cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A349150 Heinz numbers of integer partitions with at most one odd part.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 9, 11, 13, 14, 15, 17, 18, 19, 21, 23, 26, 27, 29, 31, 33, 35, 37, 38, 39, 41, 42, 43, 45, 47, 49, 51, 53, 54, 57, 58, 59, 61, 63, 65, 67, 69, 71, 73, 74, 77, 78, 79, 81, 83, 86, 87, 89, 91, 93, 95, 97, 98, 99, 101, 103, 105, 106, 107, 109
Offset: 1

Views

Author

Gus Wiseman, Nov 10 2021

Keywords

Comments

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are numbers with at most one odd prime index.
Also Heinz numbers of partitions with conjugate alternating sum <= 1.

Examples

			The terms and their prime indices begin:
      1: {}          23: {9}         49: {4,4}
      2: {1}         26: {1,6}       51: {2,7}
      3: {2}         27: {2,2,2}     53: {16}
      5: {3}         29: {10}        54: {1,2,2,2}
      6: {1,2}       31: {11}        57: {2,8}
      7: {4}         33: {2,5}       58: {1,10}
      9: {2,2}       35: {3,4}       59: {17}
     11: {5}         37: {12}        61: {18}
     13: {6}         38: {1,8}       63: {2,2,4}
     14: {1,4}       39: {2,6}       65: {3,6}
     15: {2,3}       41: {13}        67: {19}
     17: {7}         42: {1,2,4}     69: {2,9}
     18: {1,2,2}     43: {14}        71: {20}
     19: {8}         45: {2,2,3}     73: {21}
     21: {2,4}       47: {15}        74: {1,12}

Crossrefs

The case of no odd parts is A066207, counted by A000041 up to 0's.
Requiring all odd parts gives A066208, counted by A000009.
These partitions are counted by A100824, even-length case A349149.
These are the positions of 0's and 1's in A257991.
The conjugate partitions are ranked by A349151.
The case of one odd part is A349158, counted by A000070 up to 0's.
A056239 adds up prime indices, row sums of A112798.
A122111 is a representation of partition conjugation.
A300063 ranks partitions of odd numbers, counted by A058695 up to 0's.
A316524 gives the alternating sum of prime indices (reverse: A344616).
A325698 ranks partitions with as many even as odd parts, counted by A045931.
A340932 ranks partitions whose least part is odd, counted by A026804.
A345958 ranks partitions with alternating sum 1.
A349157 ranks partitions with as many even parts as odd conjugate parts.
Cf. A000290, A000700, A001222, A027187, A027193, A028260, A035363, A047993, A215366, A257992, A277579, A326841.

Programs

  • Mathematica
    primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Count[Reverse[primeMS[#]],_?OddQ]<=1&]

Formula

Union of A066207 (no odd parts) and A349158 (one odd part).

A351979 Numbers whose prime factorization has all odd prime indices and all even prime exponents.

Original entry on oeis.org

1, 4, 16, 25, 64, 100, 121, 256, 289, 400, 484, 529, 625, 961, 1024, 1156, 1600, 1681, 1936, 2116, 2209, 2500, 3025, 3481, 3844, 4096, 4489, 4624, 5329, 6400, 6724, 6889, 7225, 7744, 8464, 8836, 9409, 10000, 10609, 11881, 12100, 13225, 13924, 14641, 15376
Offset: 1

Views

Author

Gus Wiseman, Mar 11 2022

Keywords

Comments

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798, sum A056239, length A001222.
A number's prime signature is the sequence of positive exponents in its prime factorization, which is row n of A124010, length A001221, sum A001222.
Also Heinz numbers of integer partitions with all odd parts and all even multiplicities, counted by A035457 (see Emeric Deutsch's comment there).

