A129597 -id:A129597 - cp's OEIS frontend

A346704 Product of primes at even positions in the weakly increasing list (with multiplicity) of prime factors of n.

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

Offset: 1

Gus Wiseman, Aug 08 2021

			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.

Robert Israel, Table of n, a(n) for n = 1..10000

Positions of first appearances are A129597.
Positions of 1's are A008578.
Positions of primes are A168645.
The sum of prime indices of a(n) is A346698(n).
The odd version is A346703 (sum: A346697).
The odd reverse version is A346701 (sum: A346699).
The reverse version appears to be A329888 (sum: A346700).
A001221 counts distinct prime factors.
A001222 counts all prime factors.
A027187 counts partitions of even length, ranked by A028260.
A056239 adds up prime indices, row sums of A112798.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A316524 gives the alternating sum of prime indices (reverse: A344616).
A335433/A335448 rank separable/inseparable partitions.
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.

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:
map(f, [$1..100]); # Robert Israel, Aug 12 2024

Table[Times@@Last/@Partition[Flatten[Apply[ConstantArray,FactorInteger[n],{1}]],2],{n,100}]

a(n) * A346703(n) = n.

A056239(a(n)) = A346698(n).

A129595 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.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, May 01 2007, based on Marc LeBrun's Jan 11 2006 message on SeqFan mailing list

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.

			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.

A. Karttunen, Table of n, a(n) for n = 1..1275

Cf. A129593, A129596, A129597, A129598.

A122111 gives the involution of natural numbers induced when partition conjugation (see A129594) is applied to the same encoding.

A342768 a(n) = A342767(n, n).

Original entry on oeis.org

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

Offset: 1

Rémy Sigrist, Apr 02 2021

nonn

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

			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.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, PARI program for A342768
Index entries for sequences related to dismal (or lunar) arithmetic

Cf. A007947, A087019, A342767.
The sum of prime indices of a(n) is 2*A056239(n) - A061395(n).
The version for even indices is A129597(n) = 2*a(n) for n > 1.
The sorted version is A346635.
These are the positions of first appearances in A346701 and in A346703.
A001221 counts distinct prime factors.
A001222 counts prime factors with multiplicity.
A027193 counts partitions of odd length, ranked by A026424.
A209281 adds up the odd bisection of standard compositions (even: A346633).
A346697 adds up the odd bisection of prime indices (reverse: A346699).
Cf. A000290, A006530, A033942, A037143, A344653, A345957, A346698, A346700, A346704.

Table[n^2/FactorInteger[n][[-1,1]],{n,100}] (* Gus Wiseman, Aug 09 2021 *)

PARI
```
See Links section.
```

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.
A007947(a(n)) = A007947(n).
A001222(a(n)) = 2*A001222(n) - 1 for any n > 1.
From Gus Wiseman, Aug 09 2021: (Start)
A001221(a(n)) = A001221(n).
If g = A006530(n) is the greatest prime factor of n, then a(n) = n^2/g.
a(n) = A129597(n)/2.
(End)

A346635 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.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Aug 10 2021

nonn

This is the sorted version of A342768(n) = position of first appearance of n in A346701 (but A346703 works also).

			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}

Removing 1 gives a subset of A026424.
The unsorted even version is A129597.
The unsorted version is A342768(n) = A342767(n,n).
Except the first term, the even version is 2*a(n).
A000290 lists squares.
A001221 counts distinct prime factors.
A001222 counts all prime factors.
A006530 gives the greatest prime factor.
A061395 gives the greatest prime index.
A027193 counts partitions of odd length.
A056239 adds up prime indices, row sums of A112798.
A209281 = odd bisection sum of standard compositions (even: A346633).
A316524 = alternating sum of prime indices (sign: A344617, rev.: A344616).
A325534 counts separable partitions, ranked by A335433.
A325535 counts inseparable partitions, ranked by A335448.
A344606 counts alternating permutations of prime indices.
A346697 = odd bisection sum of prime indices (weights of A346703).
A346699 = odd bisection sum of reversed prime indices (weights of A346701).
Cf. A028260, A033942, A035363, A037143, A341446, A344653, A345957, A345958, A345959, A346698, A346700, A346704.

filter:= proc(n) issqr(n/max(numtheory:-factorset(n))) end proc:
filter(1):= true:
select(filter, [$1..200]); # Robert Israel, Nov 26 2022

sqrQ[n_]:=IntegerQ[Sqrt[n]];
Select[Range[100],sqrQ[#*FactorInteger[#][[-1,1]]]&]

isok(m) = (m==1) || issquare(m/vecmax(factor(m)[,1])); \\ Michel Marcus, Aug 12 2021

a(n) = A129597(n)/2 for n > 1.

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A346704 Product of primes at even positions in the weakly increasing list (with multiplicity) of prime factors of n.

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A129595 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.

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A342768 a(n) = A342767(n, n).

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Mathematica

PARI

Formula

A346635 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.

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Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Mathematica

PARI

Formula