A233692
The smallest prime that produces a set of n primes such that every prime after the first one is equal to the previous plus the product of its nonzero digits.
Original entry on oeis.org
2, 23, 23, 239, 239, 239, 3413, 14249, 524219, 4167379, 324550981, 2589767209, 346333812907
Offset: 1
For n=3, initial prime=23, set {23, 29, 47} because 23 -> 23+2*3=29 -> 29+2*9=47.
For n=2 to 9, the sets are
23, 29;
23, 29, 47;
239, 293, 347, 431;
239, 293, 347, 431, 443;
239, 293, 347, 431, 443, 491;
3413, 3449, 3881, 4073, 4157, 4297, 4801;
14249, 14537, 14957, 16217, 16301, 16319, 16481, 16673;
524219, 524939, 534659, 550859, 559859, 640859, 649499, 719483, 725531.
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checkp(p, n) = {ok = isprime(p); for (i=1, n, print1(p, ", "); digs = digits(p); np = p + prod(i=1, #digs, if (d=digs[i], d, 1)); p = np;if (i != n, ok = ok && isprime(p));); ok;} \\ Michel Marcus, Dec 15 2013
A235680
The smallest first term of a sequence of n primes such that, after the first, each is equal to the previous prime plus the sum of all of its digits, plus the product of all of its nonzero digits.
Original entry on oeis.org
2, 191, 163, 151, 127, 1644997, 36778597, 935715673, 50682890749, 16390560362269, 63334172492839
Offset: 1
Example for n=8: a(8)=935715673 because after it the seven primes are 936311069, 936337351, 936490481, 936677149, 938391809, 938811763 and 939029537, with 936311069 = 935715673 + 9*3*5*7*1*5*6*7*3+(9+3+5+7+1+5+6+7+3) and so on...
A300326
Sum of the largest possible permutations that can be written without repetition of digits in each base from binary to n+1.
Original entry on oeis.org
0, 2, 23, 251, 3181, 47971, 848638, 17283462, 398650506, 10275193716, 292733747621, 9135147415313, 309906954656231, 11356162260536389, 447015900139452604, 18811774444632517324, 842820629057975778516, 40053081963609542635686, 2012366504118798707101875
Offset: 0
Let us consider the numbers: 0[1], 10[2], 210[3], 3210[4], 43210[5], and 543210[6];
Their respective decimal representations are the first six terms of A062813: 0, 2, 21, 228, 2930, 44790. The partial sums for those terms are 0, 2, 23, 251, 3181, and 47971; after 0, the following 4 sums are primes, but 47971 is not prime. The same is true for subsequent partial sums, whence the conjecture in COMMENTS.
Cf.
A233783 for the occurrence of the ordered triple (2,23,251) in a different context.
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