A342768 -id:A342768 - 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).

A346703 Product of primes at odd positions in the weakly increasing list (with multiplicity) of prime factors of n.

Original entry on oeis.org

1, 2, 3, 2, 5, 2, 7, 4, 3, 2, 11, 6, 13, 2, 3, 4, 17, 6, 19, 10, 3, 2, 23, 4, 5, 2, 9, 14, 29, 10, 31, 8, 3, 2, 5, 6, 37, 2, 3, 4, 41, 14, 43, 22, 15, 2, 47, 12, 7, 10, 3, 26, 53, 6, 5, 4, 3, 2, 59, 6, 61, 2, 21, 8, 5, 22, 67, 34, 3, 14, 71, 12, 73, 2, 15, 38

Offset: 1

Gus Wiseman, Aug 08 2021

nonn

			The prime factors of 108 are (2,2,3,3,3), with odd bisection (2,3,3), with product 18, so a(108) = 18.
The prime factors of 720 are (2,2,2,2,3,3,5), with odd bisection (2,2,3,5), with product 60, so a(720) = 60.

Positions of 2's are A001747.
Positions of primes are A037143 (complement: A033942).
The even reverse version appears to be A329888.
Positions of first appearances are A342768.
The sum of prime indices of a(n) is A346697(n), reverse: A346699.
The reverse version is A346701.
The even version is A346704.
A001221 counts distinct prime factors.
A001222 counts all prime factors.
A056239 adds up prime indices, row sums of A112798.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A209281 (shifted) adds up the odd bisection of standard compositions.
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.
A346698 gives the sum of the even bisection of prime indices.
A346700 gives the sum of the even bisection of reversed prime indices.
Cf. A025047, A027187, A027193, A053738, A097805, A106356, A341446, A344653, A345957, A345958, A345959.

Table[Times@@First/@Partition[Append[Flatten[Apply[ConstantArray,FactorInteger[n],{1}]],0],2],{n,100}]

a(n) * A346704(n) = n.

A056239(a(n)) = A346697(n).

A346701 Heinz number of the odd bisection (odd-indexed parts) of the integer partition with Heinz number n.

Original entry on oeis.org

1, 2, 3, 2, 5, 3, 7, 4, 3, 5, 11, 6, 13, 7, 5, 4, 17, 6, 19, 10, 7, 11, 23, 6, 5, 13, 9, 14, 29, 10, 31, 8, 11, 17, 7, 6, 37, 19, 13, 10, 41, 14, 43, 22, 15, 23, 47, 12, 7, 10, 17, 26, 53, 9, 11, 14, 19, 29, 59, 10, 61, 31, 21, 8, 13, 22, 67, 34, 23, 14, 71

Offset: 1

Gus Wiseman, Aug 03 2021

nonn

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

			The partition (2,2,2,1,1) has Heinz number 108 and odd bisection (2,2,1) with Heinz number 18, so a(108) = 18.
The partitions (3,2,2,1,1), (3,2,2,2,1), (3,3,2,1,1) have Heinz numbers 180, 270, 300 and all have odd bisection (3,2,1) with Heinz number 30, so a(180) = a(270) = a(300) = 30.

Positions of last appearances are A000290 without the first term 0.
Positions of primes are A037143 (complement: A033942).
The even version is A329888.
Positions of first appearances are A342768.
The sum of prime indices of a(n) is A346699(n), non-reverse: A346697.
The non-reverse version is A346703.
The even non-reverse version is A346704.
A001221 counts distinct prime factors.
A001222 counts all prime factors.
A056239 adds up prime indices, row sums of A112798.
A103919 counts partitions by sum and alternating sum, reverse A344612.
A209281 (shifted) adds up the odd bisection of standard compositions.
A316524 gives the alternating sum of prime indices, reverse A344616.
A325534 counts separable partitions, ranked by A335433.
A325535 counts inseparable partitions, ranked by A335448.
A344606 counts alternating permutations of prime indices.
A344617 gives the sign of the alternating sum of prime indices.
A346700 gives the sum of the even bisection of reversed prime indices.
Cf. A025047, A027187, A027193, A053738, A106356, A277103, A341446, A344653, A345957, A345958, A345959, A346698, A346702.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Times@@Prime/@First/@Partition[Append[Reverse[primeMS[n]],0],2],{n,100}]

a(n) * A329888(n) = n.

A056239(a(n)) = A346699(n).

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.

A129597 Central diagonal of array A129595.

Original entry on oeis.org

1, 4, 6, 16, 10, 24, 14, 64, 54, 40, 22, 96, 26, 56, 90, 256, 34, 216, 38, 160, 126, 88, 46, 384, 250, 104, 486, 224, 58, 360, 62, 1024, 198, 136, 350, 864, 74, 152, 234, 640, 82, 504, 86, 352, 810, 184, 94, 1536, 686, 1000, 306, 416, 106, 1944, 550, 896, 342

Offset: 1

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

nonn

These are the positions of first appearances of each positive integer in A346704. - Gus Wiseman, Oct 16 2021

Antti Karttunen, Table of n, a(n) for n = 1..10000

a(n) = A129595(n,n).
The sum of prime indices of a(n) is 2*A056239(n) - A061395(n) + 1 for n > 1.
The version for odd indices is A342768(n) = a(n)/2 for n > 1.
Except the first term, the sorted version is 2*A346635.
These are the positions of first appearances in A346704.
A001221 counts distinct prime factors.
A001222 counts prime factors with multiplicity.
A027187 counts partitions of even length, ranked by A028260.
A346633 adds up the even bisection of standard compositions (odd: A209281).
A346698 adds up the even bisection of prime indices (reverse: A346699).
Cf. A000290, A006530, A037143, A329888, A344606, A345957, A346697, A346700, A346701.

