A070826 -id:A070826 - cp's OEIS frontend

A096547 Primes p such that primorial(p)/2 - 2 is prime.

5, 7, 11, 13, 17, 19, 23, 31, 41, 53, 71, 103, 167, 431, 563, 673, 727, 829, 1801, 2699, 4481, 6121, 7283, 9413, 10321, 12491, 17807, 30307, 31891, 71917, 172517

Offset: 1

Hugo Pfoertner, Jun 27 2004

Some of the results were computed using the PrimeFormGW (PFGW) primality-testing program. - Hugo Pfoertner, Nov 14 2019

a(32) > 180000. - Tyler Busby, Mar 29 2024

			Prime 7 is a term because primorial(7)/2 - 2 = A034386(7)/2 - 2 = 2*3*5*7/2 - 2 = 103 is prime.

Cf. A070826, A096177 primes p such that primorial(p)/2+2 is prime, A096178 primes of the form primorial(p)/2+2, A014545 primorial primes, A087398.

Cf. A034386.

b:= proc(n) b(n):= `if`(n=0, 1, `if`(isprime(n), n, 1)*b(n-1)) end:
q:= p-> isprime(p) and isprime(b(p)/2-2):
select(q, [$1..500])[];

k = 1; Do[k *= Prime[n]; If[PrimeQ[k - 2], Print[Prime[n]]], {n, 2, 3276}] (* Ryan Propper, Oct 25 2005 *)
Prime[#]&/@Flatten[Position[FoldList[Times,Prime[Range[1000]]]/2-2,?PrimeQ]] (* _Harvey P. Dale, Jun 09 2023 *)

5 more terms from Ryan Propper, Oct 25 2005

a(29)-a(31) from Tyler Busby, Mar 16 2024

A098012 Triangle read by rows in which the k-th term in row n (n >= 1, k = 1..n) is Product_{i=0..k-1} prime(n-i).

Original entry on oeis.org

2, 3, 6, 5, 15, 30, 7, 35, 105, 210, 11, 77, 385, 1155, 2310, 13, 143, 1001, 5005, 15015, 30030, 17, 221, 2431, 17017, 85085, 255255, 510510, 19, 323, 4199, 46189, 323323, 1616615, 4849845, 9699690, 23, 437, 7429, 96577, 1062347, 7436429, 37182145, 111546435, 223092870

Offset: 1

Alford Arnold, Sep 09 2004

Also, square array A(m,n) in which row m lists all products of m consecutive primes (read by falling antidiagonals). See also A248164. - M. F. Hasler, May 03 2017

			2
3 3*2
5 5*3 5*3*2
7 7*5 7*5*3 7*5*3*2
Or, as an infinite square array:
     2     3     5     7  ... : row 1 = A000040,
     6    15    35    77  ... : row 2 = A006094,
    30   105   385  1001  ... : row 3 = A046301,
   210  1155  5005 17017  ... : row 4 = A046302,
   ..., with col.1 = A002110, col.2 = A070826, col.3 = A059865\{1}. - _M. F. Hasler_, May 03 2017

Reinhard Zumkeller, Rows n = 1..125 of triangle, flattened

Cf. A000040, A002110, A006094, A046301, A046302, A046303.

Cf. A060381 (central terms), A104887, A248147.

P:=Filtered([1..200],IsPrime);;
T:=Flat(List([1..9],n->List([1..n],k->Product([0..k-1],i->P[n-i])))); # Muniru A Asiru, Mar 16 2019

a098012 n k = a098012_tabl !! (n-1) !! (k-1)
a098012_row n = a098012_tabl !! (n-1)
a098012_tabl = map (scanl1 (*)) a104887_tabl
-- Reinhard Zumkeller, Oct 02 2014

T:=(n,k)->mul(ithprime(n-i),i=0..k-1): seq(seq(T(n,k),k=1..n),n=1..9); # Muniru A Asiru, Mar 16 2019

Flatten[ Table[ Product[ Prime[i], {i, n, j, -1}], {n, 9}, {j, n, 1, -1}]] (* Robert G. Wilson v, Sep 21 2004 *)

T098012(n,k)=prod(i=0,k-1,prime(n-i)) \\ "Triangle" variant
A098012(m,n)=prod(i=0,m-1,prime(n+i)) \\ "Square array" variant. - M. F. Hasler, May 03 2017

n-th row = partial products of row n in A104887. - Reinhard Zumkeller, Oct 02 2014

More terms from Robert G. Wilson v, Sep 21 2004

A285993 Largest odd abundant number (A005231) equal to the product of n consecutive primes.

