A129995 -id:A129995 - cp's OEIS frontend

A049614 n! divided by its squarefree kernel.

1, 1, 1, 1, 4, 4, 24, 24, 192, 1728, 17280, 17280, 207360, 207360, 2903040, 43545600, 696729600, 696729600, 12541132800, 12541132800, 250822656000, 5267275776000, 115880067072000, 115880067072000, 2781121609728000, 69528040243200000, 1807729046323200000

Offset: 0

Labos Elemer

nonn

Also product of composite numbers less than or equal to n. - Benoit Cloitre, Aug 18 2002
Also n! divided by n primorial (or n!/n#). - Cino Hilliard, Mar 26 2006
From Alexander R. Povolotsky and Peter J. C. Moses, Aug 27 2007: (Start)
It appears that a(n) = smallest positive number m such that the sequence b(n) = { m (i^1 + 1!) (i^2 + 2!) ... (i^n + n!) / n! : i >= 0 } takes integral values. [It would be nice to have a proof of this! - N. J. A. Sloane] Cf. A064808 (for n=2), A131682 (for n=3), A131683 (for n=4), A131527 (for n=5), A131684 (for n=6), A131528. See also A129995, A131685. (End)
It appears that every term > 4 is divisible by 24. - Alexander R. Povolotsky, Oct 18 2007
The above comment is correct since each term divides the next. - Charles R Greathouse IV, Jan 16 2012
When n is not a prime number, then a(n)=m*n, where m is some integer >0; such a(n) make up the A036691 Otherwise, when n is a prime number, then a(n)=a(k), where k is the largest nonprime number preceding n (kAlexander R. Povolotsky, Aug 21 2012

			n = 11: 11! = 39916800 = 2310*17280 and 2310=2*3*5*7*11.

Cf. A000142, A002110, A003418, A034386, A036691, A045948, A048148.

A049614:= func< n | n le 1 select 1 else Factorial(n)/(&*[NthPrime(j): j in [1..#PrimesUpTo(n)]]) >;
[A049614(n): n in [0..40]]; // G. C. Greubel, Jul 21 2023

primorial := n -> mul(k, k=select(isprime, [$1..n]));
A049614 := n -> factorial(n)/primorial(n);
seq(A049614(i),i=0..24); # Peter Luschny, Feb 16 2013

Table[n!/Product[ Prime[i], {i, PrimePi[n]}], {n, 24}]

PARI
```
a(n)=prod(i=1,n,i^if(isprime(i),0,1))
```

a(n)=n!/prod(i=1,primepi(n),prime(i)) \\ Charles R Greathouse IV, Aug 30 2012

def A049614(n): return factorial(n)/product(nth_prime(j) for j in range(1,1+prime_pi(n)))
[A049614(n) for n in range(41)] # G. C. Greubel, Jul 21 2023

a(n) = A000142(n)/A034386(n).

Edited by N. J. A. Sloane, Oct 07 2007

Offset set to 0, a(0)=1 prepended to data, Peter Luschny, Feb 16 2013

A131685 a(n) = smallest positive number m such that c(i) = m (i^1 + 1) (i^2 + 2) ... (i^n + n) / n! takes integral values for all i>=0.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 7, 7, 7, 1, 1, 1, 1, 1, 11, 11, 11, 55, 143, 13, 91, 91, 91, 91, 91, 1001, 17017, 595595, 595595, 17017, 46189, 600457, 3002285, 3002285, 3002285, 3002285, 6605027, 3002285, 726869, 726869, 726869

Offset: 1

Alexander R. Povolotsky and Peter J. C. Moses, Sep 12 2007, revised Sep 17 2007

nonn

It appears that none of the terms are divisible by 3. - Alexander R. Povolotsky, Oct 18 2007

Cyril Banderier, Table of n, a(n) for n = 1..100
Index to divisibility sequences

Cf. A000027 (for n=1), A064808 (n=2), A131509 (n=3), A129995 (n=4), A131675 (n=5), ..., A131680 (n=10).

A131509 a(n) = (n + 1)(n^2 + 2)(n^3 + 3)/6.

