, A285701, A336853.
Cf. also A348514 (fixed points of map A108228, similar to A048673).

A048673 := n -> (A003961(n)+1)/2;
A048674list := proc(upto_n) local b,i; b := [ ]; for i from 1 to upto_n do if(A048673(i) = i) then b := [ op(b), i ]; fi; od: RETURN(b); end;

Join[{1}, Reap[For[n = 1, n < 10^7, n++, ff = FactorInteger[n]; If[Times @@ Power @@@ (NextPrime[ff[[All, 1]]]^ff[[All, 2]]) == 2 n - 1, Print[n]; Sow[n]]]][[2, 1]]] (* Jean-François Alcover, Mar 04 2016 *)

A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
isA048674(n) = ((n+n)==(1+A003961(n))); \\ Antti Karttunen, Nov 26 2021

Entry revised and the names in Maple-code cleaned by Antti Karttunen, Aug 25 2014

Terms a(11) - a(14) added by Antti Karttunen, Sep 11-13 2014

A104275 Numbers k such that 2k-1 is not prime.

Original entry on oeis.org

1, 5, 8, 11, 13, 14, 17, 18, 20, 23, 25, 26, 28, 29, 32, 33, 35, 38, 39, 41, 43, 44, 46, 47, 48, 50, 53, 56, 58, 59, 60, 61, 62, 63, 65, 67, 68, 71, 72, 73, 74, 77, 78, 80, 81, 83, 85, 86, 88, 89, 92, 93, 94, 95, 98, 101, 102, 103, 104, 105, 107, 108, 109, 110, 111, 113

Offset: 1

Alexandre Wajnberg, Apr 17 2005

Same as A053726 except for the first term of this sequence.
Numbers k such that A064216(k) is not prime. - Antti Karttunen, Apr 17 2015
Union of 1 and terms of the form (u+1)*(v+1) + u*v with 1 <= u <= v. - Ralf Steiner, Nov 17 2021

			a(1) = 1 because 2*1-1=1, not prime.
a(2) = 5 because 2*5-1=9, not prime (2, 3 and 4 give 3, 5 and 7 which are primes).
From _Vincenzo Librandi_, Jan 15 2013: (Start)
As a triangular array (apart from term 1):
   5;
   8,  13;
  11,  18,  25;
  14,  23,  32,  41;
  17,  28,  39,  50,  61;
  20,  33,  46,  59,  72,  85;
  23,  38,  53,  68,  83,  98, 113;
  26,  43,  60,  77,  94, 111, 128, 145;
  29,  48,  67,  86, 105, 124, 143, 162, 181;
  32,  53,  74,  95, 116, 137, 158, 179, 200, 221; etc.
which is obtained by (2*h*k + k + h + 1) with h >= k >= 1. (End)
The above array, which contains the same terms as A053726 but in different order and with some duplicates, has its own entry A144650. - _Antti Karttunen_, Apr 17 2015

Vincenzo Librandi (first 1000 terms) & Antti Karttunen, Table of n, a(n) for n = 1..10001

Cf. A006254 (complement), A246371 (a subsequence).

Cf. A005097, A008508, A014076, A047845, A053726, A064216, A144650.

[n: n in [1..220]| not IsPrime(2*n-1)]; // Vincenzo Librandi, Jan 28 2011

remove(t -> isprime(2*t-1), [$1..1000]); # Robert Israel, Apr 17 2015

Select[Range[115], !PrimeQ[2#-1] &] (* Robert G. Wilson v, Apr 18 2005 *)

select( {is_A104275(n)=!isprime(n*2-1)}, [1..115]) \\ M. F. Hasler, Aug 02 2022

from sympy import isprime
def ok(n): return not isprime(2*n-1)
print(list(filter(ok, range(1, 114)))) # Michael S. Branicky, May 08 2021

from sympy import primepi
def A104275(n):
    if n <= 2: return ((n-1)<<2)+1
    m, k = n-1, (r:=primepi(n-1)) + n - 1 + (n-1>>1)
    while m != k:
        m, k = k, (r:=primepi(k)) + n - 1 + (k>>1)
    return r+n-1 # Chai Wah Wu, Aug 02 2024

