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Answers to Rockets with n-engines, for n = 1, 2, 3… cover all single digit nontrivial values except for seven, so...

Question: Have there been any rockets that run seven engines at the same time?

Thrusters on spacecraft already in space don't count. This should probably be a launch vehicle of some kind.

zero would be the trivial single digit non-negative valule

DylanSp
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uhoh
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    Technically, SpaceX Falcon 9 is built with 9 engines, but on the first trip to ISS, two of them failed. Because of that, for some part of the propulsed flight, it was running seven engines at the same time, it was designed to work under this circumstance despite not being the nominal case or intention. – Mefitico Mar 02 '20 at 18:20
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    The Falcon 9 can tolerate two engine failures, but only one engine failed on CRS-1. – Christopher James Huff Mar 02 '20 at 22:05
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    @ChristopherJamesHuff thanks for that. I want to say "too bad a second engine didn't fail" but I won't. Oh, wait...oops – uhoh Mar 02 '20 at 22:12
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    This is genuinely surprising to me considering how nicely seven engines close-pack on the bottom of a cylindrical launcher. – ikrase Mar 05 '20 at 07:32
  • @ikrase That surprised me as well; I'd thought that it would be one of the easiest. Then I wondered if the 1+8=9 can better tolerate the loss of any 2 engines than the 1+6=7 can. – uhoh Mar 05 '20 at 08:15
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    Not gonna lie, I suspected that on the basis of Kerbal Space Program experience. – ikrase Mar 05 '20 at 08:44

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Yes.

enter image description here

(Image source)

Some configurations of the Delta II used nine solid boosters.

In launches from both the ER and WR, the first-stage RS-27A main engine and six of the nine strap-on solid-rocket motors are ignited on the ground at liftoff. Following burnout of these six SRMs, the remaining three are ignited.

Delta II Payload Planners Guide (emphasis mine)

Organic Marble
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Besides the Delta II and PSLV-G/PSLV-XL, there are a few proposed/in-development stages that would fire 7 engines at once.

Blue Origin’s New Glenn booster is designed with 7 BE-4 engines powering its first stage.

ULA has put forth a paper design for a “Delta Super Heavy”, 6 boosters surrounding the central first stage core of the Delta IV. I don’t know if the idea was for it to be flown with all 7 engines firing together, or if it ever got beyond a vague concept stage. I note that the Delta IV is currently a horizontally integrated launcher, and that going to 4 or 6 boosters would raise difficulties with that strategy.

The Angara family has a proposed A7 configuration with 6 boosters around one core stage; this is likewise unlikely to ever be built.

Russell Borogove
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The USSR's failed Moon rocket, the N1, had 30 engines on the first stage. This is significantly more than the limit of 7 you mentioned.

Picture of the Soviet N1

Source: Aerospaceweb

Burgi
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    The question is asking about exactly 7 rockets, not at least 7 rockets. The question is rather confusingly worded, in my opinion. – isaacg Mar 03 '20 at 19:28
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    @isaacg Would "Have there been any rockets that run exactly seven engines at the same time at some point during ascent by design?" work better? I wouldn't change the title but where I restate the question in the body of the text I can clarify. I'd thought the reverence to the previous N = 1, 2, 3... question made it clear that seven meant seven because if N = 1 meant "one or more" then every rocket with engines would have answered that question and there'd be no 2nd question in the first place. – uhoh Mar 03 '20 at 23:17
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    +1 because maybe the question was unclear, and it's a cool photo ;-) – uhoh Mar 03 '20 at 23:18
  • Ah! Oops, I completely misunderstood the original question. I can remove my answer if you like. – Burgi Mar 04 '20 at 09:11