Original entry on oeis.org

15015, 255255, 4849845, 111546435, 33426748355, 1236789689135, 50708377254535, 2180460221945005, 102481630431415235, 5431526412865007455, 320460058359035439845, 19548063559901161830545, 1309720258513377842646515, 1357656019974967471687377449, 107254825578022430263302818471

Offset: 5

M. F. Hasler, Apr 30 2017

nonn

The smallest term is a(5) = 3*5*7*11*13, there is no odd abundant number (A005231) equal to the product of less than 5 consecutive primes.
The smallest odd abundant number (A005231) equal to the product of n consecutive primes is equal (when it exists, i.e., for n >= 5) to the least odd number with n (distinct) prime divisors, equal to the product of the first n odd primes = A070826(n+1) = A002110(n+1)/2.
See A188342 = (945, 3465, 15015, 692835, 22309287, ...) for the least odd primitive abundant number (A006038) with n distinct prime factors, and A275449 for the least odd primitive abundant number with n prime factors counted with multiplicity.
The terms are in general not primitive abundant numbers (A091191), in particular this cannot be the case when a(n) is a multiple of a(n-1), as is the case for most of the terms, for which a(n) = a(n-1)*A117366(a(n-1)). In the other event, spf(a(n)) = nextprime(spf(a(n-1))), and a(n) is in A007741(2,3,4...). These are exactly the primitive terms in this sequence.

			For n < 5, there is no odd abundant number equal to the product of n distinct primes.
For 5 <= n <= 8, the largest odd abundant number equal to the product of n consecutive primes is 3*...*prime(n+1).
For 9 <= n <= 17, the largest odd abundant number equal to the product of n consecutive primes is 5*...*prime(n+2).
For 18 <= n <= 30, the largest odd abundant number equal to the product of n consecutive primes is 7*...*prime(n+3).
For 31 <= n <= 45, the largest odd abundant number equal to the product of n consecutive primes is 11*...*prime(n+4).
For 46 <= n <= 66, the largest odd abundant number equal to the product of n consecutive primes is 13*...*prime(n+5).

Amiram Eldar, Table of n, a(n) for n = 5..341

Cf. A005231, A006038, A007707, A007708, A007741.

A subsequence of A112643 (odd squarefree abundant numbers); see also A108227 (~ A107705) which give indices of primitive terms = those with smallest prime factor larger than that of earlier terms.

a(r,f=vector(r,i,prime(i+1)),o)={ while(sigma(factorback(f),-1)>2, o=f; f=concat(f[^1],nextprime(f[r]+1)));factorback(o)} \\ Intentionally throws an error when n < 5.

a(n) >= a(n-1)*p where p = A117366(a(n-1)) = A151800(A006530(a(n-1))) = nextprime(gpf(a(n-1))), an odd abundant number equal to the product of n consecutive primes. We have strict inequality for n = 9, 18, 31, 46, 67, ..., in which case a(n) = a(n-1)*p*p'/q, where p' = nextprime(p), q = least prime factor of a(n-1). This is the case if a(n) is in A007741.

A307540 Irregular triangle T(n,k) such that squarefree m with gpf(m) = prime(n) in each row are arranged according to increasing values of phi(m)/m.