Examples

			The terms together with their prime indices begin:
     1: 1
     4: prime(1)^2
    16: prime(1)^4
    25: prime(3)^2
    64: prime(1)^6
   100: prime(1)^2 prime(3)^2
   121: prime(5)^2
   256: prime(1)^8
   289: prime(7)^2
   400: prime(1)^4 prime(3)^2
   484: prime(1)^2 prime(5)^2
   529: prime(9)^2
   625: prime(3)^4
   961: prime(11)^2
  1024: prime(1)^10
  1156: prime(1)^2 prime(7)^2
  1600: prime(1)^6 prime(3)^2
  1681: prime(13)^2
  1936: prime(1)^4 prime(5)^2

Links

Crossrefs

The second condition alone (exponents all even) is A000290, counted by A035363.
The distinct prime factors of terms all come from A031368.
These partitions are counted by A035457 or A000009 aerated.
The first condition alone (indices all odd) is A066208, counted by A000009.
The squarefree square roots are A258116, even A258117.
A056166 = exponents all prime, counted by A055923.
A066207 = indices all even, counted by complement of A086543.
A076610 = indices all prime, counted by A000607.
A109297 = same indices as exponents, counted by A114640.
A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
A124010 gives prime signature, sorted A118914, length A001221, sum A001222.
A162641 counts even exponents, odd A162642.
A257991 counts odd indices, even A257992.
A268335 = exponents all odd, counted by A055922.
A325131 = disjoint indices from exponents, counted by A114639.
A346068 = indices and exponents all prime, counted by A351982.
A352140 = even indices with odd exponents, counted by A055922 (aerated).
A352141 = even indices with even exponents, counted by A035444.
A352142 = odd indices and odd multiplicities, counted by A117958.
Cf. A000720, A028260, A045931, A055396, A061395, A106529, A181819, A195017, A276078, A324588, A325698, A325700.

Programs

  • Mathematica
    Select[Range[1000],#==1||And@@OddQ/@PrimePi/@First/@FactorInteger[#]&&And@@EvenQ/@Last/@FactorInteger[#]&]
  • Python
    from sympy import factorint, primepi
def ok(n):
    return all(primepi(p)%2==1 and e%2==0 for p, e in factorint(n).items())
print([k for k in range(15500) if ok(k)]) # Michael S. Branicky, Mar 12 2022

Formula

Squares of elements of A066208.
Intersection of A066208 and A000290.
A257991(a(n)) = A001222(a(n)).
A162641(a(n)) = A001221(a(n)).
A162642(a(n)) = A257992(a(n)) = 0.
Sum_{n>=1} 1/a(n) = 1/Product_{k>=1} (1 - 1/prime(2*k-1)^2) = 1.4135142... . - Amiram Eldar, Sep 19 2022

A366849 Odd numbers whose halved even prime indices are relatively prime.

Original entry on oeis.org

3, 9, 15, 21, 27, 33, 39, 45, 51, 57, 63, 69, 75, 81, 87, 91, 93, 99, 105, 111, 117, 123, 129, 135, 141, 147, 153, 159, 165, 171, 177, 183, 189, 195, 201, 203, 207, 213, 219, 225, 231, 237, 243, 247, 249, 255, 261, 267, 273, 279, 285, 291, 297, 301, 303, 309
Offset: 1

Views

Author

Gus Wiseman, Nov 01 2023

Keywords

Comments

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

Examples

			The even prime indices of 91 are {4,6}, halved {2,3}, which are relatively prime, so 91 is in the sequence.
The prime indices of 665 are {3,4,8}, even {4,8}, halved {2,4}, which are not relatively prime, so 665 is not in the sequence.
The terms together with their prime indices begin:
   3: {2}
   9: {2,2}
  15: {2,3}
  21: {2,4}
  27: {2,2,2}
  33: {2,5}
  39: {2,6}
  45: {2,2,3}
  51: {2,7}
  57: {2,8}
  63: {2,2,4}
  69: {2,9}
  75: {2,3,3}
  81: {2,2,2,2}
  87: {2,10}
  91: {4,6}
  93: {2,11}
  99: {2,2,5}