[n for n in (1..250) if not is_prime(2*n-1)] # G. C. Greubel, Oct 17 2023

(define (A104275 n) (if (= 1 n) 1 (A053726 (- n 1)))) ;; More code in A053726. - Antti Karttunen, Apr 17 2015

a(n) = A047845(n-1) + 1.
For n > 1, a(n) = A053726(n-1) = n + A008508(n-1). - Antti Karttunen, Apr 17 2015
a(n) = (A014076(n)+1)/2. - Robert Israel, Apr 17 2015

More terms from Robert G. Wilson v, Apr 18 2005

A246351 Numbers k such that A048673(k) < k.

Original entry on oeis.org

5, 7, 11, 13, 17, 19, 22, 23, 29, 31, 34, 37, 38, 41, 43, 46, 47, 51, 53, 55, 58, 59, 61, 62, 65, 67, 71, 73, 74, 77, 79, 82, 83, 85, 86, 87, 89, 94, 95, 97, 101, 103, 106, 107, 109, 111, 113, 115, 118, 119, 121, 122, 123, 127, 129, 131, 133, 134, 137, 139, 141, 142, 143, 145, 146, 149, 151, 155, 157, 158, 159

Offset: 1

Antti Karttunen, Aug 24 2014

nonn

The growth rate of the sequence seems to converge:
       a(100) = 217
      a(1000) = 2231
     a(10000) = 21535
    a(100000) = 214647
   a(1000000) = 2155903
  a(10000000) = 21553153
Please see comments in A246282.

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

Complement: A246352.
Setwise difference of A246281 and A048674.
Cf. A048673, A246371, A246282.

default(primelimit, 2^22);
A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From Michel Marcus
A048673(n) = (A003961(n)+1)/2;
isA246351(n) = (A048673(n) < n);
n = 0; i = 0; while(i < 10000, n++; if(isA246351(n), i++; write("b246351.txt", i, " ", n)));

;; With Antti Karttunen's IntSeq-library.
(define A246351 (MATCHING-POS 1 1 (lambda (n) (< (A048673 n) n))))

A246361 Numbers n such that if 2n-1 = product_{k >= 1} (p_k)^(c_k), then n >= product_{k >= 1} (p_{k-1})^(c_k), where p_k indicates the k-th prime, A000040(k).

Original entry on oeis.org

1, 2, 3, 5, 8, 11, 13, 14, 17, 18, 23, 25, 26, 28, 32, 33, 38, 39, 41, 43, 50, 53, 58, 59, 61, 63, 68, 73, 74, 77, 83, 86, 88, 93, 94, 95, 98, 104, 113, 116, 122, 123, 128, 131, 137, 138, 140, 143, 149, 158, 163, 167, 172, 173, 176, 179, 182, 185, 188, 193, 194, 200, 203, 212, 213, 215, 218, 221, 228, 230, 233

Offset: 1

Antti Karttunen, Aug 24 2014

nonn

Numbers n such that A064216(n) <= n.
Numbers n such that A064989(2n-1) <= n.
The sequence grows as:
      a(100) = 332
     a(1000) = 3207
    a(10000) = 34213
   a(100000) = 340703
  a(1000000) = 3388490
suggesting that overall, less than one third of natural numbers appear in this sequence, and more than two thirds in the complement, A246362. See also comments in the latter.

			1 is present, as 2*1 - 1 = empty product = 1.
12 is not present, as (2*12)-1 = 23 = p_9, and p_8 = 19, with 12 < 19.
14 is present, as (2*14)-1 = 27 = p_2^3 = 8, and 14 >= 8.

