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.

Showing 1-10 of 339 results. Next

A293214 a(n) = Product_{d|n, dA019565(d).

Original entry on oeis.org

1, 2, 2, 6, 2, 36, 2, 30, 12, 60, 2, 2700, 2, 180, 120, 210, 2, 7560, 2, 6300, 360, 252, 2, 661500, 20, 420, 168, 94500, 2, 23814000, 2, 2310, 504, 132, 600, 43659000, 2, 396, 840, 2425500, 2, 187110000, 2, 207900, 352800, 1980, 2, 560290500, 60, 194040, 264, 485100, 2, 115259760, 840, 254677500, 792, 4620, 2, 264737261250000, 2, 13860
Offset: 1

Views

Author

Antti Karttunen, Oct 03 2017

Keywords

Links

Crossrefs

Cf. A001065, A002110, A019565, A048675, A091954, A292257, A293215 (restricted growth sequence transform).
Cf. also A293216, A293221, A293222, A293225, A293231, A293442, A300830, A300831, A300832, A300833, A300834.

Programs

  • PARI
    A019565(n) = {my(j,v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ From A019565
A293214(n) = { my(m=1); fordiv(n,d,if(d < n,m *= A019565(d))); m; };

Formula

a(n) = Product_{d|n, dA019565(d).
a(n) = A300830(n) * A300831(n) * A300832(n). - Antti Karttunen, Mar 16 2018
Other identities.
For n >= 0, a(2^n) = A002110(n).
For n >= 1:
A048675(a(n)) = A001065(n).
A001222(a(n)) = A292257(n).
A007814(a(n)) = A091954(n).
A087207(a(n)) = A218403(n).
A248663(a(n)) = A227320(n).

A293442 Multiplicative with a(p^e) = A019565(e).

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 6, 3, 4, 2, 6, 2, 4, 4, 5, 2, 6, 2, 6, 4, 4, 2, 12, 3, 4, 6, 6, 2, 8, 2, 10, 4, 4, 4, 9, 2, 4, 4, 12, 2, 8, 2, 6, 6, 4, 2, 10, 3, 6, 4, 6, 2, 12, 4, 12, 4, 4, 2, 12, 2, 4, 6, 15, 4, 8, 2, 6, 4, 8, 2, 18, 2, 4, 6, 6, 4, 8, 2, 10, 5, 4, 2, 12, 4, 4, 4, 12, 2, 12, 4, 6, 4, 4, 4, 20, 2, 6, 6, 9, 2, 8, 2, 12, 8
Offset: 1

Views

Author

Antti Karttunen, Oct 31 2017

Keywords

Comments

From Peter Munn, Apr 06 2021: (Start)
a(n) is determined by the prime signature of n.
Compare with the multiplicative, self-inverse A225546, which also maps 2^e to the squarefree number A019565(e). However, this sequence maps p^e to the same squarefree number for every prime p, whereas A225546 maps the e-th power of progressively larger primes to progressively greater powers of A019565(e).
Both sequences map powers of squarefree numbers to powers of squarefree numbers.
(End)

Links

Crossrefs

Sequences used in a definition of this sequence: A000188, A003961, A019565, A028234, A059895, A067029, A162642.
Sequences with related definitions: A225546, A293443, A293444.
Cf. also A293214.
Sequences used to express relationship between terms of this sequence: A006519, A007913, A008833, A064989, A334747.
Sequences related via this sequence: (A001222, A048675, A064547), (A007814, A162642), (A087207, A267116), (A248663, A268387).

Programs

  • Mathematica
    f[n_] := If[n == 1, 1, Apply[Times, Prime@ Flatten@ Position[Reverse@ IntegerDigits[Last@ #, 2], 1]] * f[n/Apply[Power, #]] &@ FactorInteger[n][[1]]]; Array[f, 105] (* Michael De Vlieger, Oct 31 2017 *)

