A186443 -id:A186443 - cp's OEIS frontend

A087029 Number of lunar divisors of n (unbounded version).

9, 8, 7, 6, 5, 4, 3, 2, 1, 18, 90, 16, 14, 12, 10, 8, 6, 4, 2, 16, 16, 72, 14, 12, 10, 8, 6, 4, 2, 14, 14, 14, 56, 12, 10, 8, 6, 4, 2, 12, 12, 12, 12, 42, 10, 8, 6, 4, 2, 10, 10, 10, 10, 10, 30, 8, 6, 4, 2, 8, 8, 8, 8, 8, 8, 20, 6, 4, 2, 6, 6, 6, 6, 6, 6, 6, 12, 4, 2, 4, 4, 4, 4

Offset: 1

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

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

			The 18 divisors of 10 are 1, 2, ..., 9, 10, 20, 30, ..., 90, so a(10) = 18.

D. Applegate, Table of n, a(n) for n = 1..100000
D. Applegate, C program for lunar arithmetic and number theory
D. Applegate, M. LeBrun and N. J. A. Sloane, Dismal Arithmetic, arxiv:1107.1130 [math-NT], July 2011. [Note: we have now changed the name from "dismal arithmetic" to "lunar arithmetic" - the old name was too depressing]
D. Applegate, M. LeBrun, N. J. A. Sloane, Dismal Arithmetic, J. Int. Seq. 14 (2011) # 11.9.8.
Brady Haran and N. J. A. Sloane, Primes on the Moon (Lunar Arithmetic), Numberphile video, Nov 2018.
Index entries for sequences related to dismal (or lunar) arithmetic

Cf. A087062 (lunar product).
Cf. A087028, A087083, A186443, A186510. See A189506 for the actual divisors.
See A067399 for the base-2 version.

(Uses programs from A087062. This crude program is valid for n <= 99.) dd2 := proc(n) local t1,t2,i,j; t1 := []; for i from 1 to 99 do for j from i to 99 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;

A087029(n)=#A189506_row(n) \\ To be optimized. - M. F. Hasler, Nov 15 2018

More terms from David Applegate, Nov 07 2003

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

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

N. J. A. Sloane, Feb 22 2011

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

N. J. A. Sloane, Table of n, a(n) for n = 1..50
D. Applegate, M. LeBrun and N. J. A. Sloane, Dismal Arithmetic [Note: we have now changed the name from "dismal arithmetic" to "lunar arithmetic" - the old name was too depressing]
Index entries for sequences related to dismal (or lunar) arithmetic

Cf. A087029, A186443.

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

A186970 The oex analog of the Euler phi-function for the oex prime power factorization of positive integers.

Original entry on oeis.org

1, 1, 2, 3, 4, 3, 6, 4, 8, 5, 10, 7, 12, 8, 9, 12, 16, 11, 18, 12, 14, 14, 22, 9, 24, 16, 18, 18, 28, 13, 30, 16, 22, 21, 25, 24, 36, 24, 27, 17, 40, 17, 42, 30, 33, 29, 46, 27, 48, 32, 36, 36, 52, 24, 42, 25, 40, 37, 58, 28, 60, 40, 49, 48, 50, 30, 66, 48, 49, 35, 70, 32, 72, 48, 54, 54, 61, 36

Offset: 1

Vladimir Shevelev, Mar 01 2011

nonn

Oex divisors d of an integer n are defined in A186443: those divisors d which are either 1 or numbers such that d^k || n (the highest power of d dividing n) has odd exponent k.
A positive number is called an oex prime if it has only two oex divisors; since every n >= 2 has at least two oex divisors, 1 and n, an oex prime q has only oex divisors 1 and q. A000430 is the sequence of oex primes q, i.e., A186643(q) = 2 iff q is an entry in A000430.
A unique factorization, called an oex prime power factorization, of integers n is introduced as follows: each factor p^e in the conventional prime power factorization n = Product(p^e) is written as (p^2)^(e/2) if e is even, and as (p^2)^floor(e/2)*p if e is odd. This represents n as a product of oex primes of the type q=p^2, with unconstrained exponents e/2, and of oex primes of the type q=p with exponents 0 or 1. (This is similar to splitting n into its squarefree part A007913(n) times A008833(n), followed by an ordinary prime factorization in both parts separately.)
Let n = q_1^a_1*q_2^a_2*... and m = q_1^b_1*q_2^b_2*..., a_i,b_i >= 0 be the oex prime power factorizations of n and m. Define the oex GCD of n and m as [n,m] = q_1^min(a_1,b_1) * q_2^min(a_2,b_2) * .... Then a(n) = Sum_{m=1..n, [m,n]=1} 1, the oex analog of the Euler-phi function.

