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I'm looking for a concise term to say a number must be exactly one higher than a previous number.

None of "subsequent", "incremental", or "next" seem to convey the restriction that it must be contiguous to the previous number. "Contiguous" does not convey that it must be larger rather than smaller.

Is there a specific term for this or am I going to have to use a phrase like "contiguous to and larger than"?

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I would suggest that you try asking this on the Mathematics stack exchange, except you're not allowed to cross-post... – 40XUserNotFound Jul 4 '11 at 3:53
"Successor" is the correct term for integers. "Next" is a more general term which can apply to many varied list items. – zenbike Jul 4 '11 at 7:03
up vote 36 down vote accepted

The number that's one greater than a given natural number n is called the successor of n. See the Peano axioms, which are a standard foundational definition of the set of natural numbers.

This terminology strictly is only defined for natural numbers, but I think it can be safely extended to integers without confusion.

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+1 for accuracy, and for referencing real math on the English.SE site. – zenbike Jul 4 '11 at 7:02
+1 and I'm deleting my own answer, which simply said 2 is the successor of 1. The link to Peano axioms is good in this context. – FumbleFingers Jul 4 '11 at 12:53
Just what I was looking for! – Ergwun Jul 4 '11 at 13:08
The extension of Peano axioms from natural numbers to integers is covered in this 1961 paper jstor.org/pss/2311096 – Stephen Jul 4 '11 at 14:46

If it's clear you're talking about integers, then you could say successive if you don't simply want to say next.

But I think the word next clearly implies no intervening values in relation to integers.

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Exactly right. I would just say next. It's less technical than successor, yet no less precise. Ask practically anyone on the street, elementary school classroom, or university which integer is the next one, and they will all say the same thing. – John Y Jul 4 '11 at 4:48
Yes OP didn't state whether they wanted a technical term, nontechnical term or whether integers would already be clear from the context and "next" is the most concise, which they did specify. – hippietrail Jul 4 '11 at 8:59
+1 for "successive". I was avoiding "next", as in a situation where I have a sequence of numbers, mostly contiguous but with some missing, the "next" number after 5 (for example) would be 7 if 6 were missing. I think that "next" is too overloaded. – Ergwun Jul 4 '11 at 13:06

You could say "the next consecutive integer".

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This is precise and not overly technical but the OP asked for a "concise" term. – hippietrail Jul 4 '11 at 9:00
+1 as it's better than my original fall-back phrase. – Ergwun Jul 4 '11 at 13:07

"The increment" would be technically correct, but it really depends on the structure of the sequence, the context in which you are writing/speaking, and the assumptions that can safely be made about the audience's understanding.

If the structure can be unambiguously understood as an ordinal sequence (such as the index of an array or a series), then you can use words like increment or next the same way you'd be free to use the subscript n+1 in a formulaic description or ++i in a computer program. If the description of the series/sequence is not apparent to your audience (or there is a chance that they might confuse the index/subscript with its associated value) then you can initially define the increment and then use terms like next, subsequent and so forth.

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