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 1-10 of 10 results.

A186443 Records in A087029.

Original entry on oeis.org

9, 18, 90, 180, 819, 1638, 7461, 14922, 67968
Offset: 1

Views

Author

N. J. A. Sloane, Feb 21 2011

Keywords

Links

Crossrefs

Cf. A186510.

A087083 Take unbounded lunar divisors of n as defined in A087029, add them using normal addition. See A087416 for their lunar sum.

Original entry on oeis.org

45, 44, 42, 39, 35, 30, 24, 17, 9, 495, 4500, 168, 154, 138, 120, 100, 78, 54, 28, 484, 492, 3916, 224, 198, 170, 140, 108, 74, 38, 462, 469, 476, 3276, 258, 220, 180, 138, 94, 48, 429, 435, 441, 447, 2613, 270, 220, 168, 114, 58, 385, 390, 395
Offset: 1

Views

Author

Marc LeBrun and N. J. A. Sloane, Oct 19 2003

Keywords

Links

Extensions

More terms from David Applegate, Nov 07 2003

A087416 Take unbounded lunar divisors of n as defined in A087029, add them using lunar addition. See A087083 for their conventional sum.

Original entry on oeis.org

9, 9, 9, 9, 9, 9, 9, 9, 9, 99, 99, 19, 19, 19, 19, 19, 19, 19, 19, 99, 99, 99, 29, 29, 29, 29, 29, 29, 29, 99, 99, 99, 99, 39, 39, 39, 39, 39, 39, 99, 99, 99, 99, 99, 49, 49, 49, 49, 49, 99, 99, 99, 99, 99, 99, 59, 59, 59, 59, 99, 99, 99, 99, 99, 99, 99, 69, 69, 69
Offset: 1

Views

Author

Marc LeBrun and N. J. A. Sloane, Oct 19 2003

Keywords

Comments

Two comments from David Applegate on lunar perfect numbers, Nov 08 2003: (Start)
If we define a perfect number by "n is lunarly perfect if Sum_{d|n} d == 2*n (both sum and * lunar)", no such numbers exist because 9|n, so the lunar sum of divisors ends in 9, but 2*n ends in 2.
If we define a perfect number by "n is lunarly perfect if lunar Sum_{d|n, d != n} d == n", no such numbers exist. For suppose n is perfect. n != 9 (since 9 is 9's only divisor). Then 9|n and 9 != n, so Sum_{d|n, d!=n} d ends in 9 and thus so does n. But 9ish numbers are not divisible by any single digit < 9. Thus n has no divisors of the same length as n, other than n itself. So Sum_{d|n, d!=n} d is one digit shorter than n. (End)

Links

Extensions

More terms from David Applegate, Nov 07 2003

A186508 Number of lunar divisors (A087029) of the decimal numbers 1, 10, 11, 100, 101, 110, 111, 1000, ... .

Original entry on oeis.org

9, 19, 90, 27, 90, 180, 819, 36, 90, 180, 738, 270, 738, 1638, 7641, 45, 90, 180, 738, 270, 819, 1476, 6570, 360, 738, 1476, 6732, 2457, 6570, 14922, 67968
Offset: 1

Views

Author

N. J. A. Sloane, Feb 22 2011

Keywords

Comments

This is for lunar arithmetic in base 10.

Examples

			The lunar divisors of 1 are 1,2,3,4,5,6,7,8,9, so a(1)=9.
The lunar divisors of 10 are 1...9 and 10, 20, 30, 40, ..., 90, so a(2) = 18.

Links

Crossrefs

Cf. A087029, A186443, A186510.

A087028 Number of bounded (<=n) lunar divisors of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 10, 10, 9, 8, 7, 6, 5, 4, 3, 2, 9, 9, 9, 8, 7, 6, 5, 4, 3, 2, 8, 8, 8, 8, 7, 6, 5, 4, 3, 2, 7, 7, 7, 7, 7, 6, 5, 4, 3, 2, 6, 6, 6, 6, 6, 6, 5, 4, 3, 2, 5, 5, 5, 5, 5, 5, 5, 4, 3, 2, 4, 4, 4, 4, 4, 4, 4, 4, 3, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 19, 10, 9, 8, 7, 6, 5, 4, 3, 2, 100, 91, 17, 15, 13, 11, 9, 7, 5, 3, 25, 25, 81, 22, 19, 16, 13, 10, 7, 4, 22, 22, 22, 64, 19, 16, 13, 10, 7, 4, 19, 19, 19, 19, 49, 16, 13, 10, 7, 4, 16, 16, 16, 16, 16, 36, 13, 10, 7, 4, 13, 13, 13, 13, 13, 13, 25, 10, 7, 4, 10, 10, 10, 10, 10, 10, 10, 16, 7, 4, 7, 7, 7, 7, 7, 7, 7, 7, 9, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 17
Offset: 1

Views

Author

Marc LeBrun and N. J. A. Sloane, Oct 19 2003

Keywords

Comments

Number of d, 1 <= d <= n, such that there exists an e, 1 <= e <= n, with d*e = n, where * is lunar multiplication.

Examples

			The 10 divisors of 10 <= 10 are 1, 2, ..., 9, 10.
a(100) = 19, since the lunar divisors of 100 <= 100 are 1, 2, ..., 9, 10, 20, ..., 90, 100.

Links

Crossrefs

Cf. A087029, A087082.

Programs

  • Maple
    (Uses programs from A087062) dd1 := proc(n) local t1,t2,i,j; t1 := []; for i from 1 to n do for j from i to n do if dmul(i,j) = n then t1 := [op(t1),i,j]; fi; od; od; t1 := convert(t1,set); t2 := sort(convert(t1,list)); nops(t2); end;

A186510 Number of lunar divisors of the decimal (lunar) number 111...1 (with n 1's).

