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.

Previous Showing 11-17 of 17 results.

A141090 Integral quotients of products of first k consecutive composites divided by their sums: products (dividends).

Original entry on oeis.org

4, 1728, 2903040, 12541132800, 115880067072000, 69528040243200000, 1807729046323200000, 43295255277764345856000000, 20188846756043686829592191472500736000000000, 989253491046140654650017382152536064000000000
Offset: 1

Views

Author

Enoch Haga, Jun 01 2008

Keywords

Comments

Based on A141092.
Compare with A140761 A159578 A140763 A116536.
Take the first k composite numbers. If their product divided by their sum results in an integer, their product is a term of the sequence. - Harvey P. Dale, Apr 29 2018

Examples

			a(3) = 2903040 because 4*6*8*9*10*12*14 = 2903040 and 4+6+8+9+10+12+14 = 63; 2903040/63 = 46080, integral -- 2903040 is added to the sequence.

Links

Crossrefs

Cf. A196415, A141089, A141091, A141092.

Programs

  • Mathematica
    With[{c=Select[Range[100],CompositeQ]},Table[If[IntegerQ[ Times@@Take[ c,n]/Total[ Take[ c,n]]], Times@@ Take[ c,n],0],{n,Length[c]}]]/.(0-> Nothing) (* Harvey P. Dale, Apr 29 2018 *)

Formula

Find the products and sums of first k consecutive composites. When the product divided by the sum produces an integral quotient, add product to sequence.

Extensions

Checked by N. J. A. Sloane, Oct 02 2011
Edited by N. J. A. Sloane, Apr 29 2018

A141091 Integral quotients of products of consecutive composites divided by their sums: sums (divisors).

Original entry on oeis.org

4, 27, 63, 112, 175, 224, 250, 400, 847, 896, 2368, 2448, 2695, 3596, 4300, 4624, 4961, 5076, 5546, 6032, 6156, 6664, 8750, 9048, 9200, 9976, 10295, 11620, 12312, 13572, 14697, 15872, 16275, 18139, 18352, 23572, 24304, 25544, 26814, 27072, 29986
Offset: 1

Views

Author

Enoch Haga, Jun 01 2008

Keywords

Examples

			a(3) = 63 because 4*6*8*9*10*12*14 = 2903040 and 4+6+8+9+10+12+14 = 63; 2903040/63 = 46080, integral -- 63 is added to the sequence.

Links

Crossrefs

Cf. A196415, A141089, A141090, A141092.
Compare with A140761, A159578, A140763, A116536.

Programs

Formula

Find the products and sums of consecutive composites. When the products divided by the sums produce integral quotients, add terms to sequence.

Extensions

a(37) corrected by Amiram Eldar, Jan 12 2020

A141089 Integral quotients of products of consecutive composites divided by their sums: Last consecutive composite.

Original entry on oeis.org

4, 9, 14, 18, 22, 25, 26, 33, 48, 49, 78, 80, 84, 95, 105, 110, 114, 115, 119, 123, 124, 129, 147, 150, 152, 158, 160, 170, 175, 184, 190, 200, 202, 212, 213, 242, 245, 250, 256, 258, 272, 284, 287, 288, 291, 306, 309, 314, 319, 327, 332, 333, 336, 342, 343
Offset: 1

Views

Author

Enoch Haga, Jun 01 2008

Keywords

Examples

			a(3) = 14 because 4*6*8*9*10*12*14 = 2903040 and 4+6+8+9+10+12+14 = 63; 2903040/63 = 46080, integral -- 14 is added to the sequence.

Links

Crossrefs

Cf. A141090, A141091, A141092.
Compare with A140761, A159578, A140763, A116536.

Programs

Formula

Find the products and sums of consecutive composites. When the products divided by the sums produce integral quotients, add terms to sequence.

A322608 Values of k such that (product of squarefree numbers <= k) / (sum of squarefree numbers <= k) is an integer.

Original entry on oeis.org

1, 3, 11, 14, 17, 21, 23, 33, 34, 37, 46, 47, 55, 58, 59, 61, 62, 67, 69, 73, 82, 83, 87, 94, 95, 97, 101, 106, 107, 109, 114, 115, 119, 123, 127, 133, 134, 141, 146, 151, 157, 158, 159, 161, 165, 166, 173, 181, 187, 197, 202, 203, 210, 218, 219, 223, 226, 230
Offset: 1

Views

Author

Paolo P. Lava, Dec 20 2018

Keywords

Examples

			3 is in the sequence because (1*2*3)/(1+2+3) = 1.
11 is in the sequence because (1*2*3*5*6*7*10*11)/(1+2+3+5+6+7+10+11) = 138600/45 = 3080.

Links

Crossrefs

Cf. A005117, A051838, A111059, A116536, A141092, A173143, A322607 (values of the quotient), A347690 (numbers of terms in the numerators).