Original entry on oeis.org

1, 2, 6, 3, 30, 10, 15, 5, 210, 42, 70, 14, 105, 21, 35, 7, 2310, 330, 462, 66, 770, 110, 154, 1155, 22, 165, 231, 33, 385, 55, 77, 11, 30030, 2730, 4290, 6006, 390, 546, 858, 10010, 78, 910, 1430, 2002, 130, 15015, 182, 286, 1365, 2145, 26, 3003, 195, 273, 429

Offset: 0

Michael De Vlieger, Apr 13 2019

Let gpf(m) = A006530(m) and let phi(m) = A000010(m) for m in A005117.
Row n contains m in A005117 such that A000720(A006530(m)) = n, sorted such that phi(m)/m increases as k increases.
Let m be the squarefree kernel A007947(m') of m'. We only consider squarefree m since phi(m)/m = phi(m')/m'. Let prime p | n and prime q be a nondivisor of n.
Since m is squarefree, we might encode the multiplicities of its prime divisors in a positional notation M that is finite at n significant digits. For example, m = 42 can be encoded reverse(A067255(42)) = 1,0,1,1 = 7^1 * 5^0 * 3^1 * 2^1. It is necessary to reverse row m of A067255 (hereinafter simply A067255(m)) so as to preserve zeros in M = A067255(m) pertaining to small nondivisor primes q < p. The code M is a series of 0's and 1's since m is squarefree. Then it is clear that row n contains all m such that A067255(m) has n terms, and there are 2^(n - 1) possible terms for n >= 1.
We may use an approach that generates the binary expansion of the range 2^(n - 1) < M < 2^n - 1, or we may append 1 to the reversed (n - 1)-tuples of {1, 0} to achieve codes M -> m for each row n, which is tantamount to ordering according to A059894.
Originally it was thought that the codes M were in order of the latter algorithm, and we could avoid sorting. Observation shows that the m still require sorting by the function phi(m)/m indeed to be in increasing order in row n. Still, the latter approach is slightly more efficient than the former in generating the sequence.

			Triangle begins:
1;
2;
6, 3;
30, 10, 15, 5;
210, 42, 70, 14, 105, 21, 35, 7;
...
First terms of this sequence appear bottom to top in the chart below. The
values of n appear in the header, values m = T(n,k) followed
parenthetically by phi(m)/m appear in column n. The x axis plots
according to primepi(gpf(m)), while the y axis plots k according to
phi(m)/m:
    0       1          2             3             4
    .       .          .             .             .
--- 1 ------------------------------------------------
  (1/1)     .          .             .             .
    .       .          .             .             .
    .       .          .             .             .
    .       .          .             .             7
    .       .          .             5           (6/7)
    .       .          .           (4/5)           .
    .       .          .             .             .
    .       .          .             .            35
    .       .          3             .          (24/35)
    .       .        (2/3)           .             .
    .       .          .             .             .
    .       .          .             .             .
    .       .          .             .            21
    .       .          .             .           (4/7)
    .       .          .            15             .
    .       .          .          (8/15)           .
    .       2          .             .             .
    .     (1/2)        .             .             .
    .       .          .             .             .
    .       .          .             .            105
    .       .          .             .          (16/35)
    .       .          .             .            14
    .       .          .            10           (3/7)
    .       .          .           (2/5)           .
    .       .          .             .             .
    .       .          .             .            70
    .       .          6             .          (12/35)
    .       .        (1/3)           .             .
    .       .          .             .            42
    .       .          .            30           (2/7)
    .       .          .          (4/15)           .
    .       .          .             .            210
    .       .          .             .           (8/35)
...

Michael De Vlieger, Table of n, a(n) for n = 0..16384
Michael De Vlieger, Small plot of m in A307540 at x = pi(gpf(m)), y = phi(m)/m.
Michael De Vlieger, Enlarged plot of m in A307540 at x = pi(gpf(m)), y = phi(m)/m.

Cf. A000010, A000040, A000079, A000720, A002110, A005117, A006094, A006530, A007947, A059894, A070826, A077017, A306237.

Prepend[Array[SortBy[#, Last] &@ Map[{#1, #2, EulerPhi[#1]/#1} & @@ {Times @@  MapIndexed[Prime[First@ #2]^#1 &, Reverse@ #], FromDigits@ #} &, Map[Prepend[Reverse@ #, 1] &, Tuples[{1, 0}, # - 1]]] &, 6], {{1, 0, 1}}][[All, All, 1]] // Flatten

For n > 0, row lengths = A000079(n - 1).
T(n, 1) = A002110(n) = p_n#.
T(n, 2) = A306237(n) = p_n#/prime(n - 1).
T(n, 2^(n - 1) - 1) = A006094(n).
T(n, 2^(n - 1)) = A000040(n) = prime(n) for n >= 1.
Last even term in row n = A077017(n).
First odd term in row n = A070826(n).