Crossrefs

For odd instead of halved even prime indices we have A366848.
A version for odd indices A366846, counted by A366850.
This is the odd restriction of A366847, counted by A366845.
A000041 counts integer partitions, strict A000009 (also into odds).
A035363 counts partitions into all even parts, ranks A066207.
A112798 lists prime indices, length A001222, sum A056239.
A162641 counts even prime exponents, odd A162642.
A257992 counts even prime indices, odd A257991.
A289509 lists numbers with relatively prime prime indices, ones of A289508, counted by A000837.
A366528 adds up odd prime indices, partition triangle A113685.
A366531 = 2*A366533 adds up even prime indices, triangle A113686/A174713.
Cf. A000720, A055396, A061395, A066208, A302697, A325698, A366842, A366843.

Programs

  • Mathematica
    Select[Range[100], OddQ[#]&&GCD@@Select[PrimePi/@First/@FactorInteger[#], EvenQ]==2&]

A366852 Number of integer partitions of n into odd parts with a common divisor > 1.

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 1, 1, 0, 2, 1, 1, 2, 1, 1, 4, 0, 1, 4, 1, 2, 6, 1, 1, 6, 3, 1, 8, 2, 1, 13, 1, 0, 13, 1, 7, 15, 1, 1, 19, 6, 1, 25, 1, 2, 33, 1, 1, 32, 5, 10, 39, 2, 1, 46, 14, 6, 55, 1, 1, 77, 1, 1, 82, 0, 20, 92, 1, 2, 105, 31, 1, 122, 1, 1, 166, 2, 16, 168
Offset: 0

Views

Author

Gus Wiseman, Nov 01 2023

Keywords

Examples

			The a(n) partitions for n = 3, 9, 15, 21, 25, 27:
(3)  (9)      (15)         (21)             (25)         (27)
     (3,3,3)  (5,5,5)      (7,7,7)          (15,5,5)     (9,9,9)
              (9,3,3)      (9,9,3)          (5,5,5,5,5)  (15,9,3)
              (3,3,3,3,3)  (15,3,3)                      (21,3,3)
                           (9,3,3,3,3)                   (9,9,3,3,3)
                           (3,3,3,3,3,3,3)               (15,3,3,3,3)
                                                         (9,3,3,3,3,3,3)
                                                         (3,3,3,3,3,3,3,3,3)

Links

Crossrefs

Allowing even parts gives A018783, complement A000837.
For parts > 1 instead of gcd > 1 we have A087897.
For gcd = 1 instead of gcd > 1 we have A366843.
The strict case is A366750, with evens A303280.
The strict complement is A366844, with evens A078374.
A000041 counts integer partitions, strict A000009 (also into odd parts).
A000700 counts strict partitions into odd parts.
A113685 counts partitions by sum of odd parts, rank statistic A366528.
A168532 counts partitions by gcd.
A366842 counts partitions whose odd parts have a common divisor > 1.
Cf. A007359, A047967, A051424, A055922, A066208, A302698, A337485, A365067, A366845, A366848.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],And@@OddQ/@#&&GCD@@#>1&]],{n,15}]
  • Python
    from math import gcd
from sympy.utilities.iterables import partitions
def A366852(n): return sum(1 for p in partitions(n) if all(d&1 for d in p) and gcd(*p)>1) # Chai Wah Wu, Nov 02 2023

Extensions

More terms from Chai Wah Wu, Nov 02 2023
a(0)=0 prepended by Alois P. Heinz, Jan 11 2024

A319525 Completely multiplicative with a(prime(k)) = prime(2*k - 1) (where prime(k) denotes the k-th prime).

Original entry on oeis.org

1, 2, 5, 4, 11, 10, 17, 8, 25, 22, 23, 20, 31, 34, 55, 16, 41, 50, 47, 44, 85, 46, 59, 40, 121, 62, 125, 68, 67, 110, 73, 32, 115, 82, 187, 100, 83, 94, 155, 88, 97, 170, 103, 92, 275, 118, 109, 80, 289, 242, 205, 124, 127, 250, 253, 136, 235, 134, 137, 220
Offset: 1

Views

Author

Rémy Sigrist, Sep 22 2018

Keywords

Comments

This sequence is a permutation of A066208.
See A297002 for a similar sequence.

