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A329888 a(n) = A329900(A329602(n)); Heinz number of the even bisection (even-indexed parts) of the integer partition with Heinz number n.

%I A329888 #53 Oct 17 2021 15:40:28
%S A329888 1,1,1,2,1,2,1,2,3,2,1,2,1,2,3,4,1,3,1,2,3,2,1,4,5,2,3,2,1,3,1,4,3,2,
%T A329888 5,6,1,2,3,4,1,3,1,2,3,2,1,4,7,5,3,2,1,6,5,4,3,2,1,6,1,2,3,8,5,3,1,2,
%U A329888 3,5,1,6,1,2,5,2,7,3,1,4,9,2,1,6,5,2,3,4,1,6,7,2,3,2,5,8,1,7,3,10,1,3,1,4,5
%N A329888 a(n) = A329900(A329602(n)); Heinz number of the even bisection (even-indexed parts) of the integer partition with Heinz number n.
%C A329888 From _Gus Wiseman_, Aug 05 2021 and _Antti Karttunen_, Oct 13 2021: (Start)
%C A329888 Also the product of primes at even positions in the weakly decreasing list (with multiplicity) of prime factors of n. For example, the prime factors of 108 are (3,3,3,2,2), with even bisection (3,2), with product 6, so a(108) = 6.
%C A329888 Proof: A108951(n) gives a number with the same largest prime factor (A006530) and its exponent (A071178) as in n, and with each smaller prime p = 2, 3, 5, 7, ... < A006530(n) having as its exponent the partial sum of the exponents of all prime factors >= p present in n (with primes not present in n having the exponent 0). Then applying A000188 replaces each such "partial sum exponent" k with floor(k/2). Finally, A319626 replaces those halved exponents with their first differences (here the exponent of the largest prime present stays intact, because the next larger prime's exponent is 0 in n). It should be easy to see that if prime q is not present in n (i.e., does not divide it), then neither it is present in a(n). Moreover, if the partial sum exponent of q is odd and only one larger than the partial sum exponent of the next larger prime factor of n, then q will not be present in a(n), while in all other cases q is present in a(n). See also the last example.
%C A329888 (End)
%H A329888 Antti Karttunen, <a href="/A329888/b329888.txt">Table of n, a(n) for n = 1..16384</a>
%H A329888 Antti Karttunen, <a href="/A329888/a329888.txt">Data supplement: n, a(n) computed for n = 1..65537</a>
%H A329888 <a href="/index/He#Heinz">Index entries for sequences related to Heinz numbers</a>
%H A329888 <a href="/index/Pri#primorial_numbers">Index entries for sequences related to primorial numbers</a>
%F A329888 a(n) = A329900(A329602(n)) = A329900(A000188(A108951(n))).
%F A329888 A108951(a(n)) = A329602(n).
%F A329888 a(n^2) = n for all n >= 1.
%F A329888 a(n) * A346701(n) = n. - _Gus Wiseman_, Aug 07 2021
%F A329888 A056239(a(n)) = A346700(n). - _Gus Wiseman_, Aug 07 2021
%F A329888 _Antti Karttunen_, Sep 21 2021
%F A329888 From _Antti Karttunen_, Oct 13 2021: (Start)
%F A329888 a(n) = A319626(A329602(n)) = A319626(A000188(A108951(n))).
%F A329888 For all x in A102750, a(x) = a(A253553(x)). (End)
%e A329888 From _Gus Wiseman_, Aug 15 2021: (Start)
%e A329888 The list of all numbers with image 12 and their corresponding prime factors begins:
%e A329888   144: (3,3,2,2,2,2)
%e A329888   216: (3,3,3,2,2,2)
%e A329888   240: (5,3,2,2,2,2)
%e A329888   288: (3,3,2,2,2,2,2)
%e A329888   336: (7,3,2,2,2,2)
%e A329888   360: (5,3,3,2,2,2)
%e A329888 (End)
%e A329888 The positions from the left are indexed as 1, 2, 3, ..., etc, so e.g., for 240 we pick the second, the fourth and the sixth prime factor, 3, 2 and 2, to obtain a(240) = 3*2*2 = 12. For 288, we similarly pick the second (3), the fourth (2) and the sixth (2) to obtain a(288) = 3*2*2 = 12. - _Antti Karttunen_, Oct 13 2021
%e A329888 Consider n = 11945934 = 2*3*3*3*7*11*13*13*17. Its primorial inflation is A108951(11945934) = 96478365991115908800000 = 2^9 * 3^8 * 5^5 * 7^5 * 11^4 * 13^3 * 17^1. Applying A000188 to this halves each exponent (floored down if the exponent is odd), leaving the factors 2^4 * 3^4 * 5^2 * 7^2 * 11^2 * 13^1 = 2497294800. Then applying A319626 to this number retains the largest prime factor (and its exponent), and subtracts from the exponent of each of the rest of primes the exponent of the next larger prime, so from 2^4 * 3^4 * 5^2 * 7^2 * 11^2 * 13^1 we get 2^(4-4) * 3^(4-2) * 5^(2-2) * 7^(2-2) * 11^(2-1) * 13^1 = 3^2 * 11^1 * 13^1 = 1287 = a(11945934), which is obtained also by selecting every second prime from the list [17, 13, 13, 11, 7, 3, 3, 3, 2] and taking their product. - _Antti Karttunen_, Oct 15 2021
%t A329888 Table[Times@@Last/@Partition[Reverse[Flatten[Apply[ConstantArray,FactorInteger[n],{1}]]],2],{n,100}] (* _Gus Wiseman_, Oct 13 2021 *)
%o A329888 (PARI) A329888(n) = A329900(A329602(n));
%o A329888 (PARI) A329888(n) = if(1==n,n,my(f=factor(n),m=1,p=0); forstep(k=#f~,1,-1,while(f[k,2], m *= f[k,1]^(p%2); f[k,2]--; p++)); (m)); \\ (After Wiseman's new interpretation) - _Antti Karttunen_, Sep 21 2021
%Y A329888 Cf. A000188, A108951, A319626, A329602, A329900.
%Y A329888 A left inverse of A000290.
%Y A329888 Positions of 1's are A008578.
%Y A329888 Positions of primes are A168645.
%Y A329888 The sum of prime indices of a(n) is A346700(n).
%Y A329888 The odd version is A346701.
%Y A329888 The odd non-reverse version is A346703.
%Y A329888 The non-reverse version is A346704.
%Y A329888 The version for standard compositions is A346705, odd A346702.
%Y A329888 A001221 counts distinct prime factors.
%Y A329888 A001222 counts all prime factors.
%Y A329888 A001414 adds up prime factors, row sums of A027746.
%Y A329888 A027187 counts partitions of even length, ranked by A028260.
%Y A329888 A056239 adds up prime indices, row sums of A112798.
%Y A329888 A103919 counts partitions by sum and alternating sum (reverse: A344612).
%Y A329888 A346633 adds up the even bisection of standard compositions.
%Y A329888 A346698 adds up the even bisection of prime indices.
%Y A329888 Cf. A000097, A035363, A102750, A236913, A253553, A344606, A344617, A344653, A345957, A345958, A345959.
%K A329888 nonn
%O A329888 1,4
%A A329888 _Antti Karttunen_, Dec 22 2019
%E A329888 Name amended with _Gus Wiseman_'s new interpretation - _Antti Karttunen_, Oct 13 2021