Original entry on oeis.org

9, 90, 819, 7461, 67968, 619902, 5660208, 51746211, 473692869, 4342348692, 39865757616, 366573510504, 3376339346538, 31152497957100, 287964157783869, 2666972253081303, 24749535734382636, 230152606404800004, 2144836146346691706, 20032155489077131482, 187516277719282274940, 1759326436323972795042
Offset: 1

Views

Author

N. J. A. Sloane, Feb 22 2011

Keywords

Comments

It appears that this gives the largest number of lunar divisors of any n-digit lunar number.

Links

Crossrefs

Cf. A087029, A186443.

Extensions

More terms from N. J. A. Sloane, Feb 26 2011

A186943 Number of lunar divisors (in base 10) of the n-th number whose decimal expansion contains only 0's and 1's and begins and ends with a 1 (A099821(n)).

Original entry on oeis.org

9, 90, 90, 819, 90, 738, 738, 7461, 90, 738, 819, 6570, 738, 6732, 6570, 67968, 90, 738, 738, 6570, 738, 6570, 6732, 59868, 738, 6732, 6570, 59868, 6570, 59868, 59868, 619902, 90, 738, 738, 6570, 819, 6570, 6570, 59058, 738, 6570, 7461, 59058, 6570, 59868, 59058, 539550, 738, 6732, 6570, 59868, 6570, 59058, 59868, 538821, 6570, 59868, 59058, 538902, 59058, 538821, 539550, 5660208, 90, 738, 738, 6570, 738, 6570, 6570
Offset: 1

Views

Author

N. J. A. Sloane, Mar 01 2011

Keywords

Comments

Number of lunar divisors of A099821(n), that is, A087029(A099821(n))..

Examples

			1 has 9 divisors: 1,2,3,4,5,6,7,8,9, so a(1)=9. 11 has 90 divisors, 1 through 9 and the numbers 11 through 99 that do not end in 0, so a(2)=90.

Links

Crossrefs

Cf. A087029, A099821.

A186951 Number of lunar divisors (in base 10) of the n-th nonzero number whose decimal expansion contains only 0's and 1's (A007088(n)).

Original entry on oeis.org

9, 18, 90, 27, 90, 180, 819, 36, 90, 180, 738, 270, 738, 1638, 7461, 45, 90, 180, 738, 270, 819, 1476, 6570, 360, 738, 1476, 6732, 2457, 6570, 14922, 67968, 54, 90, 180, 738, 270, 738, 1476, 6570, 360, 738, 1638, 6570, 2214, 6732, 13140, 59868, 450, 738, 1476, 6732, 2214, 6570, 13464, 59868, 3276, 6570, 13140, 59868, 22383, 59868, 135936, 619902
Offset: 1

Views

Author

N. J. A. Sloane, Mar 01 2011

Keywords

Links

Crossrefs

A007088, A087029, A186943.

A189506 Irregular triangle read by rows in which row n (n >= 1) lists the base-10 lunar divisors of n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 2, 3, 4, 5, 6, 7, 8, 9, 3, 4, 5, 6, 7, 8, 9, 4, 5, 6, 7, 8, 9, 5, 6, 7, 8, 9, 6, 7, 8, 9, 7, 8, 9, 8, 9, 9, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 20, 30, 40, 50, 60, 70, 80, 90, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21, 22, 23, 24, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 36, 37, 38, 39, 41
Offset: 1

Views

Author

N. J. A. Sloane, Apr 25 2011

Keywords

Examples

			The first 11 rows give the divisors of 1 through 11:
1 2 3 4 5 6 7 8 9
2 3 4 5 6 7 8 9
3 4 5 6 7 8 9
4 5 6 7 8 9
5 6 7 8 9
6 7 8 9
7 8 9
8 9
9
1 2 3 4 5 6 7 8 9 10 20 30 40 50 60 70 80 90
1 2 3 4 5 6 7 8 9 11 12 13 14 15 16 17 18 19 21 22 23 24 25 26 27 28 29 31 32 33 34 35 36 37 38 39 41 ... 99 (= all zeroless 1- and 2-digit numbers).

Links

Crossrefs

Cf. A087062 (lunar product).
Row n has A087029(n) terms.

Programs

  • PARI
    A189506_row(n)={my(d=digits(n),m=vecmin(d),c=vector(#d,i,List()),K,t); for(L=1,#d,K=#d-L+1;forvec(v=vector(L,i,[max(m,i==1),9]), L<=K&& listput(c[L],fromdigits(v))&&next; t=fromdigits(v); forstep(i=#c[K],1,-1, A087062(c[K][i],t)==n||next; listput(c[L],t);break)); L>=K&&forstep(i=#c[K],1,-1,t=c[K][i]; forstep(j=#c[L],1,-1,A087062(c[L][j],t)==n&&next(2)); listpop(c[K],i))); Set(concat(c))} \\ M. F. Hasler, Nov 15 2018

Extensions

Minor edits by M. F. Hasler, Nov 15 2018

A186442 In base 2, excess of lunar sum of lunar divisors of n over number of lunar divisors of n.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 81, 9, 3, 5, 7, 9, 11, 13, 15, 17, 83, 83, 27, 15, 17, 19, 21, 23, 25, 27, 85, 85, 85, 43, 27, 29, 31, 33, 35, 37, 87, 87, 87, 87, 57, 39, 41, 43, 45, 47, 89, 89, 89, 89, 89, 69, 51, 53, 55, 57, 91, 91, 91, 91, 91, 91, 79, 63, 65, 67
Offset: 1

Views

Author

N. J. A. Sloane, Feb 21 2011

Keywords

Links

Crossrefs

Equals A087416(n) - A087029(n).
Showing 1-10 of 10 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.