Formula

a(1) = 1; for n > 1, a(n) = A019565(A067029(n)) * a(A028234(n)).
Other identities. For all n >= 1:
a(a(n)) = A293444(n).
A048675(a(n)) = A001222(n).
A001222(a(n)) = A064547(n) = A048675(A293444(n)).
A007814(a(n)) = A162642(n).
A087207(a(n)) = A267116(n).
A248663(a(n)) = A268387(n).
From Peter Munn, Mar 14 2021: (Start)
Alternative definition: a(1) = 1; a(2) = 2; a(n^2) = A003961(a(n)); a(A003961(n)) = a(n); if A059895(n, k) = 1, a(n*k) = a(n) * a(k).
For n >= 3, a(n) < n.
a(2n) = A334747(a(A006519(n))) * a(n/A006519(n)), where A006519(n) is the largest power of 2 dividing n.
a(2n+1) = a(A064989(2n+1)).
a(n) = a(A007913(n)) * a(A008833(n)) = 2^A162642(n) * A003961(a(A000188(n))).
(End)

A329332 Table of powers of squarefree numbers, powers of A019565(n) in increasing order in row n. Square array A(n,k) n >= 0, k >= 0 read by descending antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 4, 3, 1, 1, 8, 9, 6, 1, 1, 16, 27, 36, 5, 1, 1, 32, 81, 216, 25, 10, 1, 1, 64, 243, 1296, 125, 100, 15, 1, 1, 128, 729, 7776, 625, 1000, 225, 30, 1, 1, 256, 2187, 46656, 3125, 10000, 3375, 900, 7, 1, 1, 512, 6561, 279936, 15625, 100000, 50625, 27000, 49, 14
Offset: 0

Views

Author

Peter Munn, Nov 10 2019

Keywords

Comments

The A019565 row order gives the table neat relationships with A003961, A003987, A059897, A225546, A319075 and A329050. See the formula section.
Transposition of this table, that is reflection about its main diagonal, has subtle symmetries. For example, consider the unique factorization of a number into powers of distinct primes. This can be restated as factorization into numbers from rows 2^n (n >= 0) with no more than one from each row. Reflecting about the main diagonal, this factorization becomes factorization (of a related number) into numbers from columns 2^k (k >= 0) with no more than one from each column. This is also unique and is factorization into powers of squarefree numbers with distinct exponents that are powers of two. See the example section.

Examples

			Square array A(n,k) begins:
n\k |  0   1     2      3        4          5           6             7
----+------------------------------------------------------------------
   0|  1   1     1      1        1          1           1             1
   1|  1   2     4      8       16         32          64           128
   2|  1   3     9     27       81        243         729          2187
   3|  1   6    36    216     1296       7776       46656        279936
   4|  1   5    25    125      625       3125       15625         78125
   5|  1  10   100   1000    10000     100000     1000000      10000000
   6|  1  15   225   3375    50625     759375    11390625     170859375
   7|  1  30   900  27000   810000   24300000   729000000   21870000000
   8|  1   7    49    343     2401      16807      117649        823543
   9|  1  14   196   2744    38416     537824     7529536     105413504
  10|  1  21   441   9261   194481    4084101    85766121    1801088541
  11|  1  42  1764  74088  3111696  130691232  5489031744  230539333248
  12|  1  35  1225  42875  1500625   52521875  1838265625   64339296875
Reflection of factorization about the main diagonal: (Start)
The canonical (prime power) factorization of 864 is 2^5 * 3^3 = 32 * 27. Reflecting the factors about the main diagonal of the table gives us 10 * 36 = 10^1 * 6^2 = 360. This is the unique factorization of 360 into powers of squarefree numbers with distinct exponents that are powers of two.
Reflection about the main diagonal is given by the self-inverse function A225546(.). Clearly, all positive integers are in the domain of A225546, whether or not they appear in the table. It is valid to start from 360, observe that A225546(360) = 864, then use 864 to derive 360's factorization into appropriate powers of squarefree numbers as above.
(End)

Crossrefs

The range of values is A072774.
Rows (abbreviated list): A000079(1), A000244(2), A000400(3), A000351(4), A011557(5), A001024(6), A009974(7), A000420(8), A001023(9), A009965(10), A001020(16), A001022(32), A001026(64).
A019565 is column 1, A334110 is column 2, and columns that are sorted in increasing order (some without the 1) are: A005117(1), A062503(2), A062838(3), A113849(4), A113850(5), A113851(6), A113852(7).
Other subtables: A182944, A319075, A329050.
Re-ordered subtable of A297845, A306697, A329329.
A000290, A003961, A003987, A059897 and A225546 are used to express relationships between terms of this sequence.
Cf. A285322.

