A346742 Numbers that may be built from fewer ones using floor(j/k) in addition to +, -, and *.

1860043, 3198487, 4782847, 5580129, 6111571, 9300217, 9566302, 9595461, 9595462, 9654511, 10678027, 12725059, 12843157, 13551745, 14349271, 14614627, 16740391, 17094685, 18334713, 18334714, 19220449, 27900651, 28698178, 28701094, 29494975, 31620739, 32034081, 33484063, 34100797, 35872267, 37998031

Offset: 1

Author

Glen Whitney, Sep 28 2021

Keywords

• nonn

Comments

Consider an integer complexity measure c(n) which is the number of ones required to build n using +, -, *, and "floor division" which for convenience will be written in this entry (after Python notation) j//k = floor(j/k). In other words, c(n) is defined identically to A091333(n) except that this floor division is also allowed, and identically to the complexity b(n) described in A348069 except that division is extended to all pairs of natural numbers by taking the floor of the quotient. Clearly for all n, c(n) <= b(n) <= A091333(n). This sequence lists the integers k for which c(k) < A091333(k).

Because of the inequality c(n) <= b(n) <= A091333(n), every entry in A348069 will eventually appear in this sequence. For example, the first term of A348069 is 50221174 = (7*3^15)//2, so we have c(50221174) = 53, b(50221174) = 54, and A091333(50221174) = 55.

The extended domain of division means that terms of this sequence are much more frequent than A348069, but it's still quite rare for division to provide more compact expressions for natural numbers (except in the presence of exponentiation, see A348089).

Examples

			The smallest n for which c(n) as defined in the comments is strictly less than A091333(n) is 1860043, because 1860043 = (7*3^12)//2 which requires c(7) + 12*c(3) + c(2) = 6 + 12*3 + 2 = 44 ones to express with these operations, whereas A091333(1860043) = A005245(1860043) = 45 by virtue of the minimal expression 1860043 = 2(2^2*5*7(3^4(3^4+1)+1)+1)+1 requiring 2+2*2+5+6+3*4+3*4+1+1+1+1 = 45 ones. Hence, the first term in this sequence is 1860043.
The next three terms with their respective minimal expressions:
3198487 = (3^9(2^2*3^4+1))//2 [46 ones] = 2*3(3^2(2^2*3*5+1)(2^2*3^5-1)+2)+1 [47 ones] = 2*3(2(7(2^2*3+1)(2^2*3(3^5+1)+1)+1)+1)+1 [48 ones]. Thus n=319487 is the least n for which c(n) < A091333(n) < A005245(n).
4782847 = (3^5(2*3^9-1))//2 [47 ones] = 2*3(2*5(3^2(2^2*3^3(3^4+1)+1)+1)+1)+1 [48 ones]
5580129 = 3*1860043 = 3((7*3^12)//2) [47 ones] = 2^3(3*5*7(3^4(3^4+1)+1)+1)+1 [48 ones]. Note this example critically takes advantage of the fact that * and // are not associative.
		

Crossrefs

Cf. A253177 and A348069.

Cf. A091333 and A005245 (other integer complexity measures).