			The oex prime power factorization of 16 is 4^2. Since [16,i]=1 for i=1, 2, 3, 5, 6, 7, 9, 10, 11, 13, 14, and 15, a(16)=12.
The oex prime power factorization of 9 is 9. Thus a(9)=8.

Cf. A000010, A000430, A186443, A186444, A186776, A064380.

highpp := proc(n,d) local nshf,a ; if n mod d <> 0 then 0; else nshf := n ; a := 0 ; while nshf mod d = 0 do nshf := nshf /d ; a := a+1 ; end do: a; end if; end proc:
oexgcd := proc(n,m) local a,p,kn,km ; a := 1 ; for p in numtheory[factorset](n) do kn := highpp(n,p) ; km := highpp(m,p) ; if type(kn,'even') = type(km,'even') then ; else kn := 2*floor(kn/2) ; km := 2*floor(km/2) ; end if; a := a*p^min(kn,km) ; end do: a ; end proc:
A186970 := proc(n) local a,i; a := 0 ; for i from 1 to n do if oexgcd(n,i) = 1 then a := a+1 ; end if; end do: a; end proc:
seq(A186970(n),n=1..80) ; # R. J. Mathar, Mar 18 2011

Let core(n) = p_1*...*p_r = A007913(n), n/core(n) = A008833(n) = q_1^c_1*...*q_t^c_t, where q_i are squares of primes.
If core(n)=1, then a(n) = n*Product_{j=1..r} (1-1/q_i); if core(n) tends to infinity, then a(n) ~ n * core(n) * Product_{i=1..t} (1-1/q_i) / Product_{j=1..r} (1+p_j).
a(n) <= A064380(n).

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

N. J. A. Sloane, Feb 22 2011

This is for lunar arithmetic in base 10.

			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.

D. Applegate, M. LeBrun and N. J. A. Sloane, Dismal Arithmetic [Note: we have now changed the name from "dismal arithmetic" to "lunar arithmetic" - the old name was too depressing]
Index entries for sequences related to dismal (or lunar) arithmetic

Cf. A087029, A186443, A186510.

A186889 Oex perfect numbers: n such that A186644(n) = 2*n.

Original entry on oeis.org

6, 18, 20, 100, 1888, 2044928, 33099776, 35021696, 45335936, 533020672

Offset: 1

Vladimir Shevelev, Feb 28 2011

nonn

There are no squarefree infinitary perfect numbers > 6 (cf. A007357). Therefore, the second and all further terms of the sequence are infinitary deficient (A129657).
No further term between 1888 and 1440000. - R. J. Mathar, Mar 18 2011
a(11) > 3*10^10. 1471763808896 is also a term. - Donovan Johnson, Jan 30 2013

			Let n = 100 with divisors 1, 2, 4, 5, 10, 20, 25, 50, and 100. By the definition in A186643, only 1, 4, 20, 25, 50, 100 among these are oex divisors. Since 1+4+20+25+50+100 = 2*100, 100 is in the sequence.

Cf. A186443, A186444, A129657, A000396, A007357.

for(n=4, 10^9, if(isprime(n), next); d=divisors(n); s=n+1; for(j=2, numdiv(n)-1, for(k=2, 30, if(n%d[j]^k<>0, if(k%2==0, s=s+d[j]); k=30))); if(s==2*n, print(n))) /* Donovan Johnson, Jan 28 2013 */

a(6)-a(10) from Donovan Johnson, Jan 28 2013

cp's OEIS Frontend

A087029 Number of lunar divisors of n (unbounded version).

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

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A186970 The oex analog of the Euler phi-function for the oex prime power factorization of positive integers.

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

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A186889 Oex perfect numbers: n such that A186644(n) = 2*n.

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