Links

Crossrefs

Cf. A000079, A066208, A297002.
Cf. also A166651.

Programs

  • PARI
    a(n) = my (f=factor(n)); prod(i=1, #f~, prime(2*primepi(f[i, 1]) - 1)^f[i, 2])

Formula

a(n) = A064989(A297002(n)).
a(n) >= n with equality iff n is a power of 2 (A000079).

A342081 Numbers without an inferior odd divisor > 1.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 13, 14, 16, 17, 19, 20, 22, 23, 26, 28, 29, 31, 32, 34, 37, 38, 41, 43, 44, 46, 47, 52, 53, 58, 59, 61, 62, 64, 67, 68, 71, 73, 74, 76, 79, 82, 83, 86, 88, 89, 92, 94, 97, 101, 103, 104, 106, 107, 109, 113, 116, 118, 122, 124
Offset: 1

Views

Author

Gus Wiseman, Mar 06 2021

Keywords

Comments

We define a divisor d|n to be inferior if d <= n/d. Inferior divisors are counted by A038548 and listed by A161906.
Numbers n such that n is either a power of 2 or has a single odd prime factor > sqrt(n). Equivalently, numbers n such that all odd prime factors are > sqrt(n). - Chai Wah Wu, Mar 08 2021

Examples

			The divisors > 1 of 72 are {2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72}, of which {3, 9} are odd and {2, 3, 4, 6, 8} are inferior, with intersection {3}, so 72 is not in the sequence.

Links

Crossrefs

The strictly inferior version is the same with A001248 added.
Positions of 1's in A069288.
The superior version is A116882, with complement A116883.
The complement is A342082.
A006530 selects the greatest prime factor.
A020639 selects the smallest prime factor.
A038548 counts superior (or inferior) divisors, with strict case A056924.
- Odd -
A000009 counts partitions into odd parts, ranked by A066208.
A001227 counts odd divisors.
A026424 lists numbers with odd Omega.
A027193 counts odd-length partitions.
A058695 counts partitions of odd numbers.
A067659 counts strict partitions of odd length, ranked by A030059.
A340101 counts factorizations into odd factors; A340102 also has odd length.
A340854/A340855 cannot/can be factored with odd minimum factor.
A341594 counts strictly superior odd divisors
A341675 counts superior odd divisors.
- Inferior: A033676, A066839, A161906.
- Superior: A033677, A051283, A059172, A063538/A063539.
- Strictly Inferior A333805, A341674.
- Strictly Superior: A064052/A048098, A341645/A341646.
Cf. A000005, A000203, A001055, A001221, A001222, A001414, A207375, A244991, A300272, A340832, A340931.

Programs

  • Mathematica
    Select[Range[100],Function[n,Select[Divisors[n]//Rest,OddQ[#]&&#<=n/#&]=={}]]
  • PARI
    is(n) = #select(x -> x > 2 && x^2 <= n, factor(n)[, 1]) == 0; \\ Amiram Eldar, Nov 01 2024
  • Python
    from sympy import primefactors
A342081_list = [n for n in range(1,10**3) if len([p for p in primefactors(n) if p > 2 and p*p <= n]) == 0] # Chai Wah Wu, Mar 08 2021

A352143 Numbers whose prime indices and conjugate prime indices are all odd.

Original entry on oeis.org

1, 2, 5, 8, 11, 17, 20, 23, 31, 32, 41, 44, 47, 59, 67, 68, 73, 80, 83, 92, 97, 103, 109, 124, 125, 127, 128, 137, 149, 157, 164, 167, 176, 179, 188, 191, 197, 211, 227, 233, 236, 241, 257, 268, 269, 272, 275, 277, 283, 292, 307, 313, 320, 331, 332, 347, 353
Offset: 1

Views

Author

Gus Wiseman, Mar 18 2022

Keywords

Comments

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798, sum A056239, length A001222.
A number's prime signature is the sequence of positive exponents in its prime factorization, which is row n of A124010, length A001221, sum A001222.
These are the Heinz numbers of integer partitions whose parts and conjugate parts are all odd. They are counted by A053253.