A334748 Let p be the smallest odd prime not dividing the squarefree part of n. Multiply n by p and divide by the product of all smaller odd primes.

Original entry on oeis.org

3, 6, 5, 12, 15, 10, 21, 24, 27, 30, 33, 20, 39, 42, 7, 48, 51, 54, 57, 60, 35, 66, 69, 40, 75, 78, 45, 84, 87, 14, 93, 96, 55, 102, 105, 108, 111, 114, 65, 120, 123, 70, 129, 132, 135, 138, 141, 80, 147, 150, 85, 156, 159, 90, 165, 168, 95, 174, 177, 28, 183, 186, 189

Offset: 1

Peter Munn, May 09 2020

A permutation of A028983.
A007417 (which has asymptotic density 3/4) lists index n such that a(n) = 3n. The sequence maps the terms of A007417 1:1 onto A145204\{0}, defining a bijection between them.
Similarly, bijections are defined from the odd numbers (A005408) to the nonsquare odd numbers (A088828), from the positive even numbers (A299174) to A088829, from A003159 to the nonsquares in A003159, and from A325424 to the nonsquares in A036668. The latter two bijections are between sets where membership depends on whether a number's squarefree part divides by 2 and/or 3.

			84 = 21*4 has squarefree part 21 (and square part 4). The smallest odd prime absent from 21 = 3*7 is 5 and the product of all smaller odd primes is 3. So a(84) = 84*5/3 = 140.

Permutation of A028983.
Row 3, and therefore column 3, of A331590. Cf. A334747 (row 2).
A007913, A034386, A225546, A284723 are used in formulas defining the sequence.
The formula section details how the sequence maps the terms of A003961, A019565, A070826; and how f(a(n)) relates to f(n) for f = A008833, A048675, A267116; making use of A003986.
Subsequences: A016051, A145204\{0}, A329575.
Bijections are defined that relate to A003159, A005408, A007417, A036668, A088828, A088829, A299174, A325424.

a(n) = {my(c=core(n), m=n); forprime(p=3, , if(c % p, m*=p; break, m/=p)); m;} \\ Michel Marcus, May 22 2020

a(n) = n * p / (A034386(p-1)/2), where p = A284723(A007913(n)).
a(n) = A334747(A334747(n)).
a(n) = A331590(3, n) = A225546(4 * A225546(n)).
a(2*n) = 2 * a(n).
a(A019565(n)) = A019565(n+2).
a(k * m^2) = a(k) * m^2.
a(A003961(n)) = A003961(A334747(n)).
a(A070826(n)) = prime(n+1).
A048675(a(n)) = A048675(n) + 2.
A008833(a(n)) = A008833(n).
A267116(a(n)) = A267116(n) OR 1, where OR denotes the bitwise operation A003986.
a(A007417(n)) = A145204(n+1) = 3 * A007417(n).

A096178 Primes of the form primorial(p)/2+2.

Original entry on oeis.org

3, 5, 17, 107, 15017, 3234846617, 100280245067, 3710369067407, 307444891294245707, 961380175077106319537, 139867498408927468089138080936033904837498617

Offset: 1

Hugo Pfoertner, Jun 27 2004

nonn

Primes of the form A070826(n)+2.

			a(4) = 107 because 107 is a prime of the form primorial(7)/2 + 2 = A070826(4) + 2 = 2*3*5*7/2 + 2.

Amiram Eldar, Table of n, a(n) for n = 1..18

Cf. A070826, A096177 (primorial(p)/2+2 is prime), A096547 (primorial(p)/2-2 is prime), A067024 (smallest p+2 that has n distinct prime factors), A014545 (primorial primes), A087398.

for(n=1,30,p=prod(k=1,n,prime(k))/2+2;if(ispseudoprime(p),print1(p,", "))) \\ Hugo Pfoertner, Dec 26 2019

a(n) = A070826(A096177(n)) + 2. - Amiram Eldar, Dec 26 2019

a(1) inserted by Amiram Eldar, Dec 26 2019

A097273 Least integer with each "mod 2 prime signature".

