A020725 -id:A020725 - cp's OEIS frontend

A335738 Factorize each integer m >= 2 as the product of powers of nonunit squarefree numbers with distinct exponents that are powers of 2. The sequence lists m such that the factor with the largest exponent is a power of 2.

2, 4, 8, 12, 16, 20, 24, 28, 32, 40, 44, 48, 52, 56, 60, 64, 68, 76, 80, 84, 88, 92, 96, 104, 112, 116, 120, 124, 128, 132, 136, 140, 144, 148, 152, 156, 160, 164, 168, 172, 176, 184, 188, 192, 204, 208, 212, 220, 224, 228, 232, 236, 240, 244, 248, 256, 260, 264, 268, 272

Offset: 1

Peter Munn, Jun 20 2020

nonn

2 is the only term not divisible by 4. All powers of 2 are present. Every term divisible by an odd square is divisible by 16, the first such being 144.
The defined factorization is unique. Every positive number is a product of at most one squarefree number (A005117), at most one square of a squarefree number (A062503), at most one 4th power of a squarefree number (A113849), at most one 8th power of a squarefree number, and so on.
Iteratively map m using A000188, until 1 is reached, as A000188^k(m), for some k >= 1. m is in the sequence if and only if the preceding number, A000188^(k-1)(m), is 2. k can be shown to be A299090(m).
Closed under squaring, but not closed under multiplication: 12 = 3^1 * 2^2 and 432 = 3^1 * 3^2 * 2^4 are in the sequence, but 12 * 432 = 5184 = 3^4 * 2^6 = 2^2 * 6^4 is not.
The asymptotic density of this sequence is Sum_{k>=0} (d(2^(k+1)) - d(2^k))/2^(2^(k+1)-1) = 0.21363357193921052068..., where d(k) = 2^(k-1)/((2^k-1)*zeta(k)) is the asymptotic density of odd k-free numbers for k >= 2, and d(1) = 0. - Amiram Eldar, Feb 10 2024

			6 is a squarefree number, so its factorization for the definition (into powers of nonunit squarefree numbers with distinct exponents that are powers of 2) is the trivial "6^1". 6^1 is therefore the factor with the largest exponent, and is not a power of 2, so 6 is not in the sequence.
48 factorizes for the definition as 3^1 * 2^4. The factor with the largest exponent is 2^4, which is a power of 2, so 48 is in the sequence.
10^100 (a googol) factorizes in this way as 10^4 * 10^32 * 10^64. The factor with the largest exponent, 10^64, is a power of 10, not a power of 2, so 10^100 is not in the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Complement within A020725 of A335740.
A000188, A299090 are used in a formula defining this sequence.
Powers of squarefree numbers: A005117(1), A144338(1), A062503(2), A113849(4).
Subsequences: A000079\{1}, A001749, A181818\{1}, A273798.
Numbers in the even bisection of A336322.
Row m of A352780 essentially gives the defined factorization of m.

f[p_, e_] := p^Floor[e/2]; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; Select[Range[2, 300], FixedPointList[s, #] [[-3]] == 2 &] (* Amiram Eldar, Nov 27 2020 *)

is(n) = {my(e = valuation(n, 2), o = n >> e); if(e == 0, 0, if(o == 1, n > 1, floor(logint(e, 2)) > floor(logint(vecmax(factor(o)[,2]), 2))));} \\ Amiram Eldar, Feb 10 2024

{a(n)} = {m : m >= 2 and A000188^(k-1)(m) = 2, where k = A299090(m)}.

{a(n)} = {m : m >= 2 and A352780(m,e) = 2^(2^e), where e = A299090(m)-1}. - Peter Munn, Jun 24 2022

A085554 Greater of twin primes of the form x^2+2, x^2+4.

Original entry on oeis.org

5, 13, 229, 1093, 2029, 3253, 13693, 21613, 59053, 65029, 91813, 140629, 178933, 199813, 205213, 227533, 328333, 567013, 700573, 804613, 815413, 1071229, 2241013, 3629029, 4223029, 4347229, 4809253, 5212093, 5919493, 6185173

Offset: 1

Cino Hilliard, Jul 04 2003

Except for the first term, all a(n)=13 (mod 72) with x=3 (mod 6). The lesser of the twin prime pair is given by A253639, the x-values in A086381. - M. F. Hasler, Jan 18 2015

M. F. Hasler, Table of n, a(n) for n = 1..3044 (all terms below 10^12).

Cf. A085553, A086381, A087475, A059100.

