It looks like Parodia werneri to me, which I think is synonymous with Parodia uebelmanniana (also known as Notocactus uebelmanniana). I think there are a number of "names" for this particular cactus. If I remember correctly, the genus Notocactus was "merged" (for a lack of better words) with Parodia and is not as often used with identification. I hope this helps.

Notocactus roseoflorus or N. roseoluteus. Not P./N. uebelmanniana, which has a more darkly purple hue, IIRC.

And I learned it as Notocactus - it may be changed, as PM states, to Parodia by the nomenclutter boffins. Surely a nicely-flowering South American glob o' cactus if I've ever seen one.

I thought many Parodia were lumped into Notocactus. No?

I don't think Howard's plant is Notocactus roseoluteus. Its flowers are larger, with not so pointy petals & the pokeys stick straight out (rather than hug the plant). Blooms are lighter pink as well. Here's a photo of a friend's plant:

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Notocactus uebelmannianus blooms change color as they open. The blooms on Howard's plant are larger opened than mine ever were. But I believe that is what you have.

I had 2 similar plants. The blooms never fully opened on the 1 seen below when I took pix. The other I missed seeing blooms altogether! :(

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I believe Parodia werneri is a synonym for Notocactus uebelmannianus, as is Parodia crassigibba.

OK, when I look at the flowers and the plant, they don't look the same. I understand, just as with any plant, there can be variation from same species to same species. Look at the body of my plants: one has larger "nubbins" to form the ribs and is a darker green and the spines are different and the flowers on the right have more rounded tips as Rosemarie has noted. Granted, the plant on the right is a larger plant overall and the plant on the left may change as it gets larger and this may be the difference. I have no problem if they are the same plant. Anyway, a great big thank you to all. And Jeff, you are just too funny!

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bikerdoc5968,
You're welcome. :) Let us know if the cactus you inquired about changes at all over time (flowers or the overall appearance of the cactus), I'd be interested to find out.