The Captain reported the Cabin Pressure Warning Horn sounded and cabin pressure was noted to be approximately 13;000 feet. The flight crew immediately descended and continued to destination airport to make a precautionary landing.

Narrative

Leaving FL410 on descent to ZZZ; the Cabin Pressure Warning Horn sounded and cabin pressure was noted to be approximately 13;000 feet. We initiated an immediate descent per the AFM procedure. ATC had cleared us to descend to FL360. We requested lower. I was pilot monitoring and communicating with ATC. I told ATC that we needed a lower altitude immediately. I said we had an immediate situation; but did not specifically request priority handling. ATC asked about the nature of our problem. I told the Controller that we have a pressurization problem and need to get lower quickly. We were cleared down to 16;000 feet; then down to 4;000 feet. We were also given deviation authority around thunderstorms and direct to ZZZ when able. We arrived and ZZZ approximately XA:25 local time. We did not deviate from any FAR in response to this situation. ATC cleared us as we requested. No one was injured or suffered any adverse effects of the emergency descent. I believe the issue was caused by our descent from FL410 with full anti-ice systems operating and thrust lever inadequate to maintain cabin pressure.