Transpose[Select[Table[x^2+{2,4},{x,5000}],AllTrue[#,PrimeQ]&]][[2]] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Jan 15 2015 *)

is_A086381(x)=ispseudoprime(x^2+2)&&ispseudoprime(x^2+4) \\ or is_A067201(x)&&is_A007591(x)
A085554 = apply(A087475,select(is_A086381,vector(9999,n,n))) \\ A087475=x->x^2+4.
write(f="b085554.txt",c=1," 5"); forstep(x=3,1e6,6,is_A086381(x)&&write(f,c++" "x^2+4))
\\ M. F. Hasler, Jan 18 2015

A085554 = A087475 o A086381 = A020725^2 o A253639, i.e., a(n) = A087475(A086381(n)) = A253639(n)+2. - M. F. Hasler, Jan 18 2015

Edited by Don Reble, May 03 2006

Definition corrected by Harvey P. Dale and Franklin T. Adams-Watters, Jan 15 2015

A335740 Factorize each integer m >= 2 as the product of powers of nonunit squarefree numbers with distinct exponents that are powers of 2. The sequence lists m such that the factor with the largest exponent is not a power of 2.

Original entry on oeis.org

3, 5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 18, 19, 21, 22, 23, 25, 26, 27, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 45, 46, 47, 49, 50, 51, 53, 54, 55, 57, 58, 59, 61, 62, 63, 65, 66, 67, 69, 70, 71, 72, 73, 74, 75, 77, 78, 79, 81, 82, 83, 85, 86, 87, 89, 90

Offset: 1

Peter Munn, Jun 20 2020

nonn

Every missing number greater than 2 is a multiple of 4. Every power of 2 is missing. Every positive power of every squarefree number greater than 2 is present.
The defined factorization is unique. Every positive number is a product of at most one squarefree number (A005117), at most one square of a squarefree number (A062503), at most one 4th power of a squarefree number (A113849), at most one 8th power of a squarefree number, and so on.
Iteratively map m using A000188, until 1 is reached, as A000188^k(m), for some k >= 1. m is in the sequence if and only if the preceding number, A000188^(k-1)(m), is greater than 2. k can be shown to be A299090(m).
The asymptotic density of this sequence is 1 - Sum_{k>=0} (d(2^(k+1)) - d(2^k))/2^(2^(k+1)-1) = 0.78636642806078947931..., where d(k) = 2^(k-1)/((2^k-1)*zeta(k)) is the asymptotic density of odd k-free numbers for k >= 2, and d(1) = 0. - Amiram Eldar, Feb 10 2024

			6 is a squarefree number, so its factorization for the definition (into powers of nonunit squarefree numbers with distinct exponents that are powers of 2) is the trivial "6^1". 6^1 is therefore the factor with the largest exponent, and is not a power of 2, so 6 is in the sequence.
48 factorizes for the definition as 3^1 * 2^4. The factor with the largest exponent is 2^4, which is a power of 2, so 48 is not in the sequence.
10^100 (a googol) factorizes in this way as 10^4 * 10^32 * 10^64. The factor with the largest exponent, 10^64, is a power of 10, not a power of 2, so 10^100 is in the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Complement within A020725 of A335738.
A000188, A299090 are used in a formula defining this sequence.
Powers of squarefree numbers: A005117(1), A144338(1), A062503(2), A113849(4).
Subsequences: A042968\{1,2}, A182853, A268390.
With {1}, numbers in the odd bisection of A336322.

f[p_, e_] := p^Floor[e/2]; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; Select[Range[2, 100], FixedPointList[s, #] [[-3]] > 2 &] (* Amiram Eldar, Nov 27 2020 *)

is(n) = {my(e = valuation(n, 2), o = n >> e); if(e == 0, n > 1, if(o == 1, e < 1, floor(logint(e, 2)) <= floor(logint(vecmax(factor(o)[,2]), 2))));} \\ Amiram Eldar, Feb 10 2024

{a(n)} = {m : m >= 2 and A000188^(k-1)(m) > 2, where k = A299090(m)}.

A175430 a(n) = (n-1)! * (n+1)!.