Original entry on oeis.org

1, 4, 33, 220, 1005, 3456, 9709, 23528, 50985, 101260, 187561, 328164, 547573, 877800, 1359765, 2044816, 2996369, 4291668, 6023665, 8303020, 11260221, 15047824, 19842813, 25849080, 33300025, 42461276, 53633529, 67155508, 83407045, 102812280, 125842981

Offset: 0

Alexander R. Povolotsky, Aug 13 2007, Aug 25 2007

See also A131685(k) = smallest positive number m such that c(i) = m (i^1 + 1) (i^2 + 2) ... (i^k+ k) / k! takes integral values for all i>=0. For k=3, A131685(k)=1, which implies that this is a well defined integer sequence. - Alexander R. Povolotsky, May 18 2015

M. F. Hasler, Table of n, a(n) for n = 0..100
Index to divisibility sequences
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A000027 (k=1), A064808 (k=2), this sequence (k=3), A129995 (k=4), A131675 (k=5), ..., A131680 (k=10).

[(n^1 + 1)*(n^2 + 2)*(n^3 + 3)/6: n in [0..30]]; // Vincenzo Librandi, Apr 25 2015

p:=proc(n,i) mul( n^j+j, j=1..i)/i!; end; [seq(p(n,3),n=0..30)];
seq((1/6)*(n+1)*(n^2+2)*(n^3+3),n=0..25); # Emeric Deutsch, Aug 23 2007

Table[x = 3; Product[(n^k) + k, {k, x}]/6, {n, 0, 27}] (* Michael De Vlieger, Apr 24 2015 *)
LinearRecurrence[{7,-21,35,-35,21,-7,1},{1,4,33,220,1005,3456,9709},40] (* Harvey P. Dale, Oct 18 2016 *)

A131509(n):=(n^1 + 1)*(n^2 + 2)*(n^3 + 3)/6$
makelist(A131509(n),n,0,30); /* Martin Ettl, Nov 03 2012 */

vector(20,n,n--;(n+1)*(n^2+2)*(n^3+3)/3!) \\ Derek Orr, Apr 25 2015

A131509(n)=(n+1)*(n^2+2)*(n^3+3)/6 \\ M. F. Hasler, May 02 2015

G.f.: (1 -3x +26x^2 +38x^3 +53x^4 +5x^5)/(1-x)^7. - Emeric Deutsch, Aug 23 2007

Corrected and extended by R. J. Mathar and Emeric Deutsch, Aug 21 2007

A131675 a(n) = (Product_{i=1..5} n^i+i)/5!.

Original entry on oeis.org

1, 6, 1221, 231880, 13443885, 340203456, 4910472385, 47565216504, 342540938025, 1962871989130, 9382270310061, 38701449021984, 141297910237237, 465502930269300, 1404867737405385, 3930816255364816, 10296122969028753, 25448298063869070, 59744930256741205

Offset: 0

Alexander R. Povolotsky, Sep 15 2007

See A131685 about well-definedness. - M. F. Hasler, May 02 2015

T. D. Noe, Table of n, a(n) for n = 0..1000
Index to divisibility sequences
Index entries for linear recurrences with constant coefficients, signature (16,-120,560,-1820,4368,-8008,11440,-12870,11440,-8008,4368,-1820,560,-120,16,-1).

[((n+1)*(n^2+2)*(n^3+3)*(n^4+4)*(n^5+5))/Factorial(5): n in [0..20]]; // Vincenzo Librandi, Apr 25 2015

Table[x = 5; Product[(n^k) + k, {k, x}]/x!, {n, 0, 17}] (* Michael De Vlieger, Apr 24 2015 *)
LinearRecurrence[{16,-120,560,-1820,4368,-8008,11440,-12870,11440,-8008,4368,-1820,560,-120,16,-1},{1,6,1221,231880,13443885,340203456,4910472385,47565216504,342540938025,1962871989130,9382270310061,38701449021984,141297910237237,465502930269300,1404867737405385,3930816255364816},20] (* Harvey P. Dale, Oct 04 2024 *)

