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Product of primes at even positions in the weakly increasing list (with multiplicity) of prime factors of n.
+10
17
1, 1, 1, 2, 1, 3, 1, 2, 3, 5, 1, 2, 1, 7, 5, 4, 1, 3, 1, 2, 7, 11, 1, 6, 5, 13, 3, 2, 1, 3, 1, 4, 11, 17, 7, 6, 1, 19, 13, 10, 1, 3, 1, 2, 3, 23, 1, 4, 7, 5, 17, 2, 1, 9, 11, 14, 19, 29, 1, 10, 1, 31, 3, 8, 13, 3, 1, 2, 23, 5, 1, 6, 1, 37, 5, 2, 11, 3, 1, 4, 9
EXAMPLE
The prime factors of 108 are (2,2,3,3,3), with even bisection (2,3), with product 6, so a(108) = 6.
The prime factors of 720 are (2,2,2,2,3,3,5), with even bisection (2,2,3), with product 12, so a(720) = 12.
MAPLE
f:= proc(n) local F, i;
F:= ifactors(n)[2];
F:= sort(map(t -> t[1]$t[2], F));
mul(F[i], i=2..nops(F), 2)
end proc:
MATHEMATICA
Table[Times@@Last/@Partition[Flatten[Apply[ConstantArray, FactorInteger[n], {1}]], 2], {n, 100}]
CROSSREFS
Positions of first appearances are A129597. The sum of prime indices of a(n) is A346698(n). A001221 counts distinct prime factors.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A316524 gives the alternating sum of prime indices (reverse: A344616).
A344606 counts alternating permutations of prime indices.
A344617 gives the sign of the alternating sum of prime indices.
A346633 adds up the even bisection of standard compositions.
Cf. A026424, A035363, A209281, A236913, A342768, A344653, A345957, A345958, A345960, A345961, A345962.
Array A(i,j): A(1,1), A(2,1), A(1,2), A(3,1), A(2,2), A(1,3), ... of elementwise sums of partitions encoded in the prime factorizations of i and j.
+10
9
1, 2, 2, 3, 4, 3, 4, 9, 9, 4, 5, 8, 6, 8, 5, 6, 25, 27, 27, 25, 6, 7, 18, 15, 16, 15, 18, 7, 8, 49, 12, 125, 125, 12, 49, 8, 9, 16, 35, 54, 10, 54, 35, 16, 9, 10, 27, 81, 343, 45, 45, 343, 81, 27, 10, 11, 50, 18, 32, 21, 24, 21, 32, 18, 50, 11, 12, 121, 30, 81, 625, 175
COMMENTS
As described by Marc LeBrun, we can map integers 1-to-1 to partitions in a "crazy" order: factor n, take the (finite) tuple of exponents, add 1 to the first, use the rest as successive differences between parts and finally subtract 1 from the last part, thus we get the following partitions (elements in ascending order): 2 -> [1] -> 1, 3 -> [0,1] -> 1+1, 4 -> [2] -> 2, 5 -> [0,0,1] -> 1+1+1, 6 -> [1,1] -> 2+2, 7 -> [0,0,0,1] -> 1+1+1+1, 8 -> [3] -> 3, 9 -> [0,2] -> 1+2, 10 -> [1,0,1] -> 2+2+2, etc. Inverse process: from a sorted (elements in ascending order) partition of n, subtract 1 from the first part, then take the first differences of parts and add 1 to the last (of differences or the first part if a singular partition) and use them as the exponents for A000040(1), A000040(2), etc. and multiply. This array is obtained when we encode in such a way the partition obtained as an element-wise sum of two partitions encoded by i and j. The element-wise addition begins from the largest elements of the partitions, continuing towards the smaller elements and if the partitions do not contain the same number of elements, the shorter is prepended with as many zeros as needed to make them of equal length.
On what condition does A(i,j) = i*j ? E.g., A(3,5)=15, A(3,10)=30, A(5,11)=55. However A(3,7)=35 and A(5,7)=21.
EXAMPLE
a(54) = A(9,2) = 27 because when we add element-wise partition 1+2 encoded by 9 to a singular partition 1 encoded by 2, we get partition 1+3, which maps to exponent tuple [0,3] and 27 = 2^0 * 3^3.
CROSSREFS
A122111 gives the involution of natural numbers induced when partition conjugation (see A129594) is applied to the same encoding.