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 9, 12, 15, 16, 18, 24, 27, 30, 32, 36, 45, 48, 54, 60, 64, 72, 81, 90, 96, 105, 108, 120, 128, 135, 144, 162, 180, 192, 210, 216, 225, 240, 243, 256, 270, 288, 315, 324, 360, 384, 405, 420, 432, 450, 480, 486, 512, 540, 576, 630, 648, 675, 720

Offset: 1

Ray Chandler, Aug 22 2004

nonn

For n = 2^e_0 * p_1^e_1 * ... * p_n^e_n where p_i is odd prime and e_1 >= e_2 >= ... >= e_n, define "mod 2 prime signature" to be ordered prime exponents (e_0,e_1,...,e_n).
Least integer with a given mod 2 prime signature is obtained by replacing p_1 with 3, p_2 with 5,..., p_n with n-th odd prime.
A097272 sorted and duplicates removed.
Numbers k such that A097272(k) = k.
Verified up to a(68) = 972, 2*a(n) is also the order of a dihedral group D such that the lattice of normal subgroups of D is not isomorphic to the lattice of normal subgroups of any dihedral group of order less than 2*a(n). - Miles Englezou, May 18 2025

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A097272, A097274, A097275.

Cf. A025487, A070826, A147516.

lpsQ[n_] := n==1 || (Max@ Differences[(f = FactorInteger[n])[[;;,2]]] < 1 && f[[-1,1]] == Prime[Length[f] + 1]); Select[Range[1000], lpsQ[# / 2^IntegerExponent[#, 2]] &] (* Amiram Eldar, Jul 23 2024 *)

Sum_{n>=1} 1/a(n) = 2 * Product_{n>=2} 1/(1 - 1/A070826(n)) = 3.2482341898... . - Amiram Eldar, Jul 23 2024

Offset corrected by Amiram Eldar, Jul 23 2024

A098348 Triangular array read by rows: a(n, k) = number of ordered factorizations of a "hook-type" number with n total prime factors and k distinct prime factors. "Hook-type" means that only one prime can have multiplicity > 1.

Original entry on oeis.org

1, 2, 3, 4, 8, 13, 8, 20, 44, 75, 16, 48, 132, 308, 541, 32, 112, 368, 1076, 2612, 4683, 64, 256, 976, 3408, 10404, 25988, 47293, 128, 576, 2496, 10096, 36848, 116180, 296564, 545835, 256, 1280, 6208, 28480, 120400, 454608, 1469892, 3816548

Offset: 1

Alford Arnold, Sep 04 2004

The first three columns are A000079, A001792 and A098385.
The first two diagonals are A000670 and A005649.
A070175 gives the smallest representative of each hook-type prime signature, so this sequence is a rearrangement of A074206(A070175).

			a(4, 2) = 20 because 24=2*2*2*3 has 20 ordered factorizations and so does any other number with the same prime signature.

Cf. A050324, A070175, A070826, A074206, A095705. A098349 gives the row sums. A098384.

a(n, k) = 1 + (Sum_{i=1..k-1} binomial(k-1, i)*a(i, i)) + (Sum_{j=1..k} Sum_{i=j..j+n-k-1} binomial(k-1, j-1)*a(i, j)) + (Sum_{j=1..k-1} binomial(k-1,j-1)*a(j+n-k, j)). - David Wasserman, Feb 21 2008
a(n, k) = A074206(2^(n+1-k)*A070826(k)). - David Wasserman, Feb 21 2008
The following conjectural formula for the triangle entries agrees with the values listed above: T(n,k) = Sum_{j = 0..n-k} 2^(n-k-j)*binomial(n-k,j)*a(k,j), where a(k,j) = 2^j*Sum_{i = j+1..k+1} binomial(i,j+1)*(i-1)!*Stirling2(k+1,i). See A098384 for related conjectures. - Peter Bala, Apr 20 2012

Edited and extended by David Wasserman, Feb 21 2008

A104462 Convert the binary strings in A101305 to decimal.

