cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-8 of 8 results.

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

Views

Author

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

Keywords

Comments

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

Links

Crossrefs

Cf. A000027 (for n=1), A064808 (n=2), A131509 (n=3), A129995 (n=4), A131675 (n=5), ..., A131680 (n=10).
See also A049614.

Programs

  • Maple
    # Maple program from Cyril Banderier, Sep 18 2007:
List:=NULL: for n from 1 to 1000 do m:=1: #running till n=50 will last 2 min.
for i from 1 to numtheory[pi](n) do div:=ithprime(i): d:=1: e:=0: oldmini:=-1:mini:=0:
while oldmini<>mini do e:=e+1: #the last time consuming loop could be skipped by proving e<=floor(ln(n)/ln(div)):
d:=d*div;for x from 0 to d-1 do [seq((x &^k mod d)+k mod d,k=1..n)]:contrib[d,x]:=nops(select(has,%,0)): od:
L:=seq(add(contrib[div^j,x mod div^j],j=1..e),x=0..div^e-1); oldmini:=mini: mini:=min(L): od:
if mini

Extensions

More terms from Cyril Banderier, Sep 17 2007

A129995 a(n) = (n + 1)*(n^2 + 2)*(n^3 + 3)*(n^4 + 4)/4!.

Original entry on oeis.org

1, 5, 165, 4675, 65325, 543456, 3155425, 14146210, 52259625, 166192975, 469090061, 1201490445, 2839166005, 6268589250, 13060542825, 25881747316, 49095506065, 89615392425, 158091087925, 270522770375, 450420100221, 731644012660, 1162094343345, 1808433948150
Offset: 0

Views

Author

Alexander R. Povolotsky, Aug 19 2007, Aug 25 2007

Keywords

Comments

Following my conjecture, computations by Peter J. C. Moses, mediation by Clark Kimberling and helpful comments from George E. Andrews, it is now known that a(n) = (n^1 + 1)*(n^2 + 2)*(n^3 + 3)*...*(n^k + k)/k! is an integer-valued sequence if and only if k belongs to {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 17, 18, 19, 20, 21}; this is the case for k=4; see generalization in A131685.

Links

Crossrefs

Cf. A000027 (k=1), A064808 (k=2), A131509 (k=3), this sequence (k=4), A131675 (k=5), ..., A131680 (k=10).
See A131685 for a generalization.

Programs

  • Magma
    [(n^1 + 1)*(n^2 + 2)*(n^3 + 3)*(n^4 + 4)/24: n in [0..30]]; // Vincenzo Librandi, Apr 25 2015
  • Maple
    p:=proc(n,i) mul( n^j+j, j=1..i)/i!; end; [seq(p(n,4),n=0..30)];
seq((n+1)*(n^2+2)*(n^3+3)*(n^4+4)/factorial(4), n = 0 .. 20) # Emeric Deutsch, Aug 26 2007
  • Mathematica
    Table[x = 4; Product[(n^k) + k, {k, x}]/x!, {n, 0, 23}] (* Michael De Vlieger, Apr 24 2015 *)
LinearRecurrence[{11,-55,165,-330,462,-462,330,-165,55,-11,1},{1,5,165,4675,65325,543456,3155425,14146210,52259625,166192975,469090061},30] (* Harvey P. Dale, Dec 07 2021 *)
  • PARI
    vector(20,n,n--;(n+1)*(n^2+2)*(n^3+3)*(n^4+4)/4!) \\ Derek Orr, Apr 25 2015
  • PARI
    A129995(n)=(n+1)*(n^2+2)*(n^3+3)*(n^4+4)/12 \\ M. F. Hasler, May 02 2015

Formula

G.f.: (1-6x+165x^2+2970x^3+22480x^4+55969x^5+51511x^6+16490x^7+1595x^8+25x^9)/(1-x)^11. - Emeric Deutsch, Aug 26 2007
G.f.: -(1 + x*(-6 + x*(165 + x*(2970 + x*(22480 + x*(55969 + x*(51511 + 5*x*(3298 + x*(319 + 5*x))))))))) / (x - 1)^11. - Peter J. C. Moses, Aug 29 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

Views

Author

Alexander R. Povolotsky, Sep 15 2007

Keywords

Comments

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

Links

Programs

  • Magma
    [((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
  • Mathematica
    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 *)
  • PARI
    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
  • PARI
    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

Formula

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

Extensions

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

Views

Author

Alexander R. Povolotsky, Sep 15 2007

Keywords

Comments

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

Links

Programs

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

Extensions

Definition made explicit by M. F. Hasler, May 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

Views

Author

Alexander R. Povolotsky, Sep 15 2007

Keywords

Links

Crossrefs

Cf. A131685.

Programs

Formula

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

Extensions

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

Views

Author

Alexander R. Povolotsky, Sep 15 2007

Keywords

Comments

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

Programs

Extensions

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

A131677 a(n) = (Product_{i=1..7} n^i+i) / 7!.

Original entry on oeis.org

1, 8, 274725, 8903032600, 21521701559085, 9892478959203456, 1527238784041075105, 109733832449349303000, 4483781212288588835625, 118795734924428077310080, 2233888850312257843810061, 31811523551546985038211552, 359951182400070234774044725
Offset: 0

Views

Author

Alexander R. Povolotsky, Sep 15 2007

Keywords

Comments

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

Links

Programs

Extensions

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

A131678 a(n) = (Product_{i=1..8} n^i+i) / 8!.

Original entry on oeis.org

1, 9, 9065925, 7310502643675, 176327300873583405, 483041091658815453456, 320648364425775841520065, 79074323113562613259765875, 9403175220694650942397475625, 639220975955961365494757841040, 27923612862792073359883606310061, 852385355738368243011331354210716, 19346552845649626158477975728463925
Offset: 0

Views

Author

Alexander R. Povolotsky, Sep 15 2007

Keywords

Comments

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

Links

Programs

Extensions

Definition made explicit by M. F. Hasler, May 02 2015
Showing 1-8 of 8 results.

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