Examples

			The terms together with their prime indices begin:
   1: {}
   2: {1}
   5: {3}
   8: {1,1,1}
  11: {5}
  17: {7}
  20: {1,1,3}
  23: {9}
  31: {11}
  32: {1,1,1,1,1}
  41: {13}
  44: {1,1,5}
  47: {15}
  59: {17}
  67: {19}
  68: {1,1,7}
  73: {21}
  80: {1,1,1,1,3}

Crossrefs

The restriction to primes is A031368.
These partitions appear to be counted by A053253.
The even version is A066207^2.
For even instead of odd conjugate parts we get A066208^2.
The first condition alone (all odd indices) is A066208, counted by A000009.
The second condition alone is A346635, counted by A000009.
A055922 counts partitions with odd multiplicities, ranked by A268335.
A066207 = indices all even, counted by A035363 (complement A086543).
A109297 = same indices as exponents, counted by A114640.
A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
A124010 gives prime signature, sorted A118914, length A001221, sum A001222.
A162642 counts odd prime exponents, even A162641.
A238745 gives the Heinz number of the conjugate prime signature.
A257991 counts odd indices, even A257992.
A258116 ranks strict partitions with all odd parts, even A258117.
A351979 = odd indices and even multiplicities, counted by A035457.
A352140 = even indices and odd multiplicities, counted by A055922 aerated.
A352141 = even indices and even multiplicities, counted by A035444.
A352142 = odd indices and odd multiplicities, counted by A117958.
Cf. A000290, A000701, A000720, A028260, A045931, A046682, A055396, A061395, A195017, A241638, A325698, A325700.

Programs

  • Mathematica
    primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
Select[Range[100],And@@OddQ/@primeMS[#]&&And@@OddQ/@conj[primeMS[#]]&]

Formula

Intersection of A066208 and A346635.

A356604 Number of integer compositions of n into odd parts covering an initial interval of odd positive integers.

Original entry on oeis.org

1, 1, 1, 1, 3, 4, 5, 9, 13, 24, 40, 61, 101, 160, 257, 415, 679, 1103, 1774, 2884, 4656, 7517, 12165, 19653, 31753, 51390, 83134, 134412, 217505, 351814, 569081, 920769, 1489587, 2409992, 3899347, 6309059, 10208628, 16518910, 26729830, 43254212, 69994082
Offset: 0

Views

Author

Gus Wiseman, Aug 30 2022

Keywords

Examples

			The a(1) = 1 through a(8) = 13 compositions:
  (1)  (11)  (111)  (13)    (113)    (1113)    (133)      (1133)
                    (31)    (131)    (1131)    (313)      (1313)
                    (1111)  (311)    (1311)    (331)      (1331)
                            (11111)  (3111)    (11113)    (3113)
                                     (111111)  (11131)    (3131)
                                               (11311)    (3311)
                                               (13111)    (111113)
                                               (31111)    (111131)
                                               (1111111)  (111311)
                                                          (113111)
                                                          (131111)
                                                          (311111)
                                                          (11111111)
The a(9) = 24 compositions:
  (135)  (11133)  (1111113)  (111111111)
  (153)  (11313)  (1111131)
  (315)  (11331)  (1111311)
  (351)  (13113)  (1113111)
  (513)  (13131)  (1131111)
  (531)  (13311)  (1311111)
         (31113)  (3111111)
         (31131)
         (31311)
         (33111)

Crossrefs

The case of partitions is A053251, ranked by A356232 and A356603.
These compositions are ranked by the intersection of A060142 and A333217.
This is the odd initial case of A107428.
This is the odd restriction of A107429.
This is the normal/covering case of A324969 (essentially A000045).
The non-initial version is A356605.
A000041 counts partitions, compositions A011782.
A055932 lists numbers with prime indices covering an initial interval.
A066208 lists numbers with all odd prime indices, counted by A000009.
Cf. A001221, A001222, A001227, A005408, A061395, A066205, A073493, A137921, A356224.