Original entry on oeis.org

2, 6, 48, 720, 17280, 604800, 29030400, 1828915200, 146313216000, 14485008384000, 1738201006080000, 248562743869440000, 41758540970065920000, 8142915489162854400000, 1824013069572479385600000, 465123332740982243328000000, 133955519829402886078464000000

Offset: 1

Jaroslav Krizek, May 10 2010

Also the number of automorphisms in the (n+1)-alternating group graph for n > 1. - Eric W. Weisstein, Jul 30 2018

			a(5) = (5-1)! * (5+1)! = 4! * 6! = 24 * 720 = 17280.
a(5) = ((1*2) * (2*3) * (3*4) * (4*5) * (5*6)) / 5 = 17280.

Amiram Eldar, Table of n, a(n) for n = 1..253
Eric Weisstein's World of Mathematics, Alternating Group Graph.
Eric Weisstein's World of Mathematics, Graph Automorphism.

Cf. A020725, A129464, A179442, A229020.

#[[1]] #[[3]] & /@ Partition[Range[0, 20]!, 3, 1] (* Harvey P. Dale, Jun 20 2017 *)
Table[(n - 1)! (n + 1)!, {n, 20}] (* or *) Table[Gamma[n] Gamma[n + 2], {n, 20}] (* Eric W. Weisstein, Jul 30 2018 *)

a(n) = (n-1)! * (n+1)!; \\ Michel Marcus, Feb 03 2016

a(n) = (Product_{k=1..n} k*A020725(k)) / n;
a(n) = (Product_{k=1..n} k*(k+1)) / n.
a(n) = (-1) * A129464(n) = A179442(n) * n. [Jaroslav Krizek, Jul 22 2010]
a(n) = 2*A010791(n-1). - Michel Marcus, Feb 03 2016
Sum_{n>=1} 1/a(n) = A229020 = BesselI(2,2). - Amiram Eldar, Oct 07 2020

A067665 The start of a record-setting run of consecutive integers i with distinct A001222(i).

Original entry on oeis.org

1, 6, 15, 60, 726, 6318, 189375, 755968, 683441871, 33714015615

Offset: 1

G. L. Honaker, Jr., Feb 03 2002

The list of indices of record terms in A068796;
n is in the sequence if A068796(n) is larger than A068796(m) when n is larger than m. For the known terms, f(a(n)) = n+1. Is that true for all n? In other words, is the monotonic subsequence of A068796 identical to A020725?
a(11) > 10^13. - Giovanni Resta, Jan 08 2014

			The values of f(n) for n=1 to 15 are 2,1,2,2,2,3,3,2,1,3,2,3,2,1,4. Records occur at f(1)=2, f(6)=3 and f(15)=4.

J.-M. De Koninck, J. B. Friedlander, and F. Luca, On strings of consecutive integers with a distinct number of prime factors, Proc. Amer. Math. Soc., 137 (2009), 1585-1592.

Cf. A001222, A067650, A068069, A068796, A068797.

bigomega[n_] := Plus@@Last/@FactorInteger[n]; f[n_] := For[k=1; s={bigomega[n]}, True, k++, If[MemberQ[s, z=bigomega[n+k]], Return[k], AppendTo[s, z]]]; For[n=1; max=0, True, n++, If[f[n]>max, Print[n, " ", max=f[n]]]]

a(n,lim=1e12,startAt=1)={
  forstep(i=startAt-1,lim,10^6-n,
    my(v=vectorsmall(min(10^6,lim\1-i),j,bigomega(j+i)));
    for(j=n,#v,if(#vecsort(v[j-n+1..j],,8)==n,return(j+i-n+1)))
  )
}; \\ Charles R Greathouse IV, Jul 03 2013

More terms from Shyam Sunder Gupta, Feb 08 2002
Edited by Robert G. Wilson v, Feb 20 2002
Edited by Dean Hickerson, Mar 05 2002
a(10) from Donovan Johnson, Oct 15 2008

A155705 Triangle read by rows where T(m,n) = 2mn + m + n + 2.