Original entry on oeis.org

0, 2, 20, 328, 10512, 672800, 86118464, 22046326912, 11287719379200, 11558624644301312, 23672063271529088000, 96960771160183144450048, 794302637344220319334797312, 13013854410247705711981319168000, 426437981314996820770203866497040384

Offset: 0

Jorge Coveiro, Apr 23 2005

The a(n)-th composition in standard order is (2,3,..,n+1), where the k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. Moreover, the binary indices of a(n) are row n of A193973. Including 1 gives A164894, reverse A246534. - Gus Wiseman, Jun 28 2022

			From _Gus Wiseman_, Jun 28 2022: (Start)
The terms together with their standard compositions begin:
      0: ()
      2: (2)
     20: (2,3)
    328: (2,3,4)
  10512: (2,3,4,5)
(End)

Michael S. Branicky, Table of n, a(n) for n = 0..80

Cf. A101305.
A version for prime indices is A070826.
Cf. A000120, A002110, A029931, A066099, A070939, A164894, A193973, A233564, A246534, A272919, A333218, A333255.

convert(10,decimal,binary); convert(10100,decimal,binary); convert(101001000,decimal,binary); convert(10100100010000,decimal,binary); convert(10100100010000100000,decimal,binary);

stcinv[q_]:=Total[2^Accumulate[Reverse[q]]]/2;
Table[stcinv[Range[2,n]],{n,8}] (* Gus Wiseman, Jun 28 2022 *)

def a(n): return 0 if n==0 else int("".join("1"+"0"*(i+1) for i in range(n)), 2)
print([a(n) for n in range(15)]) # Michael S. Branicky, Jun 28 2022

a(14) and beyond from Michael S. Branicky, Jun 28 2022

A122842 Square roots of the odd indexed terms of A038547.

Original entry on oeis.org

1, 3, 9, 27, 15, 243, 729, 45, 6561, 19683, 135, 177147, 225, 105, 4782969, 14348907, 1215, 675, 387420489, 3645, 3486784401, 10460353203, 315, 94143178827, 3375, 32805, 2541865828329, 6075, 98415, 68630377364883

Offset: 1

Alexander Adamchuk, Sep 13 2006, Sep 25 2006

nonn

A038547(n) is the least number with exactly n odd divisors. For odd n these are perfect squares.

			a(1) = 1 because A038547(1) = 1.
a(2) = 3 because A038547(3) = 9.
a(5) = 15 because A038547(9) = 225.

Amiram Eldar, Table of n, a(n) for n = 1..2100 (terms 1..1000 from R. J. Mathar)

Cf. A038547, A121858, A001227, A005179, A119265.

Cf. A007051, A070826, A002110.

(* Function a038547[ ] is defined in A038547 *)
a122842[n_]:=Sqrt[a038547[2n-1]]
Map[a122842,Range[30]] (* Hartmut F. W. Hoft, Feb 07 2023 *)

a(n) = sqrt(A038547(2*n-1)).
a(n) = sqrt(A119265(2*n-1, 2*n-1)).
a(n) = 3^(n-1) for n = (p+1)/2, where p is an odd prime.
a(3*n+2) = 5*3^n for n = (p-1)/2, where p is an odd prime.

More terms from R. J. Mathar, Sep 20 2006

Edited by Hartmut F. W. Hoft, Feb 07 2023

cp's OEIS Frontend

A096547 Primes p such that primorial(p)/2 - 2 is prime.

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A098012 Triangle read by rows in which the k-th term in row n (n >= 1, k = 1..n) is Product_{i=0..k-1} prime(n-i).

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A285993 Largest odd abundant number (A005231) equal to the product of n consecutive primes.

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A307540 Irregular triangle T(n,k) such that squarefree m with gpf(m) = prime(n) in each row are arranged according to increasing values of phi(m)/m.

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A334748 Let p be the smallest odd prime not dividing the squarefree part of n. Multiply n by p and divide by the product of all smaller odd primes.

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PARI

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A096178 Primes of the form primorial(p)/2+2.

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PARI

Formula

Extensions

A097273 Least integer with each "mod 2 prime signature".

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