Programs

  • Mathematica
    normQ[m_]:=Or[m=={},Union[m]==Range[Max[m]]];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],normQ[(#+1)/2]&]],{n,0,15}]

Extensions

More terms from Alois P. Heinz, Sep 01 2022

A356846 Number of integer compositions of n into parts not covering an interval of positive integers.

Original entry on oeis.org

0, 0, 0, 0, 2, 5, 11, 25, 57, 115, 236, 482, 978, 1986, 4003, 8033, 16150, 32402, 64943, 130207, 260805, 522123, 1045168, 2091722, 4185431, 8374100, 16753538, 33515122, 67042865, 134106640, 268246886, 536549760, 1073194999, 2146553011, 4293391411, 8587283895
Offset: 0

Views

Author

Gus Wiseman, Sep 03 2022

Keywords

Examples

			The a(0) = 0 through a(6) = 8 compositions:
  .  .  .  .  (13)  (14)   (15)
              (31)  (41)   (24)
                    (113)  (42)
                    (131)  (51)
                    (311)  (114)
                           (141)
                           (411)
                           (1113)
                           (1131)
                           (1311)
                           (3111)

Crossrefs

The complement is counted by A107428, initial A107429.
The case of partitions is A239955, ranked by A073492, initial A053251, complement A034296.
These compositions are ranked by A356842, complement A356841.
A000041 counts partitions, compositions A011782.
A066208 lists numbers with all odd prime indices, counted by A000009.
A073491 lists numbers with gapless prime indices, initial A055932.
Cf. A001227, A060142, A080259, A188575, A239327, A356604, A356605.

Programs

  • Mathematica
    gappyQ[m_]:=And[m!={},Union[m]!=Range[Min[m],Max[m]]];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],gappyQ]],{n,0,15}]

Formula

a(n) = A011782(n) - A107428(n).

A366321 Numbers m whose prime indices have even sum k such that k/2 is not a prime index of m.

Original entry on oeis.org

1, 3, 7, 10, 13, 16, 19, 21, 22, 27, 28, 29, 34, 36, 37, 39, 43, 46, 48, 52, 53, 55, 57, 61, 62, 64, 66, 71, 75, 76, 79, 81, 82, 85, 87, 88, 89, 90, 91, 94, 100, 101, 102, 107, 108, 111, 113, 115, 116, 117, 118, 120, 129, 130, 131, 133, 134, 136, 138, 139, 144
Offset: 0

Views

Author

Gus Wiseman, Oct 13 2023

Keywords

Comments

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

Examples

			The prime indices of 84 are y = {1,1,2,4}, with even sum 8; but 8/2 = 4 is in y, so 84 is not in the sequence.
The terms together with their prime indices begin:
    1: {}
    3: {2}
    7: {4}
   10: {1,3}
   13: {6}
   16: {1,1,1,1}
   19: {8}
   21: {2,4}
   22: {1,5}
   27: {2,2,2}
   28: {1,1,4}
   29: {10}
   34: {1,7}
   36: {1,1,2,2}

Crossrefs

Partitions of this type are counted by A182616, strict A365828.
A066207 lists numbers with all even prime indices, odd A066208.
A086543 lists numbers with at least one odd prime index, counted by A366322.
A300063 ranks partitions of odd numbers.
A366319 ranks partitions of n not containing n/2.
A366321 ranks partitions of 2k that do not contain k.
Cf. A000041, A006827, A047967, A320924, A339662, A365825, A365920, A366318, A366528, A366530.

Programs

  • Mathematica
    prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],EvenQ[Total[prix[#]]]&&FreeQ[prix[#],Total[prix[#]]/2]&]
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