Original entry on oeis.org

6, 9, 14, 12, 19, 26, 15, 24, 33, 42, 18, 29, 40, 51, 62, 21, 34, 47, 60, 73, 86, 24, 39, 54, 69, 84, 99, 114, 27, 44, 61, 78, 95, 112, 129, 146, 30, 49, 68, 87, 106, 125, 144, 163, 182, 33, 54, 75, 96, 117, 138, 159, 180, 201, 222, 36, 59, 82, 105, 128, 151

Offset: 1

Vincenzo Librandi, Jan 25 2009

2*T(m,n)-3 = (2*m+1)*(2*n+1) is not prime, obviously. Also: first column: 3*A020725; second column: A016897; third column: A017041; fourth column: 3*A016789. - Vincenzo Librandi, Nov 20 2012

			Triangle begins:
6;
9,  14;
12, 19, 26;
15, 24, 33, 42;
18, 29, 40, 51, 62;
21, 34, 47, 60, 73,  86;
24, 39, 54, 69, 84,  99,  114;
27, 44, 61, 78, 95,  112, 129, 146;
30, 49, 68, 87, 106, 125, 144, 163, 182;
33, 54, 75, 96, 117, 138, 159, 180, 201, 222; etc.

Vincenzo Librandi, Rows n = 1..100, flattened

Cf. A153043, A020725, A016897, A017041.

[2*n*k + n + k + 2: k in [1..n],  n in [1..11]]; // Vincenzo Librandi, Nov 20 2012

t[n_,k_]:=2 n*k + n + k + 2; Table[t[n, k], {n, 11}, {k, n}]//Flatten (* Vincenzo Librandi, Nov 20 2012 *)

A210976 Square array T(n,k), n>=1, k>=1, read by antidiagonals in which column k lists the positive integers whose number of divisors is not k.

Original entry on oeis.org

2, 3, 1, 4, 4, 1, 5, 6, 2, 1, 6, 8, 3, 2, 1, 7, 9, 5, 3, 2, 1, 8, 10, 6, 4, 3, 2, 1, 9, 12, 7, 5, 4, 3, 2, 1, 10, 14, 8, 7, 5, 4, 3, 2, 1, 11, 15, 10, 9, 6, 5, 4, 3, 2, 1, 12, 16, 11, 11, 7, 6, 5, 4, 3, 2, 1, 13, 18, 12, 12, 8, 7, 6, 5, 4, 3, 2, 1, 14, 20

Offset: 1

Omar E. Pol, Aug 11 2012

Column k lists the numbers j such that A000005(j) <> k.

			Column 2 lists the nonprimes A018252 because they are the positive integers whose number of divisors is not 2.
Array begins:
2,   1,  1,  1,  1,  1,  1,  1,  1,  1,
3,   4,  2,  2,  2,  2,  2,  2,  2,
4,   6,  3,  3,  3,  3,  3,  3,
5,   8,  5,  4,  4,  4,  4,
6,   9,  6,  5,  5,  5,
7,  10,  7,  7,  6,
8,  12,  8,  9,
9,  14, 10,
10, 15,
11,

Columns (1-4): A020725, A018252, A213367, A210994.

A179442 a(n) = ((n-1)! * (n+1)!) / n.

Original entry on oeis.org

2, 3, 16, 180, 3456, 100800, 4147200, 228614400, 16257024000, 1448500838400, 158018273280000, 20713561989120000, 3212195459235840000, 581636820654489600000, 121600871304831959040000

Offset: 1

Jaroslav Krizek, Jul 14 2010

nonn

			a(5) = ((5-1)! * (5+1)!) / 5 = (4! * 6!) / 5 = (24 * 720) / 5 = 17280 / 5 = 3456.
a(5) = ((5 -1)!^2) * (5+1) = 24^2 * 6 = 3456.

Cf. A001044, A020725, A175430, A229020, A261879.

Table[(n - 1)!*(n + 1)!/n, {n, 1, 15}] (* Amiram Eldar, Jan 18 2021 *)

a(n) = (n-1)!^2*(n+1) \\ Charles R Greathouse IV, Oct 23 2023

a(n) = Product_{k=1..n} (k * A020725(k)) / (n^2) = Product_{k=1..n} (k * (k+1)) / (n^2).
a(n) = A175430(n) / n = A001044(n-1) * (n+1) = ((n -1)^2)! * (n+1).
G.f.: 1 + G(0), where G(k)= 1 + x*(k+1)/(1 - (k+2)/(k+2 + 1/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jul 08 2013
From Amiram Eldar, Jan 18 2021: (Start)
Sum_{n>=1} 1/a(n) = BesselI(2,2) + BesselI(3,2) = A229020 + A261879.
Sum_{n>=1} (-1)^(n+1)/a(n) = BesselJ(2,2) - BesselJ(3,2). (End)

A229212 Square array of numerators of t(n,k) = (1+1/(k*n))^n, read by descending antidiagonals.