Vec((135*x^14 +86852*x^13 +5864611*x^12 +109724496*x^11 +782427151*x^10 +2468818430*x^9 +3704965659*x^8 +2710222344*x^7 +952834509*x^6 +152249688*x^5 +9878785*x^4 +212504*x^3 +1245*x^2 -10*x +1) / (x -1)^16 + O(x^100)) \\ Colin Barker, Apr 24 2015

A131675(n,k=5)=prod(i=1,k,(n^i+i))/k! \\ Changing the optional 2nd argument allows one to produce A000027 (k=1), A064808 (k=2), A131509 (k=3), A129995 (k=4), A131676 (k=6), ..., A131680 (k=10). - M. F. Hasler, May 02 2015

G.f.: (135*x^14 +86852*x^13 +5864611*x^12 +109724496*x^11 +782427151*x^10 +2468818430*x^9 +3704965659*x^8 +2710222344*x^7 +952834509*x^6 +152249688*x^5 +9878785*x^4 +212504*x^3 +1245*x^2 -10*x +1) / (x -1)^16. - Colin Barker, Apr 24 2015

Definition made explicit by M. F. Hasler, May 02 2015

A131680 a(n) = (Product_{i=1..10} n^i+i)/10!.

Original entry on oeis.org

1, 11, 54266008005, 94467113468457039310, 538562285352301951109430061, 102370328298891480707678565453456, 2171004564341130364494477279762016705, 10015112821822553484101305268477882115400, 15057116321451208557735379863635553426467625, 9594364176429126945241161642390324911313805168

Offset: 0

Alexander R. Povolotsky, Sep 15 2007

See also A131685(k) = smallest positive number m such that c(i) = m (i^1 + 1) (i^2 + 2) ... (i^k+ k) / k! takes integral values for all i>=0: For k=10, A131685(k)=1, which implies that this is a well defined integer sequence. - Alexander R. Povolotsky, Apr 24 2015; corrected by M. F. Hasler, May 02 2015

Index to divisibility sequences

[((n+1)*(n^2+2)*(n^3+3)*(n^4+4)*(n^5+5)*(n^6+6)*(n^7+7)*(n^8+8)*(n^9+9)*(n^10+10))/Factorial(10): n in [0..10]]; // Vincenzo Librandi, Apr 25 2015

Table[x = 10; Product[(n^k) + k, {k, x}]/x!, {n, 0, 9}] (* Michael De Vlieger, Apr 24 2015 *)

A131680(n,k=10)=prod(i=1,k,(n^i+i))/k! \\ Changing the optional 2nd argument allows one to produce A000027 (k=1), A064808 (k=2), A131509 (k=3), A129995 (k=4), A131675(k=5), ..., A131679 (k=9). - M. F. Hasler, May 02 2015

Definition made explicit by M. F. Hasler, May 02 2015

A131189 Numbers n >= 0 such that d(n) = (n^1 + 1)(n^2 + 2) ... (n^14 + 14) / 14!, e(n) = (n^1 + 1)(n^2 + 2) ... (n^15 + 15) / 15!, and f(n) = (n^1 + 1)*(n^2 + 2) ... (n^16 + 16) / 16! take nonintegral values.

Original entry on oeis.org

2, 9, 16, 23, 30, 37, 51, 58, 65, 72, 79, 86, 100, 107, 114, 121, 128, 135, 149, 156, 163, 170, 177, 184, 198, 205, 212, 219, 226, 233, 247, 254, 261, 268, 275, 282, 296, 303, 310, 317, 324, 331, 345, 352, 359, 366, 373, 380, 394, 401, 408, 415, 422, 429