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

Complement: A246362.
Union of A246371 and A048674.
Subsequence: A246360.
Cf. A000040, A064216, A064989, A246281.

default(primelimit, 2^30);
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A064216(n) = A064989((2*n)-1);
isA246361(n) = (A064216(n) <= n);
n = 0; i = 0; while(i < 10000, n++; if(isA246361(n), i++; write("b246361.txt", i, " ", n)));
(Scheme, with Antti Karttunen's IntSeq-library)
(define A246361 (MATCHING-POS 1 1 (lambda (n) (<= (A064216 n) n))))

A246372 Numbers n such that 2n-1 = product_{k >= 1} (p_k)^(c_k), then n <= product_{k >= 1} (p_{k-1})^(c_k), where p_k indicates the k-th prime, A000040(k).

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 9, 10, 12, 15, 16, 19, 20, 21, 22, 24, 25, 26, 27, 29, 30, 31, 33, 34, 35, 36, 37, 40, 42, 44, 45, 46, 47, 48, 49, 51, 52, 54, 55, 56, 57, 60, 62, 64, 65, 66, 67, 69, 70, 71, 72, 75, 76, 78, 79, 80, 81, 82, 84, 85, 87, 89, 90, 91, 92, 93, 96, 97, 99, 100, 101, 102, 103, 105, 106, 107, 108, 109, 110

Offset: 1

Antti Karttunen, Aug 24 2014

nonn

Numbers n such that A064216(n) >= n.

Numbers n such that A064989(2n-1) >= n.

			1 is present, as 2*1 - 1 = empty product = 1.
2 is present, as 2*2 - 1 = 3 = p_2, and p_{2-1} = p_1 = 2 >= 2.
3 is present, as 2*3 - 1 = 5 = p_3, and p_{3-1} = p_2 = 3 >= 3.
5 is not present, as 2*5 - 1 = 9 = p_2 * p_2, and p_1 * p_1 = 4, with 4 < 5.
6 is present, as 2*6 - 1 = 11 = p_5, and p_{5-1} = p_4 = 7 >= 6.
25 is present, as 2*25 - 1 = 49 = p_4^2, and p_3^2 = 5*5 = 25 >= 25.
35 is present, as 2*35 - 1 = 69 = 3*23 = p_2 * p_9, and p_1 * p_8 = 2*19 = 38 >= 35.

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

Complement: A246371
Union of A246362 and A048674.
Subsequences: A006254 (A111333), A246373 (the primes present in this sequence).
Cf. A000040, A064216, A064989, A246352.

default(primelimit, 2^30);
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A064216(n) = A064989((2*n)-1);
isA246372(n) = (A064216(n) >= n);
n = 0; i = 0; while(i < 10000, n++; if(isA246372(n), i++; write("b246372.txt", i, " ", n)));
(Scheme, with Antti Karttunen's IntSeq-library)
(define A246372 (MATCHING-POS 1 1 (lambda (n) (>= (A064216 n) n))))

A053726 "Flag numbers": number of dots that can be arranged in successive rows of K, K-1, K, K-1, K, ..., K-1, K (assuming there is a total of L > 1 rows of size K > 1).

Original entry on oeis.org

5, 8, 11, 13, 14, 17, 18, 20, 23, 25, 26, 28, 29, 32, 33, 35, 38, 39, 41, 43, 44, 46, 47, 48, 50, 53, 56, 58, 59, 60, 61, 62, 63, 65, 67, 68, 71, 72, 73, 74, 77, 78, 80, 81, 83, 85, 86, 88, 89, 92, 93, 94, 95, 98, 101, 102, 103, 104, 105, 107, 108, 109, 110, 111, 113, 116

Offset: 1

Dan Asimov, asimovd(AT)aol.com, Apr 09 2003

Numbers of the form F(K, L) = KL+(K-1)(L-1), K, L > 1, i.e. 2KL - (K+L) + 1, sorted and duplicates removed.
If K=1, L=1 were allowed, this would contain all positive integers.
Positive numbers > 1 but not of the form (odd primes plus one)/2. - Douglas Winston (douglas.winston(AT)srupc.com), Sep 11 2003
In other words, numbers n such that 2n-1, or equally, A064216(n) is a composite number. - Antti Karttunen, Apr 17 2015
Note: the following comment was originally applied in error to the numerically similar A246371. - Allan C. Wechsler, Aug 01 2022
From Matthijs Coster, Dec 22 2014: (Start)
Also area of (over 45 degree) rotated rectangles with sides > 1. The area of such rectangles is 2ab - a - b + 1 = 1/2((2a-1)(2b-1)+1).
Example: Here a = 3 and b = 5. The area = 23.
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(End)
The smallest integer > k/2 and coprime to k, where k is the n-th odd composite number. - Mike Jones, Jul 22 2024
Numbers k such that A193773(k-1) > 1. - Allan C. Wechsler, Oct 22 2024