1, 2, 3, 8, 5, 12, 7, 32, 27, 20, 11, 48, 13, 28, 45, 128, 17, 108, 19, 80, 63, 44, 23, 192, 125, 52, 243, 112, 29, 180, 31, 512, 99, 68, 175, 432, 37, 76, 117, 320, 41, 252, 43, 176, 405, 92, 47, 768, 343, 500, 153, 208, 53, 972, 275, 448, 171, 116, 59, 720
COMMENTS
This sequence has similarities with A087019. These are the positions of first appearances of each positive integer in A346701, and also in A346703. - Gus Wiseman, Aug 09 2021
FORMULA
a(n) = n iff n = 1 or n is a prime number.
a(p^k) = p^(2*k-1) for any k > 0 and any prime number p.
If g = A006530(n) is the greatest prime factor of n, then a(n) = n^2/g. (End)
EXAMPLE
For n = 42:
- 42 = 2 * 3 * 7, so:
2 3 7
x 2 3 7
-------
2 3 7
2 3 3
+ 2 2 2
-----------
2 2 3 3 7
- hence a(42) = 2 * 2 * 3 * 3 * 7 = 252.
MATHEMATICA
Table[n^2/FactorInteger[n][[-1, 1]], {n, 100}] (* Gus Wiseman, Aug 09 2021 *)
PROG
(PARI) See Links section.
CROSSREFS
The version for even indices is A129597(n) = 2*a(n) for n > 1. These are the positions of first appearances in A346701 and in A346703. A001221 counts distinct prime factors.
A001222 counts prime factors with multiplicity.
A209281 adds up the odd bisection of standard compositions (even: A346633).
A346697 adds up the odd bisection of prime indices (reverse: A346699).
Numbers whose division (or multiplication) by their greatest prime factor yields a perfect square. Numbers k such that k* A006530(k) is a perfect square.
+10
7
1, 2, 3, 5, 7, 8, 11, 12, 13, 17, 19, 20, 23, 27, 28, 29, 31, 32, 37, 41, 43, 44, 45, 47, 48, 52, 53, 59, 61, 63, 67, 68, 71, 73, 76, 79, 80, 83, 89, 92, 97, 99, 101, 103, 107, 108, 109, 112, 113, 116, 117, 124, 125, 127, 128, 131, 137, 139, 148, 149, 151, 153
COMMENTS
This is the sorted version of A342768(n) = position of first appearance of n in A346701 (but A346703 works also).
EXAMPLE
The terms together with their prime indices begin:
1: {} 31: {11} 71: {20}
2: {1} 32: {1,1,1,1,1} 73: {21}
3: {2} 37: {12} 76: {1,1,8}
5: {3} 41: {13} 79: {22}
7: {4} 43: {14} 80: {1,1,1,1,3}
8: {1,1,1} 44: {1,1,5} 83: {23}
11: {5} 45: {2,2,3} 89: {24}
12: {1,1,2} 47: {15} 92: {1,1,9}
13: {6} 48: {1,1,1,1,2} 97: {25}
17: {7} 52: {1,1,6} 99: {2,2,5}
19: {8} 53: {16} 101: {26}
20: {1,1,3} 59: {17} 103: {27}
23: {9} 61: {18} 107: {28}
27: {2,2,2} 63: {2,2,4} 108: {1,1,2,2,2}
28: {1,1,4} 67: {19} 109: {29}
29: {10} 68: {1,1,7} 112: {1,1,1,1,4}
MAPLE
filter:= proc(n) issqr(n/max(numtheory:-factorset(n))) end proc:
filter(1):= true:
MATHEMATICA
sqrQ[n_]:=IntegerQ[Sqrt[n]];
Select[Range[100], sqrQ[#*FactorInteger[#][[-1, 1]]]&]
PROG
(PARI) isok(m) = (m==1) || issquare(m/vecmax(factor(m)[, 1])); \\ Michel Marcus, Aug 12 2021
CROSSREFS
Removing 1 gives a subset of A026424. The unsorted even version is A129597. Except the first term, the even version is 2*a(n).
A001221 counts distinct prime factors.
A006530 gives the greatest prime factor.
A061395 gives the greatest prime index.
A027193 counts partitions of odd length.
A344606 counts alternating permutations of prime indices.
A346699 = odd bisection sum of reversed prime indices (weights of A346701).
Cf. A028260, A033942, A035363, A037143, A341446, A344653, A345957, A345958, A345959, A346698, A346700, A346704.
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