Offset: 1

Alexander R. Povolotsky, Sep 25 2007

Initial terms were calculated by Peter J. C. Moses; see comment in A129995.
From Max Alekseyev: (Start)
To check whether 14! divides (n^1 + 1)*(n^2 + 2)*(n^3 + 3)*...*(n^14 +14) it is enough to check whether every prime power q from the prime factorization of 14! divides (n^1 + 1)*(n^2 + 2)*(n^3 + 3)*...*(n^14 + 14) for n=0,1,...,q-1.
Note that 14! = 2^11 * 3^5 * 5^2 * 7^2 * 11 * 13.
It is easy to verify that:
i) 2^11 divides (n^1 + 1)*(n^2 + 2)*(n^3 + 3)*...*(n^14 + 14) for every n=0,1,...,2^11-1;
ii) 3^5 divides (n^1 + 1)*(n^2 + 2)*(n^3 + 3)*...*(n^14 + 14) for every n=0,1,...,3^5-1;
iii) 7^2 divides (n^1 + 1)*(n^2 + 2)*(n^3 + 3)*...*(n^14 + 14) for every n=0,1,...,7^2-1,
except for n=2, 9, 16, 23, 30, 37 (i.e., n of the form 7m+2 but not 49m+44);
iv) 11 divides (n^1 + 1)*(n^2 + 2)*(n^3 + 3)*...*(n^14 + 14) for every n=0,1,...,11-1;
v) 13 divides (n^1 + 1)*(n^2 + 2)*(n^3 + 3)*...*(n^14 + 14) for every n=0,1,...,13-1.
This proves that for k=14, a(n) is nonintegral only for n in the set difference { 7m+2} \ { 49m+44 }.
In simple cases like { 7m+2 } \ { 49m+44 }, one can get an explicit formula.
Note that { 7m+2 } \ { 49m+44 } can be viewed as the arithmetic sequence 7m + 2 where m cannot be 6 modulo 7.
The number of nonnegative integers equal to 6 modulo 7 not exceeding m is equal to floor((m+1)/7).
Therefore in our sequence the (m-floor((m+1)/7))-th term equals 7m+2 when m is minimum possible.
Let us find the minimum m satisfying m - floor((m+1)/7) = n.
Let t = (m+1) mod 7. Then m - ((m+1)-t)/7 = n or 6m+t-1 = 7n, implying that m = (7n-t+1)/6.
Depending on the value of t we have m = floor((7n+1)/6) or m = floor((7n+1)/6) - 1.
Since m must be the minimum possible, we have the second case when floor((7n+1)/6) mod 7 > (7n+1) mod 6; and the first case otherwise.
In other words, the explicit formula for m is m = floor((7n+1)/6) - 1, if floor((7n+1)/6) mod 7 > (7n+1) mod 6; m = floor((7n+1)/6), otherwise.
We can also consider all possible residues of n modulo 6*7 and find out that the inequality floor((7n+1)/6) mod 7 > (7n+1) mod 6 holds for n mod 42 in { 5, 11, 17, 23, 29, 35, 41 } or alternatively n mod 6 = 5.
Therefore m = floor((7n+1)/6) - 1, if n mod 6 = 5; m = floor((7n+1)/6), otherwise.
So for the sequence {2,9,16,23,30,37,51,58,65,72,79,86,100,...} we can give formula a(n) = 7*floor((7n+1)/6) - 5, if n mod 6 = 5; a(n) = 7*floor((7n+1)/6) + 2, otherwise.
(End)

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,1,-1).

a:= n-> `if`((n-1) mod 6 = 5, 7*floor((7*n-6)/6)-5, 7*floor((7*n-6)/6)+2):
seq(a(n), n=1..80);  # Alois P. Heinz, Aug 08 2013

LinearRecurrence[{1, 0, 0, 0, 0, 1, -1}, {2, 9, 16, 23, 30, 37, 51}, 50] (* G. C. Greubel, Feb 19 2017 *)

x='x+O('x^50); Vec(x*(12*x^6+7*x^5+7*x^4+7*x^3+7*x^2+7*x+2) / ((x-1)^2*(x+1)*(x^2-x+1)*(x^2+x+1))) \\ G. C. Greubel, Feb 19 2017

a(n) = 7*floor((7n+1)/6) - 5, if n mod 6 = 5; a(n) = 7*floor((7n+1)/6) + 2, otherwise. For e(n) and g(n), proof is similar; see also the comments. - Max Alekseyev

G.f.: x*(12*x^6+7*x^5+7*x^4+7*x^3+7*x^2+7*x+2) / ((x-1)^2*(x+1)*(x^2-x+1)*(x^2+x+1)). - Colin Barker, Aug 08 2013

More terms from Colin Barker, Aug 08 2013

A131191 Numbers n>=0 such that d(n) = (n^1 + 1) (n^2 + 2) ... (n^22 + 22) / 22!, e(n) = (n^1 + 1) (n^2 + 2) ... (n^23 + 23) / 23!, and f(n) = (n^1 + 1) (n^2 + 2) ... (n^24 + 24) / 24! take nonintegral values.

