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productive numbers
A number  $n$  is said to be (prime) productive if  $n+1$  is prime, as well as all the numbers obtained inserting one '×' operator among its digits and adding one.

For example, 256 is a prime productive number since  $256+1$,  $2\cdot56+1$, and  $25\cdot6+1$ are all prime numbers.

There are only 208 productive numbers below  $5\cdot 10^{14}$, the largest being 8220001387336.

By definition, each productive number is one less than a prime number, so they are all even (except for 1) and do not end with the digit '4' (except for 4 itself).

The first prime productive numbers are 1, 2, 4, 6, 12, 16, 22, 28, 36, 52, 58, 66, 82, 106, 112, 136, 166, 178, 256, 306, 336, 352, 448 more terms

Pictorial representation of remainders (mod 2, 3, ...,11) frequency. For a table of values and more details click here
remainders of productive numbers

A graph displaying how many productive numbers are multiples of the primes p from 2 to 71. In black the ideal line 1/p.
divisibility of productive numbers
Useful links OEIS, Sequence A089395
Productive numbers can also be... (you may click on names or numbers and on + to get more values)

aban 12 + 652 658 718 982 abundant 12 + 5280616 5538316 11830336 28056616 admirable 12 + 2002 202636 405412 73226356 alternating 12 + 1498 3412 49018 49612 amenable 12 + 65521528 73226356 94150168 114769048 apocalyptic 508 + 19402 25228 25366 28516 arithmetic 22 + 8114332 8486602 9116608 9598162 Bell 52 binomial 28 + 66 136 2002 211876 c.heptagonal 22 106 c.nonagonal 28 136 c.pentagonal 16 106 c.triangular 136 166 canyon 106 + 7606 8536 9748 75268 congruent 22 + 7929268 8114332 9109468 9116608 constructible 12 16 136 256 Cunningham 28 82 2026 Curzon 306 d-powerful 1498 + 60352 227386 1935442 2673556 decagonal 52 deficient 16 + 8610442 9109468 9116608 9598162 dig.balanced 12 + 762226 7929268 8486602 60270676 Duffinian 16 36 256 3136 eban 36 52 66 2002 66036 economical 16 + 1909408 1935442 11311126 11830336 emirpimes 58 + 119758 770026 1651942 1935442 enlightened 256 equidigital 16 + 1909408 1935442 11311126 11830336 esthetic 12 Eulerian 66 502 evil 12 + 181508542 214162078 656835886 876128326 fibodiv 28 2998 Friedman 276556 frugal 256 gapful 352 + 25228 7663156 11830336 94150168 happy 28 + 762226 7663156 7929268 9116608 harmonic 28 Harshad 12 + 96052 917728 1433512 2673556 heptagonal 112 616 hexagonal 28 66 highly composite 12 36 hoax 22 + 211876 770026 39933892 73226356 house 652 hyperperfect 28 iban 12 + 2002 3412 10312 43312 idoneal 12 + 22 28 58 112 impolite 16 256 inconsummate 7126 interprime 12 Jordan-Polya 12 16 36 256 junction 616 + 917728 1610428 44883532 65521528 katadrome 52 + 652 982 8542 9532 Lynch-Bell 12 36 magnanimous 12 + 58 112 136 556 metadrome 12 + 58 136 178 256 modest 1018 51226666 Moran 25366 mountain 352 + 586 1498 2686 3592 nialpdrome 22 + 982 8542 8662 9532 nude 12 + 66 112 336 448 O'Halloran 12 36 oban 12 + 586 616 658 718 odious 16 + 73226356 94150168 114769048 394166542 palindromic 22 66 616 2002 pancake 16 + 106 352 562 2776 panconsummate 12 36 166 partition 22 pentagonal 12 22 1162 6112 211876 perfect 28 pernicious 12 + 5538316 7663156 8114332 9598162 Perrin 12 22 plaindrome 12 + 448 556 1228 11578 power 16 36 256 3136 powerful 16 36 256 3136 practical 12 + 276556 917728 967708 5280616 prim.abundant 12 + 2002 202636 405412 73226356 pronic 12 306 pseudoperfect 12 + 621628 702268 917728 967708 repdigit 22 66 repfigit 28 Ruth-Aaron 16 self 5506 + 11830336 22067518 36046162 114769048 self-describing 22 semiprime 22 + 4764778 8610442 36046162 51226666 sliding 52 502 Smith 22 + 79978 770026 967708 6614248 sphenic 66 + 2460178 11311126 19835602 23261206 square 16 36 256 3136 super Niven 12 36 306 super-d 106 + 335206 1715866 8610442 9116608 superabundant 12 36 tau 12 + 66036 455992 11830336 46544968 triangular 28 36 66 136 tribonacci 3136 uban 12 + 52 58 66 82 Ulam 16 + 6112 76366 241228 631582 undulating 616 untouchable 52 + 658882 702268 762226 967708 upside-down 28 82 wasteful 12 + 8610442 9109468 9116608 9598162 Zuckerman 12 36 112 Zumkeller 12 + 25228 52768 60352 66036 zygodrome 22 66