%I #58 Dec 03 2022 19:40:45
%S 1,2,1,2,1,3,1,2,1,2,1,3,1,2,1,2,1,3,1,2,1,2,1,3,1,2,1,2,1,4,1,2,1,2,
%T 1,3,1,2,1,2,1,3,1,2,1,2,1,3,1,2,1,2,1,3,1,2,1,2,1,4,1,2,1,2,1,3,1,2,
%U 1,2,1,3,1,2,1,2,1,3,1,2,1,2,1,3,1,2,1,2,1,4,1,2,1,2,1,3
%N Least gap in the partition having Heinz number n; index of the least prime not dividing n.
%C The "least gap" of a partition is the least positive integer that is not a part of the partition. For example, the least gap of the partition [7,4,2,2,1] is 3.
%C We define the Heinz number of a partition p = [p_1, p_2, ..., p_r] as Product(p_j-th prime, j=1...r) (concept used by _Alois P. Heinz_ in A215366 as an "encoding" of a partition). For example, for the partition [1, 1, 2, 4, 10] we get 2*2*3*7*29 = 2436.
%C In the Maple program the subprogram B yields the partition with Heinz number n.
%C Sum of least gaps of all partitions of m = A022567(m).
%C From _Antti Karttunen_, Aug 22 2016: (Start)
%C Index of the least prime not dividing n. (After a formula given by Heinz.)
%C Least k such that A002110(k) does not divide n.
%C One more than the number of trailing zeros in primorial base representation of n, A049345.
%C (End)
%C The least gap is also called the mex (minimal excludant) of the partition. - _Gus Wiseman_, Apr 20 2021
%D G. E. Andrews and K. Eriksson, Integer Partitions, Cambridge Univ. Press, 2004, Cambridge.
%D Miklós Bóna, A Walk Through Combinatorics, World Scientific Publishing Co., 2002.
%H Alois P. Heinz, <a href="/A257993/b257993.txt">Table of n, a(n) for n = 1..20000</a>
%H George E. Andrews and David Newman, <a href="https://doi.org/10.1007/s00026-019-00427-w">Partitions and the Minimal Excludant</a>, Annals of Combinatorics, Volume 23, May 2019, Pages 249-254.
%H P. J. Grabner and A. Knopfmacher, <a href="https://www.math.tugraz.at/fosp/pdfs/tugraz_0087.pdf">Analysis of some new partition statistics</a>, Ramanujan J., 12, 2006, 439-454.
%H Brian Hopkins, James A. Sellers, and Dennis Stanton, <a href="https://arxiv.org/abs/2009.10873">Dyson's Crank and the Mex of Integer Partitions</a>, arXiv:2009.10873 [math.CO], 2020.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Mex_(mathematics)">Mex (mathematics)</a>.
%H <a href="/index/Pri#primorialbase">Index entries for sequences related to primorial base</a>.
%F a(n) = A000720(A053669(n)). - _Alois P. Heinz_, May 18 2015
%F From _Antti Karttunen_, Aug 22-30 2016: (Start)
%F a(n) = 1 + A276084(n).
%F a(n) = A055396(A276086(n)).
%F A276152(n) = A002110(a(n)).
%F (End)
%F Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1 + Sum_{k>=1} 1/A002110(k) = 1.705230... (1 + A064648). - _Amiram Eldar_, Jul 23 2022
%F a(n) << log n/log log n. - _Charles R Greathouse IV_, Dec 03 2022
%e a(18) = 3 because the partition having Heinz number 18 = 2*3*3 is [1,2,2], having least gap equal to 3.
%p with(numtheory): a := proc (n) local B, q: B := proc (n) local nn, j, m: nn := op(2, ifactors(n)): for j to nops(nn) do m[j] := op(j, nn) end do: [seq(seq(pi(op(1, m[i])), q = 1 .. op(2, m[i])), i = 1 .. nops(nn))] end proc: for q while member(q, B(n)) = true do end do: q end proc: seq(a(n), n = 1 .. 150);
%p # second Maple program:
%p a:= n-> `if`(n=1, 1, (s-> min({$1..(max(s)+1)} minus s))(
%p {map(x-> numtheory[pi](x[1]), ifactors(n)[2])[]})):
%p seq(a(n), n=1..100); # _Alois P. Heinz_, May 09 2016
%p # faster:
%p A257993 := proc(n) local p, c; c := 1; p := 2;
%p while n mod p = 0 do p := nextprime(p); c := c + 1 od: c end:
%p seq(A257993(n), n=1..100); # _Peter Luschny_, Jun 04 2017
%t A053669[n_] := For[p = 2, True, p = NextPrime[p], If[CoprimeQ[p, n], Return[p]]]; a[n_] := PrimePi[A053669[n]]; Array[a, 100] (* _Jean-François Alcover_, Nov 28 2016 *)
%t Table[k = 1; While[! CoprimeQ[Prime@ k, n], k++]; k, {n, 100}] (* _Michael De Vlieger_, Jun 22 2017 *)
%o (Scheme)
%o (define (A257993 n) (let loop ((n n) (i 1)) (let* ((p (A000040 i)) (d (modulo n p))) (if (not (zero? d)) i (loop (/ (- n d) p) (+ 1 i))))))
%o ;; _Antti Karttunen_, Aug 22 2016
%o (Python)
%o from sympy import nextprime, primepi
%o def a053669(n):
%o p = 2
%o while True:
%o if n%p!=0: return p
%o else: p=nextprime(p)
%o def a(n): return primepi(a053669(n)) # _Indranil Ghosh_, May 12 2017
%o (PARI) a(n) = forprime(p=2,, if (n % p, return(primepi(p)))); \\ _Michel Marcus_, Jun 22 2017
%Y Cf. A215366, A002110, A022567, A049345, A053669, A055396, A064648, A276086, A276094, A276152.
%Y Positions of 1's are A005408.
%Y Positions of 2's are A047235.
%Y The number of gaps is A079067.
%Y The version for crank is A257989.
%Y The triangle counting partitions by this statistic is A264401.
%Y One more than A276084.
%Y The version for greatest difference is A286469 or A286470.
%Y A maximal instead of minimal version is A339662.
%Y Positions of even terms are A342050.
%Y Positions of odd terms are A342051.
%Y A000070 counts partitions with a selected part.
%Y A006128 counts partitions with a selected position.
%Y A056239 adds up prime indices, row sums of A112798.
%Y A073491 lists numbers with gap-free prime indices.
%Y A238709 counts partitions by sum and least difference.
%Y A333214 lists positions of adjacent unequal prime gaps.
%Y A339737 counts partitions by sum and greatest gap.
%Y Cf. A001223, A001511, A005117, A026794, A029707, A072233, A079068, A098743, A124010, A279945, A325351, A325352.
%K nonn
%O 1,2
%A _Emeric Deutsch_, May 18 2015
%E A simpler description added to the name by _Antti Karttunen_, Aug 22 2016