Original entry on oeis.org

7, 18, 29, 40, 51, 62, 73, 84, 95, 106, 128, 139, 150, 161, 172, 183, 194, 205, 216, 227, 249, 260, 271, 282, 293, 304, 315, 326, 337, 348, 370, 381, 392, 403, 414, 425, 436, 447, 458, 469, 491, 502, 513, 524, 535, 546, 557, 568, 579, 590, 612, 623, 634, 645, 656, 667, 678, 689, 700, 711, 733, 744

Offset: 1

Alexander R. Povolotsky, Sep 25 2007

nonn

If n is in this sequence, then so is n+121. - Max Alekseyev, Feb 02 2015

Cf. A017473, A129995

Notice that 22! = 2^19 * 3^9 * 5^4 * 7^3 * 11^2 * 13 * 17 * 19. All these prime powers divide (n^1 + 1)*(n^2 + 2)*(n^3 +3)*...*(n^22 + 22), except for 11^2. 11^2 does not divide (n^1 + 1)*(n^2 + 2)*(n^3 + 3)*...*(n^22 + 22) for n = 7, 18, 29, 40, 51, 62, 73, 84, 95, 106 modulo 121. That is, d(n) is nonintegral for n the form 11m+7 but not 121m+117, and so are e(n) and f(n). - Max Alekseyev, Nov 10 2007

Initial terms were calculated by Peter J. C. Moses; see comment in A129995.

More terms from Max Alekseyev, Feb 02 2015

A131676 a(n) = (Product_{i=1..6} n^i+i) / 6!.

Original entry on oeis.org

1, 7, 14245, 28405300, 9191136045, 886286703456, 38188743738145, 932714257963020, 14966184483875625, 173860405001195185, 1563721100613810061, 11427034989921521488, 70319024498214551605, 374482754394635213250, 1763001772206469563945, 7462412915610398239816

Offset: 0

Alexander R. Povolotsky, Sep 15 2007

Index entries for linear recurrences with constant coefficients, signature (22, -231, 1540, -7315, 26334, -74613, 170544, -319770, 497420, -646646, 705432, -646646, 497420, -319770, 170544, -74613, 26334, -7315, 1540, -231, 22, -1).

Cf. A131685.

Table[Product[ n^i + i, {i, 1, 6}]/6!, {n, 0, 15}] (* Michael De Vlieger, Jan 03 2016 *)

A131676(n, k=6)=prod(i=1, k, (n^i+i))/k! \\ Changing the optional 2nd argument allows one to produce A000027 (k=1), A064808 (k=2), A131509 (k=3), A129995 (k=4), A131675 (k=5), ..., A131680 (k=10). - M. F. Hasler, May 02 2015

G.f.: (1 - 15*x + 14322*x^2 + 28091987*x^3 + 8569506575*x^4 + 690621422337*x^5 + 20769948618958*x^6 + 283347184706283*x^7 + 1969675285865562*x^8 + 7493939424807955*x^9 + 16292973927985678*x^10 + 20712738704664489*x^11 + 15498276638623618*x^12 + 6765765599122915*x^13 + 1679542499740050*x^14 + 226176197184209*x^15 + 15278037714093*x^16 + 454493699352*x^17 + 4732512736*x^18 + 10869320*x^19 + 1575*x^20)/(1 - x)^22. - M. F. Hasler, May 02 2015

Definition made explicit by M. F. Hasler, May 02 2015

A131679 a(n) = (Product_{i=1..9} n^i+i) / 9!.

