A188548 -id:A188548 - cp's OEIS frontend

A067399 Number of divisors of n in OR-numbral arithmetic.

1, 2, 2, 3, 2, 4, 3, 4, 2, 4, 2, 6, 2, 6, 5, 5, 2, 4, 2, 6, 3, 4, 2, 8, 2, 4, 4, 9, 2, 10, 8, 6, 2, 4, 2, 6, 2, 4, 2, 8, 2, 6, 2, 6, 4, 4, 4, 10, 2, 4, 4, 6, 2, 8, 4, 12, 2, 4, 4, 15, 4, 16, 14, 7, 2, 4, 2, 6, 2, 4, 2, 8, 3, 4, 2, 6, 2, 4, 2, 10, 2, 4, 2, 9, 5, 4, 2, 8, 2, 8, 4, 6, 2, 8, 6, 12, 2, 4, 4, 6

Offset: 1

Jens Voß, Jan 23 2002

nonn

See A048888 for the definition of OR-numbral arithmetic. The example shows that this sequence is not multiplicative.

In other words, number of lunar divisors of n in base 2.

			a(15)=5 since [15] has the 5 OR-numbral divisors [1], [3], [5], [7] and [15].
If written as a triangle with rows of lengths 1,2,4,8,16,...:
1,
2, 2,
3, 2, 4, 3,
4, 2, 4, 2, 6, 2, 6, 5,
5, 2, 4, 2, 6, 3, 4, 2, 8, 2, 4, 4, 9, 2, 10, 8,
6, 2, 4, 2, 6, 2, 4, 2, 8, 2, 6, 2, 6, 4, 4, 4, 10, 2, 4, 4, 6, 2, 8, 4, 12, 2, 4, 4, 15, 4, 16, 14,
...,
the last terms in each row give A079500(n). The penultimate terms in the rows give 2*A079500(n-1). - _N. J. A. Sloane_, Mar 05 2011

N. J. A. Sloane, Table of n, a(n) for n = 1..1024
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]
D. Applegate, M. LeBrun, N. J. A. Sloane, Dismal Arithmetic, J. Int. Seq. 14 (2011) # 11.9.8.
A. Frosini and S. Rinaldi, On the Sequence A079500 and Its Combinatorial Interpretations, J. Integer Seq., Vol. 9 (2006), Article 06.3.1.
Index entries for sequences related to dismal (or lunar) arithmetic

A079500 is the subsequence a(2^k-1). - N. J. A. Sloane, Feb 23 2011
Cf. A003986, A007059, A048888, A067138, A067139, A067398, A067400, A067401.
See A188548 for the sum of the divisors.

A190149 Even numbers n (written in binary) such that in base-2 lunar arithmetic, the sum of the divisors of n is a number containing a 0 (in binary).

Original entry on oeis.org

10010, 100010, 100110, 110010, 1000010, 1000100, 1000110, 1001010, 1001110, 1010010, 1100010, 1100110, 1110010, 10000010, 10000100, 10000110, 10001010, 10001100, 10001110, 10010010, 10010110, 10011010, 10011110, 10100010, 10100110, 10110010, 11000010, 11000100, 11000110, 11001010, 11001110, 11010010, 11100010, 11100110, 11110010, 100000010, 100000100

Offset: 1

N. J. A. Sloane, May 05 2011

As remarked in A188548, if n is even then most of the time A188548(n) = 111...111 (that is, a number of the form 2^k-1). This sequence lists the exceptions.

			In base-2 lunar arithmetic, the divisors of 10010 are 1, 10, 1001 and 10010, whose sum is 11011.

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]
D. Applegate, M. LeBrun, and N. J. A. Sloane, Dismal Arithmetic, J. Int. Seq. 14 (2011) # 11.9.8.
Index entries for sequences related to dismal (or lunar) arithmetic

Cf. A188548, A067399. See A190150 and A190151 for the base-10 representation of these numbers.

A190150 A190149 converted to base 10.

Original entry on oeis.org

18, 34, 38, 50, 66, 68, 70, 74, 78, 82, 98, 102, 114, 130, 132, 134, 138, 140, 142, 146, 150, 154, 158, 162, 166, 178, 194, 196, 198, 202, 206, 210, 226, 230, 242, 258, 260, 262, 264, 266, 268, 270, 274, 276, 278, 282, 284, 286, 290, 294, 298, 302, 306, 310, 314, 318, 322, 324, 326, 330, 334, 338, 354, 358, 370, 386, 388, 390, 394, 396, 398, 402, 406

Offset: 1

N. J. A. Sloane, May 05 2011

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]
D. Applegate, M. LeBrun, N. J. A. Sloane, Dismal Arithmetic, J. Int. Seq. 14 (2011) # 11.9.8.
Index entries for sequences related to dismal (or lunar) arithmetic

Cf. A188548, A190149, A190151.