Formula

A(n,k) = A019565(n)^k.
A(k,n) = A225546(A(n,k)).
A(n,2k) = A000290(A(n,k)) = A(n,k)^2.
A(2n,k) = A003961(A(n,k)).
A(n,2k+1) = A(n,2k) * A(n,1).
A(2n+1,k) = A(2n,k) * A(1,k).
A(A003987(n,m), k) = A059897(A(n,k), A(m,k)).
A(n, A003987(m,k)) = A059897(A(n,m), A(n,k)).
A(2^n,k) = A319075(k,n+1).
A(2^n, 2^k) = A329050(n,k).
A(n,k) = A297845(A(n,1), A(1,k)) = A306697(A(n,1), A(1,k)), = A329329(A(n,1), A(1,k)).
Sum_{n>=0} 1/A(n,k) = zeta(k)/zeta(2*k), for k >= 2. - Amiram Eldar, Dec 03 2022

A285332 a(0) = 1, a(1) = 2, a(2n) = A019565(a(n)), a(2n+1) = A065642(a(n)).

Original entry on oeis.org

1, 2, 3, 4, 6, 9, 5, 8, 15, 12, 14, 27, 10, 25, 7, 16, 210, 45, 35, 18, 105, 28, 462, 81, 21, 20, 154, 125, 30, 49, 11, 32, 10659, 420, 910, 75, 78, 175, 33, 24, 3094, 315, 385, 56, 780045, 924, 374, 243, 110, 63, 55, 40, 4389, 308, 170170, 625, 1155, 60, 286, 343, 42, 121, 13, 64, 54230826, 31977, 28405, 630, 1330665, 1820, 714
Offset: 0

Views

Author

Antti Karttunen, Apr 17 2017

Keywords

Comments

Note the indexing: the domain starts from 0, while the range excludes zero.
This sequence can be represented as a binary tree. Each left hand child is produced as A019565(n), and each right hand child as A065642(n), when the parent node contains n >= 2:
1
|
...................2...................
3 4
6......../ \........9 5......../ \........8
/ \ / \ / \ / \
/ \ / \ / \ / \
/ \ / \ / \ / \
15 12 14 27 10 25 7 16
210 45 35 18 105 28 462 81 21 20 154 125 30 49 11 32
etc.
Where will 38 appear in this tree? It is a reasonable assumption that by iterating A087207 starting from 38, as A087207(38) = 129, A087207(129) = 8194, A087207(8194) = 1501199875790187, ..., we will eventually hit a prime A000040(k), most likely with a largish index k. This prime occurs at the penultimate edge at right, as a(A000918(k)) = a((2^k)-2), and thus 38 occurs somewhere below it as a(m) = 38, m > k. All the numbers that share prime factors with 38, namely 76, 152, 304, 608, 722, ..., occur similarly late in this tree, as they form the rightward branch starting from 38. Alternatively, by iterating A285330 (each iteration moves one step towards the root) starting from 38, we might instead first hit some power of 3, or say, one of the terms of A033845 (the rightward branch starting from 6), in which case the first prime encountered would be a(2)=3 and 38 would appear on the left-hand side instead of the right-hand side subtree.
As long as it remains conjecture that A019565 has no cycles, it is certainly also an open question whether this is a permutation of the natural numbers: If A019565 has any cycles, then neither any of the terms in those cycles nor any A065642-trajectories starting from those terms (that is, numbers sharing same prime factors) may occur in this tree.
Sequence exhibits some outrageous swings, for example, a(703) = 224, but a(704) is 1427 decimal digits (4739 binary digits) long, thus it no longer fits into a b-file.
However, the scatter plot of A286543 gives some flavor of the behavior of this sequence even after that point. - Antti Karttunen, Dec 25 2017

Links

Crossrefs

Inverse: A285331.
Cf. A007947, A019565, A033845, A048675, A065642, A087207, A109162, A285319, A285320, A285328, A285330, A285333, A286542, A286543.
Compare also to permutation A285112 and array A285321.