Original entry on oeis.org

1, 10, 524816325, 15995379784360900, 5136081211768056707885, 104827108835105429096703456, 359044402823940369662885183425, 354548318931625565271233374406000, 140230322081790179721500725877795625, 27516367648544953143193233240569070880, 3102623679344954347223585172112606310061

Offset: 0

Alexander R. Povolotsky, Sep 15 2007

See A131685 about well-definedness. - M. F. Hasler, May 02 2015

Table[Product[ n^i + i, {i, 1, 9}]/9!, {n, 0, 10}] (* Michael De Vlieger, Jan 03 2016 *)

A131679(n,k=9)=prod(i=1,k,(n^i+i))/k! \\ Changing the optional 2nd argument allows one to produce A000027 (k=1), A064808 (k=2), A131509 (k=3), A129995 (k=4), A131675 (k=5), ..., A131680 (k=10). - M. F. Hasler, May 02 2015

Definition made explicit by M. F. Hasler, May 02 2015

A131190 Numbers n>=0 such that d(n) = (n^1 + 1) (n^2 + 2) ... (n^25 + 25) / 25! is nonintegral.

Original entry on oeis.org

2, 7, 12, 18, 22, 27, 29, 37, 40, 47, 51, 52, 62, 72, 73, 77, 84, 87, 95, 97, 102, 106, 112, 122, 127, 128, 137, 139, 147, 150, 152, 161, 162, 172, 177, 183, 187, 194, 197, 202, 205, 212, 216, 222, 227, 237, 247, 249, 252, 260, 262, 271, 272, 277, 282, 287, 293, 297, 302, 304, 312, 315, 322, 326, 327, 337

Offset: 1

Alexander R. Povolotsky, Sep 25 2007

nonn

If n is in this sequence the so is n+6050. - Max Alekseyev, Feb 02 2015

Cf. A129995, A131189, A131191, A131192, A131685

{ is_A131190(n) = setsearch([2,12,22,27,37,47],n%50) || ( (n%11)==7 && (n%121)!=117 ) } /* Max Alekseyev, Feb 02 2015 */

Notice that 25! = 2^22 * 3^10 * 5^6 * 7^3 * 11^2 * 13 * 17 * 19 * 23. The value of (n^1+1)(n^2+2)...(n^25+25) is always divisible by all these prime powers, except 5^6 and 11^2. There is no divisibility by 5^6 for n in {50m+2, 50m+12, 50m+22, 50m+27, 50m+37, 50m+47} and by 11^2 for n in {11m+7} \ {121m+117}. Therefore, the sequence is the union {50m+2} U {50m+12} U {50m+22} U {50m+27} U {50m+37} U {50m+47} U ( {11m+7} \ {121m+117} ). - Max Alekseyev, Nov 10 2007

Initial terms were calculated by Peter J. C. Moses; see comment in A129995.

More terms from Max Alekseyev, Feb 02 2015

cp's OEIS Frontend

A049614 n! divided by its squarefree kernel.

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A131685 a(n) = smallest positive number m such that c(i) = m (i^1 + 1) (i^2 + 2) ... (i^n + n) / n! takes integral values for all i>=0.

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A131509 a(n) = (n + 1)*(n^2 + 2)*(n^3 + 3)/6.

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A131675 a(n) = (Product_{i=1..5} n^i+i)/5!.

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A131680 a(n) = (Product_{i=1..10} n^i+i)/10!.

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A131189 Numbers n >= 0 such that d(n) = (n^1 + 1)*(n^2 + 2) ... (n^14 + 14) / 14!, e(n) = (n^1 + 1)*(n^2 + 2) ... (n^15 + 15) / 15!, and f(n) = (n^1 + 1)*(n^2 + 2) ... (n^16 + 16) / 16! take nonintegral values.

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A131509 a(n) = (n + 1)(n^2 + 2)(n^3 + 3)/6.

A131189 Numbers n >= 0 such that d(n) = (n^1 + 1)(n^2 + 2) ... (n^14 + 14) / 14!, e(n) = (n^1 + 1)(n^2 + 2) ... (n^15 + 15) / 15!, and f(n) = (n^1 + 1)*(n^2 + 2) ... (n^16 + 16) / 16! take nonintegral values.