A190151 A190149 converted to base 10 and halved.

Original entry on oeis.org

9, 17, 19, 25, 33, 34, 35, 37, 39, 41, 49, 51, 57, 65, 66, 67, 69, 70, 71, 73, 75, 77, 79, 81, 83, 89, 97, 98, 99, 101, 103, 105, 113, 115, 121, 129, 130, 131, 132, 133, 134, 135, 137, 138, 139, 141, 142, 143, 145, 147, 149, 151, 153, 155, 157, 159, 161, 162, 163, 165, 167, 169, 177, 179, 185, 193, 194, 195, 197, 198, 199, 201, 203, 205, 207, 209, 211

Offset: 1

N. J. A. Sloane, May 05 2011

Take the even numbers n such that in base 2 lunar arithmetic, the sum of the divisors of n is not of the form 2^k-1, and divide them (in ordinary arithmetic) by 2 (cf. A190149, A190150)

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]
D. Applegate, M. LeBrun, N. J. A. Sloane, Dismal Arithmetic, J. Int. Seq. 14 (2011) # 11.9.8.
Index entries for sequences related to dismal (or lunar) arithmetic

Cf. A188548, A190149, A190150.

A190376 a(n) = sum (in ordinary arithmetic) of A067399(k), for k from 2^n to 2^(n+1)-1.

Original entry on oeis.org

1, 4, 12, 31, 75, 175, 393, 864, 1868, 3978, 8394

Offset: 0

N. J. A. Sloane, May 09 2011

I was hoping this would turn out to be a known sequence, in which case we would learn something about the average values of A067399.

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. A067399, A188548.

read("transforms");
numbralADD := proc(a,b) option remember; ORnos(a,b) ; end proc:
numbralMUL := proc(a,b) option remember; local p,bshf,s ; p := 0 ; bshf := b ; for s from 0 do if bshf mod 2 <> 0 then p := numbralADD(p, 2^s*a ) ; end if; bshf := floor(bshf/2) ; if bshf = 0 then return p; end if; end do; end proc:
isnumbralDiv := proc(n,d) option remember; for e from 0 do if numbralMUL(e,d) = n then return true; elif numbralMUL(e,d) > 2*n then return false; end if; end do: end proc:
numbralDivisors := proc(n) option remember; local d,i; d := {} ; for i from 1 to n do if isnumbralDiv(n,i) then d := d union {i} ; end if; end do: d ; end proc:
A067399 := proc(n) nops(numbralDivisors(n)) ; end proc:
A190376 := proc(n) add(A067399(k),k=2^n..2^(n+1)-1) ; end proc: # R. J. Mathar, May 30 2011

A190632 In base 3 lunar arithmetic, the lunar sum of the lunar divisors of n.

Original entry on oeis.org

2, 2, 22, 22, 12, 22, 22, 22, 222, 202, 102, 222, 222, 112, 122, 122, 122, 222, 202, 202, 222, 222, 212, 222, 222, 222, 2222, 2002, 1002, 2222, 2022, 1012, 1122, 1022, 1022, 2222, 2202, 1102, 2222, 2222, 1112, 1122, 1122, 1122, 1222, 1202, 1202, 1222, 1222, 1212, 1222, 1222, 1222, 2222, 2002, 2002, 2222, 2012, 2012, 2222, 2022, 2022, 2222, 2102, 2102, 2222, 2222, 2112, 2222, 2222, 2122, 2222, 2202, 2202, 2222, 2222, 2212, 2222, 2222, 2222

Offset: 1

N. J. A. Sloane, May 14 2011

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]
D. Applegate, M. LeBrun, N. J. A. Sloane, Dismal Arithmetic, J. Int. Seq. 14 (2011) # 11.9.8.
Index entries for sequences related to dismal (or lunar) arithmetic

Cf. A007089, A188548, A087416.

cp's OEIS Frontend

A067399 Number of divisors of n in OR-numbral arithmetic.

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A190149 Even numbers n (written in binary) such that in base-2 lunar arithmetic, the sum of the divisors of n is a number containing a 0 (in binary).

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A190150 A190149 converted to base 10.

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A190151 A190149 converted to base 10 and halved.

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A190376 a(n) = sum (in ordinary arithmetic) of A067399(k), for k from 2^n to 2^(n+1)-1.

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A190632 In base 3 lunar arithmetic, the lunar sum of the lunar divisors of n.

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