Programs

  • Maple
    with(numtheory): P:=proc(q) local a,b,c,n; a:=1; b:=0; c:=[];
for n from 1 to q do if issqrfree(n) then a:=a*n; b:=b+n;
if frac(a/b)=0 then c:=[op(c),n];
fi; fi; od; op(c); end: P(60);
  • Mathematica
    seq = {}; sum = 0; prod = 1; Do[If[SquareFreeQ[n], sum += n; prod *= n; If[Divisible[prod, sum], AppendTo[seq, n]]], {n, 1, 230}]; seq (* Amiram Eldar, Mar 05 2021 *)

Extensions

Definition corrected by N. J. A. Sloane, Sep 19 2021 at the suggestion of Harvey P. Dale.

A159639 Last divisors at which integral quotients occur consecutively.

Original entry on oeis.org

154, 2183, 4002, 8635, 19203, 93017, 96298, 122414, 166762, 182090, 201354, 241237, 337645, 346495, 406813, 456729, 574678, 668811, 781635, 799006, 929442, 952150, 1014194, 1379625, 1455259, 1513549, 1558110, 1573089, 1938354, 2028842
Offset: 1

Views

Author

Enoch Haga, Apr 17 2009, Apr 21 2009

Keywords

Comments

Considering dividend/divisor=quotient, in a(1) of A116536, A159578, and A159579, 3=30/10; sometimes integral quotients appear n times in succession -- see A159580 where a(3)indicates that 5 integral quotients appear one after another. A159581 and A159639 give the first and last values of the divisors producing these integral quotients.
Many of the associated sequences submitted by this author were wrong. Should be recomputed. The UBASIC program should be regarded with suspicion. - N. J. A. Sloane, Oct 02 2011.

Examples

			a(1)=154 because it is the last or second of two divisors (125 being the first) where integral quotients are produced in succession (one after the other): 5577321750/125=44618574, integral; and 161742330750/154=1050274875, integral. See a(4) in A116536, A159578, and A159579.

Crossrefs

Cf. A116536, A159578 A159579 A159580 A159581

Programs

  • UBASIC
    10 'product of cons primes divided by sum cons primes 20 N=3:Q=2*N:R=R+N:R=R+2 30 A=3:S=sqrt(N) 40 B=N/A 50 if int(B)*A=N then 110 60 A=A+2:if A

A322607 Numbers that can be expressed as the ratio between the product and the sum of consecutive squarefree numbers starting from 1.

Original entry on oeis.org

1, 3080, 350350, 61850250, 17823180375, 6871260396000, 88909822914869880000, 2746644314348614680000, 2980109081068246927800000, 9638057975990853416623724908800000, 424217819372970387341691005411520000, 51912228216508515627667235880347808000000, 152157812632066726080765311397008321568000000
Offset: 1

Views

Author

Paolo P. Lava, Dec 20 2018

Keywords

Examples

			1 is a term because 1/1 = (1*2*3)/(1+2+3) = 1.
3080 is a term because (1*2*3*5*6*7*10*11)/(1+2+3+5+6+7+10+11) = 138600/45 = 3080.

Crossrefs

Cf. A005117, A111059, A116536, A141092, A173143, A322608.

Programs

  • Maple
    with(numtheory): P:=proc(q) local a,b,c,n; a:=1; b:=0; c:=[];
for n from 1 to q do if issqrfree(n) then a:=a*n; b:=b+n;
if frac(a/b)=0 then if n>1 then c:=[op(c),a/b];
fi; fi; fi; od; op(c); end: P(60);

Formula

a(n) = A111059(A322608(n+1))/A173143(A322608(n+1)).

A120387 c(n) mod b(n) where c(n) = (n-1)! and b(n) = Sum_{i=1..n} i.

Original entry on oeis.org

0, 1, 2, 6, 9, 15, 20, 0, 0, 45, 54, 66, 77, 0, 0, 120, 135, 153, 170, 0, 0, 231, 252, 0, 0, 0, 0, 378, 405, 435, 464, 0, 0, 0, 0, 630, 665, 0, 0, 780, 819, 861, 902, 0, 0, 1035, 1080, 0, 0, 0, 0, 1326, 1377, 0, 0, 0, 0, 1653, 1710, 1770, 1829, 0, 0, 0, 0, 2145, 2210, 0, 0, 2415
Offset: 1

Views

Author

Paolo P. Lava, Jun 30 2006

Keywords

Examples

			a(5) = (5-1)! mod (1+2+3+4+5) = 24 mod 15 = 9.

Crossrefs

Cf. A000217, A116536.
Cf. A226718, A119690.

Programs

  • Maple
    P:=proc(n) local i,k; for i from 1 by 1 to n do print((i-1)! mod sum('k','k'=0..i)); od; end: P(100);

Formula

For n>1: if neither n nor n+1 is prime, then a(n)=0. Otherwise, a(n)=n(n-1)/2 - 1 for odd n and a(n)=n(n-1)/2 for even n. - Ivan Neretin, May 29 2015
Previous Showing 11-17 of 17 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.