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A328616
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Number of digits in primorial base expansion of n that are maximal possible in their positions.
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4
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0, 1, 0, 1, 1, 2, 0, 1, 0, 1, 1, 2, 0, 1, 0, 1, 1, 2, 0, 1, 0, 1, 1, 2, 1, 2, 1, 2, 2, 3, 0, 1, 0, 1, 1, 2, 0, 1, 0, 1, 1, 2, 0, 1, 0, 1, 1, 2, 0, 1, 0, 1, 1, 2, 1, 2, 1, 2, 2, 3, 0, 1, 0, 1, 1, 2, 0, 1, 0, 1, 1, 2, 0, 1, 0, 1, 1, 2, 0, 1, 0, 1, 1, 2, 1, 2, 1, 2, 2, 3, 0, 1, 0, 1, 1, 2, 0, 1, 0, 1, 1, 2, 0, 1, 0, 1
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OFFSET
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0,6
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LINKS
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FORMULA
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EXAMPLE
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In primorial base (see A049345), the maximum digit value that can occur in the k-th position from the right (with k=1 standing for the rightmost, i.e., the least significant digit position) is A000040(k)-1, and it is for the terms of A057588 (primorial numbers minus one) where all significant digits are maximal allowed for their positions, e.g. 209 is written as "6421" because 209 = 6*30 + 4*6 + 2*2 + 1*1, thus a(209) = 4.
87 is written as "2411" because 87 = 2*A002110(3) + 4*A002110(2) + 1*A002110(1) + 1*A002110(0) = 2*30 + 4*6 + 1*2 + 1*1. Only the digit positions 1 and 3 are occupied with maximum digits allowed in those positions (that are 1 and 4, being one less than the corresponding primes, 2 and 5), thus a(87) = 2.
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MATHEMATICA
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a[n_] := Module[{k = n, p = 2, s = {}, r}, While[{k, r} = QuotientRemainder[k, p]; k != 0 || r != 0, AppendTo[s, r]; p = NextPrime[p]]; Count[Prime[Range[1, Length[s]]] - s, 1]]; a[0] = 0; Array[a, 100, 0] (* Amiram Eldar, Mar 13 2024 *)
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PROG
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(PARI) A328616(n) = { my(s=0, p=2); while(n, s += ((p-1)==(n%p)); n = n\p; p = nextprime(1+p)); (s); };
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CROSSREFS
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Cf. A057588 (positions of records, and the first occurrence of each n > 0).
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KEYWORD
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nonn,base
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AUTHOR
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STATUS
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approved
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