Table[If[n==1,1,2*n^2/FactorInteger[n][[-1,1]]],{n,100}] (* Gus Wiseman, Aug 10 2021 *)

A129597(n) = if(1==n, n, my(f=factor(n)); (2*n*n)/f[#f~, 1]); \\ Antti Karttunen, Oct 16 2021

From Gus Wiseman, Aug 10 2021: (Start)
For n > 1, A001221(a(n)) = A099812(n).
If g = A006530(n) is the greatest prime factor of n > 1, then a(n) = 2n^2/g.
a(n) = A100484(A000720(n)) = 2n iff n is prime.
a(n > 1) = 2*A342768(n).
(End)

A342767 Array T(n, k), n, k > 0, read by antidiagonals; a variant of lunar multiplication (A087062) based on prime factorizations of numbers (see Comments section for precise definition).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 4, 3, 4, 1, 1, 2, 4, 4, 2, 1, 1, 4, 3, 8, 3, 4, 1, 1, 2, 6, 4, 4, 6, 2, 1, 1, 8, 3, 8, 5, 8, 3, 8, 1, 1, 4, 8, 4, 6, 6, 4, 8, 4, 1, 1, 4, 9, 16, 5, 12, 5, 16, 9, 4, 1, 1, 2, 6, 8, 8, 6, 6, 8, 8, 6, 2, 1, 1, 8, 3, 8, 9, 16, 7, 16, 9, 8, 3, 8, 1

Offset: 1

Rémy Sigrist, Apr 02 2021

To compute T(n, k):
- write the prime factors of n and of k in ascending order with multiplicities on two lines, right aligned,
- to "multiply" two prime numbers: take the smallest,
- to "add" two prime numbers: take the largest,
- for example, for T(12, 14):
     12   ->   2 2 3
     14   -> x   2 7
             -------
               2 2 3
           + 2 2 2
           ---------
             2 2 2 3   ->  24 = T(12, 14)
This sequence is closely related to lunar multiplication (A087062):
- let n and k be two p-smooth numbers,
- let f be the function that associates to a p-smooth number, say m, the unique number whose (p+1)-base digits are prime, nondecreasing and whose product is m,
- let g be the inverse of f,
- then for any p-smooth numbers n and k, T(n, k) = g(f(n) "*" f(k)) where "*" denotes lunar product in base p+1,
- as T(n, p) = n for any prime number >= A006530(n), we don't have prime numbers here,
- however, if we consider only p-smooth numbers (for some prime number p), then p is the "unit" and the semiprimes p*q (with q <= p) are "prime".

			Array T(n, k) begins:
  n\k|  1  2   3   4   5   6   7   8   9  10  11  12  13  14
  ---+------------------------------------------------------
    1|  1  1   1   1   1   1   1   1   1   1   1   1   1   1
    2|  1  2   2   4   2   4   2   8   4   4   2   8   2   4  ->  A061142
    3|  1  2   3   4   3   6   3   8   9   6   3  12   3   6  ->  A079065
    4|  1  4   4   8   4   8   4  16   8   8   4  16   4   8
    5|  1  2   3   4   5   6   5   8   9  10   5  12   5  10
    6|  1  4   6   8   6  12   6  16  18  12   6  24   6  12
    7|  1  2   3   4   5   6   7   8   9  10   7  12   7  14
    8|  1  8   8  16   8  16   8  32  16  16   8  32   8  16
    9|  1  4   9   8   9  18   9  16  27  18   9  36   9  18
   10|  1  4   6   8  10  12  10  16  18  20  10  24  10  20
   11|  1  2   3   4   5   6   7   8   9  10  11  12  11  14
   12|  1  8  12  16  12  24  12  32  36  24  12  48  12  24
   13|  1  2   3   4   5   6   7   8   9  10  11  12  13  14
   14|  1  4   6   8  10  12  14  16  18  20  14  24  14  28

Rémy Sigrist, Table of n, a(n) for n = 1..10011
Rémy Sigrist, PARI program for A342767
Index entries for sequences related to dismal (or lunar) arithmetic

Cf. A006530, A061142, A079065, A087062, A342765, A342768.

PARI
```
See Links section.
```

T(n, k) = T(k, n).
T(n, n) = A342768(n).
T(n, 1) = 1.
T(n, 2) = A061142(n).
T(n, 3) = A079065(n).
T(n, p) = n for any prime number p >= A006530(n).

cp's OEIS Frontend

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

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A346703 Product of primes at odd positions in the weakly increasing list (with multiplicity) of prime factors of n.

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A346701 Heinz number of the odd bisection (odd-indexed parts) of the integer partition with Heinz number n.

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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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A129597 Central diagonal of array A129595.

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A342767 Array T(n, k), n, k > 0, read by antidiagonals; a variant of lunar multiplication (A087062) based on prime factorizations of numbers (see Comments section for precise definition).

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PARI

Formula