Original entry on oeis.org

2, 3, 9, 4, 25, 64, 5, 49, 343, 625, 6, 81, 1000, 6561, 7776, 7, 121, 2197, 28561, 161051, 117649, 8, 169, 4096, 83521, 1048576, 4826809, 2097152, 9, 225, 6859, 194481, 4084101, 47045881, 170859375, 43046721, 10, 289

Offset: 1

Jean-François Alcover, Sep 16 2013

Limit(t(n,k), n -> infinity) = exp(1/k).
1st row = A020725
2nd row = A016754
3rd row = A016779
4th row = A016816
5th row = A016865
1st column = A000169
2nd column = A085527

			Table of fractions begins:
   2,       3/2,        4/3,         5/4, ...
  9/4,     25/16,      49/36,       81/64, ...
64/27,   343/216,   1000/729,    2197/1728, ...
625/256, 6561/4096, 28561/20736, 83521/65536, ...
...
Table of numerators begins:
2,      3,     4,     5, ...
9,     25,    49,    81, ...
64,   343,  1000,  2197, ...
625, 6561, 28561, 83521, ...
...
Triangle of antidiagonals begins:
2;
3, 9;
4, 25, 64;
5, 49, 343, 625;
...

Cf. A229213(denominators), A016754, A016779, A016816, A016865, A000169, A085527.

t[n_, k_] := (1+1/(k*n))^n; Table[t[n-k+1, k], {n, 1, 9}, {k, n, 1, -1}] // Flatten // Numerator

A338112 Least number that is both the sum and product of n distinct positive integers.

Original entry on oeis.org

1, 3, 6, 24, 120, 720, 5040, 40320, 362880, 3628800, 39916800, 479001600, 6227020800, 87178291200, 1307674368000, 20922789888000, 355687428096000, 6402373705728000, 121645100408832000, 2432902008176640000, 51090942171709440000, 1124000727777607680000

Offset: 1

Rick L. Shepherd, Oct 10 2020

Each a(n) = n! except that a(2) = 1+2 = 3. For n > 0, only each integer >= A000217(n) is the sum of n distinct positive integers. For the integers that are products of these types, see below.

			a(1) = 1 because we define sums and products as sum(m) := prod(m) := m for all integers m in this case where these normally-binary operations only have one operand.
a(3) = 6 because 6 = 1+2+3 = 1*2*3 (with all the distinct positive integers the same in the sum and the product only for this term and a(1)).
a(5) = 120 because 120 = 1+2+3+4+110 (= ... = 22+23+24+25+26) = 1*2*3*4*5.

Cf. A000142, A000217.

Cf. Products of k distinct positive integers: A000027 (k=1), A020725 (k=2), A080257 (k=3), A122181 (k=4).

Array[If[# <= 2, (#^2 - #)/2 &[# + 1], #!] &, 22] (* Michael De Vlieger, Oct 15 2020 *)
With[{nn=30},Rest[CoefficientList[Series[x (2+x-x^2)/(2(1-x)),{x,0,nn}],x] Range[0,nn]!]] (* Harvey P. Dale, Aug 10 2021 *)

PARI
```
a(n) = if(n<1, , if(n==2, 3, n!))
```

a(n) = A000142(n) for n = 1 and n > 2; a(2) = 3.
a(n) = max(A000142(n), A000217(n)).
E.g.f.: x*(2 + x - x^2)/(2*(1 - x)). - Stefano Spezia, Oct 11 2020

cp's OEIS Frontend

A335738 Factorize each integer m >= 2 as the product of powers of nonunit squarefree numbers with distinct exponents that are powers of 2. The sequence lists m such that the factor with the largest exponent is a power of 2.

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A085554 Greater of twin primes of the form x^2+2, x^2+4.

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A335740 Factorize each integer m >= 2 as the product of powers of nonunit squarefree numbers with distinct exponents that are powers of 2. The sequence lists m such that the factor with the largest exponent is not a power of 2.

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A175430 a(n) = (n-1)! * (n+1)!.

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A067665 The start of a record-setting run of consecutive integers i with distinct A001222(i).

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A155705 Triangle read by rows where T(m,n) = 2*m*n + m + n + 2.

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A210976 Square array T(n,k), n>=1, k>=1, read by antidiagonals in which column k lists the positive integers whose number of divisors is not k.

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A179442 a(n) = ((n-1)! * (n+1)!) / n.

A155705 Triangle read by rows where T(m,n) = 2mn + m + n + 2.