  This article demonstrates a new kind of programmable logic for the
representation of an integer that can be used for the programmable Josephson
voltage standard. It can enable the numbers of junctions in most bits to be
variable integer values, which is different from normal binary logic or ternary
logic. Consequently, missing junctions due to superconducting short circuits
can be tolerated under this logic. This logic can also have nearly the same
segmentation efficiency as ternary logic. The completeness of the sequences
using this logic is proven by the recursive method in mathematics in this
paper. After that, a new algorithm for the representation of integers is
presented according to the proven process, and an analysis of the number of
fault-tolerant junctions for each bit is provided. Although the first and
second bits are not tolerant to missing junctions, bits beyond these can
tolerate one to hundreds of missing junctions. Due to the non-fixed multiples
between the bits of the sequence, this logic is called non-integer-multiple
logic. Finally, the design and fabrication of a 3 V programmable Josephson
junction array using this logic are described, and the measurements and
analysis of the characteristic parameters are presented.