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

Essentially same as A104275, but without the initial one.
A144650 sorted into ascending order, with duplicates removes.
Cf. A006254 (complement, apart from 1, which is in neither sequence).
Cf. also A000720, A008508, A064216, A071904, A199593, A250474.
Differs from its subsequence A246371 for the first time at a(8) = 20, which is missing from A246371.

select( {is_A053726(n)=n>4 && !isprime(n*2-1)}, [1..115]) \\ M. F. Hasler, Aug 02 2022

from sympy import isprime
def ok(n): return n > 1 and not isprime(2*n-1)
print(list(filter(ok, range(1, 117)))) # Michael S. Branicky, May 08 2021

from sympy import primepi
def A053726(n):
    if n == 1: return 5
    m, k = n, (r:=primepi(n)) + n + (n>>1)
    while m != k:
        m, k = k, (r:=primepi(k)) + n + (k>>1)
    return r+n # Chai Wah Wu, Aug 02 2024

;; with Antti Karttunen's IntSeq-library.
(define A053726 (MATCHING-POS 1 1 (lambda (n) (and (> n 1) (not (prime? (+ n n -1)))))))
;; Antti Karttunen, Apr 17 2015

;; with Antti Karttunen's IntSeq-library.
(define (A053726 n) (+ n (A000720 (A071904 n))))
;; Antti Karttunen, Apr 17 2015

a(n) = A008508(n) + n + 1.
From Antti Karttunen, Apr 17 2015: (Start)
a(n) = n + A000720(A071904(n)). [The above formula reduces to this. A000720(k) gives number of primes <= k, and A071904 gives the n-th odd composite number.]
a(n) = A104275(n+1). (End)
a(n) = A116922(A071904(n)). - Mike Jones, Jul 22 2024
a(n) = A047845(n+1)+1. - Amiram Eldar, Jul 30 2024

More terms from Douglas Winston (douglas.winston(AT)srupc.com), Sep 11 2003

A246342 a(0) = 12, after which, if a(n-1) = product_{k >= 1} (p_k)^(c_k), then a(n) = (1/2) * (1 + product_{k >= 1} (p_{k+1})^(c_k)), where p_k indicates the k-th prime, A000040(k).

Original entry on oeis.org

12, 23, 15, 18, 38, 35, 39, 43, 24, 68, 86, 71, 37, 21, 28, 50, 74, 62, 56, 149, 76, 104, 230, 305, 235, 186, 278, 224, 1337, 1062, 2288, 8951, 4482, 16688, 67271, 33637, 16821, 66688, 571901, 338059, 181516, 258260, 455900, 1180337, 1080207, 1817863, 1157487, 984558, 1230848, 53764115

Offset: 0

Antti Karttunen, Aug 24 2014

nonn

Iterates of A048673 starting from value 12.
All numbers 1 .. 11 are in finite cycles of A048673/A064216, thus 12 is the smallest number in this cycle, regardless of whether the cycle is infinite or finite.
This sequence soon reaches much larger values than the corresponding A246343 (iterating the same cycle in the other direction). However, with the corresponding sequences starting from 16 (A246344 & A246345), there is no such pronounced difference, and with them the bias is actually the other way.

			Start with a(0) = 12; thereafter each new term is obtained by replacing each prime factor of the previous term with the next prime, to whose product 1 is added before it is halved:
12 = 2^2 * 3 = p_1^2 * p_2 -> ((p_2^2 * p_3)+1)/2 = ((9*5)+1)/2 = 23, thus a(1) = 23.
23 = p_9 -> (p_10 + 1)/2 = (29+1)/2 = 15, thus a(2) = 15.

Antti Karttunen, Table of n, a(n) for n = 0..1001

A246343 gives the terms of the same cycle when going in the opposite direction from 12.
Cf. A048673, A064216, A246344, A246345.
Cf. also A246281, A246282, A246351, A246352, A246361, A246362, A246371, A246372.

default(primelimit, 2^30);
A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ Using code of Michel Marcus
A048673(n) = (A003961(n)+1)/2;
k = 12; for(n=0, 1001, write("b246342.txt", n, " ", k) ; k = A048673(k));
(Scheme, with memoization-macro definec)
(definec (A246342 n) (if (zero? n) 12 (A048673 (A246342 (- n 1)))))

a(0) = 12, and for n >= 1, a(n) = A048673(a(n-1)).

A246343 a(0) = 12, after which, if (2*a(n-1)) - 1 = product_{k >= 1} (p_k)^(c_k) then a(n) = product_{k >= 1} (p_{k-1})^(c_k), where p_k indicates the k-th prime, A000040(k).

Original entry on oeis.org

12, 19, 31, 59, 44, 46, 55, 107, 134, 166, 317, 398, 282, 557, 470, 622, 763, 531, 1051, 1267, 1807, 3607, 7211, 4522, 9041, 3700, 3725, 3982, 7951, 15889, 30053, 24018, 24189, 34535, 14630, 12916, 21769, 27599, 24524, 32678, 26094, 43073, 34446, 68881, 116479, 143359, 275221, 550439, 667462, 1051489

Offset: 0

Antti Karttunen, Aug 24 2014

nonn

Iterates of A064216 starting from value 12.

All numbers from 1 to 11 are in finite cycles of A048673/A064216, thus 12 is the smallest number in this cycle, regardless of whether it is infinite or finite.

			Start with a(0) = 12; then after each new term is obtained by doubling the previous term, from which one is subtracted, after which each prime factor is replaced with the previous prime:
12 -> ((2*12)-1) = 23 = p_9, and p_8 = 19, thus a(1) = 19.
19 -> ((2*19)-1) = 37 = p_12, and p_11 = 31, thus a(2) = 31.
31 -> ((2*31)-1) = 61 = p_18, and p_17 = 59, thus a(3) = 59.
59 -> ((2*59)-1) = 117 = 3*3*13 = p_2 * p_2 * p_6, and p_1 * p_1 * p_5 = 2*2*11 = 44, thus a(4) = 44.

Antti Karttunen, Table of n, a(n) for n = 0..1001

A246342 gives the terms of the same cycle when going to the opposite direction from 12.
Cf. A048673, A064216, A246344, A246345.
Cf. also A246281, A246282, A246351, A246352, A246361, A246362, A246371, A246372.

default(primelimit, 2^30);
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A064216(n) = A064989((2*n)-1);
k = 12; for(n=0, 1001, write("b246343.txt", n, " ", k); k = A064216(k));
(Scheme, with memoization-macro definec)
(definec (A246343 n) (if (zero? n) 12 (A064216 (A246343 (- n 1)))))

a(0) = 12, a(n) = A064216(a(n-1)).

cp's OEIS Frontend

A064989 Multiplicative with a(2^e) = 1 and a(p^e) = prevprime(p)^e for odd primes p.

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A064216 Replace each p^e with prevprime(p)^e in the prime factorization of odd numbers; inverse of sequence A048673 considered as a permutation of the natural numbers.

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A048674 Fixed points of A048673 and A064216: Numbers n such that if n = product_{k >= 1} (p_k)^(c_k), then Product_{k >= 1} (p_{k+1})^(c_k) = (2*n)-1, where p_k indicates the k-th prime, A000040(k).

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A104275 Numbers k such that 2k-1 is not prime.

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A246351 Numbers k such that A048673(k) < k.

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A246361 Numbers n such that if 2n-1 = product_{k >= 1} (p_k)^(c_k), then n >= product_{k >= 1} (p_{k-1})^(c_k), where p_k indicates the k-th prime, A000040(k).

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