Programs

  • Mathematica
    Block[{a = {1, 2}}, Do[AppendTo[a, If[EvenQ[i], Times @@ Prime@ Flatten@ Position[#, 1] &@ Reverse@ IntegerDigits[a[[i/2 + 1]], 2], If[# == 1, 1, Function[{n, c}, SelectFirst[Range[n + 1, n^2], Times @@ FactorInteger[#][[All, 1]] == c &]] @@ {#, Times @@ FactorInteger[#][[All, 1]]}] &[a[[(i - 1)/2 + 1]] ] ]], {i, 2, 70}]; a] (* Michael De Vlieger, Mar 12 2021 *)
  • PARI
    A019565(n) = {my(j,v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ This function from M. F. Hasler
A007947(n) = factorback(factorint(n)[, 1]); \\ From Andrew Lelechenko, May 09 2014
A065642(n) = { my(r=A007947(n)); if(1==n,n,n = n+r; while(A007947(n) <> r, n = n+r); n); };
A285332(n) = { if(n<=1,n+1,if(!(n%2),A019565(A285332(n/2)),A065642(A285332((n-1)/2)))); };
for(n=0, 4095, write("b285332.txt", n, " ", A285332(n)));
  • Python
    from operator import mul
from sympy import prime, primefactors
def a007947(n): return 1 if n<2 else reduce(mul, primefactors(n))
def a019565(n): return reduce(mul, (prime(i+1) for i, v in enumerate(bin(n)[:1:-1]) if v == '1')) if n > 0 else 1 # This function from Chai Wah Wu
def a065642(n):
    if n==1: return 1
    r=a007947(n)
    n = n + r
    while a007947(n)!=r:
        n+=r
    return n
def a(n):
    if n<2: return n + 1
    if n%2==0: return a019565(a(n//2))
    else: return a065642(a((n - 1)//2))
print([a(n) for n in range(51)]) # Indranil Ghosh, Apr 18 2017
  • Scheme
    ;; With memoization-macro definec.
(definec (A285332 n) (cond ((<= n 1) (+ n 1)) ((even? n) (A019565 (A285332 (/ n 2)))) (else (A065642 (A285332 (/ (- n 1) 2))))))

Formula

a(0) = 1, a(1) = 2, a(2n) = A019565(a(n)), a(2n+1) = A065642(a(n)).
For n >= 0, a(2^n) = A109162(2+n). [The left edge of the tree.]
For n >= 0, a(A000225(n)) = A000079(n). [Powers of 2 occur at the right edge of the tree.]
For n >= 2, a(A000918(n)) = A000040(n). [And the next vertices inwards contain primes.]
For n >= 2, a(A036563(1+n)) = A001248(n). [Whose right children are their squares.]
For n >= 0, a(A055010(n)) = A000244(n). [Powers of 3 are at the rightmost edge of the left subtree.]
For n >= 2, a(A129868(n-1)) = A062457(n).
A048675(a(n)) = A285333(n).
A046523(a(n)) = A286542(n).

A293215 Restricted growth sequence transform of A293214, where A293214(n) = Product_{d|n, dA019565(d).

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 5, 6, 7, 2, 8, 2, 9, 10, 11, 2, 12, 2, 13, 14, 15, 2, 16, 17, 18, 19, 20, 2, 21, 2, 22, 23, 24, 25, 26, 2, 27, 28, 29, 2, 30, 2, 31, 32, 33, 2, 34, 7, 35, 36, 37, 2, 38, 28, 39, 40, 41, 2, 42, 2, 43, 44, 45, 46, 47, 2, 48, 49, 50, 2, 51, 2, 52, 53, 54, 55, 56, 2, 57, 58, 59, 2, 60, 61, 62, 63, 64, 2, 65, 66, 67, 68, 69, 70, 71, 2, 72, 73
Offset: 1

Views

Author

Antti Karttunen, Oct 03 2017

Keywords

Comments

For all i, j: a(i) = a(j) => A001065(i) = A001065(j).

Links

Crossrefs

Cf. A001065, A019565, A048675, A293214, A293217 (a variant), A293223, A293224, A293226.
Differs from related A293232 for the first time at n=55, where a(55) = 28, while A293232(55) = 39.

Programs

  • PARI
    rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
write_to_bfile(start_offset,vec,bfilename) = { for(n=1, length(vec), write(bfilename, (n+start_offset)-1, " ", vec[n])); }
A019565(n) = {my(j,v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ This function from M. F. Hasler
A293214(n) = { my(m=1); fordiv(n,d,if(d < n,m *= A019565(d))); m; };
write_to_bfile(1,rgs_transform(vector(16384,n,A293214(n))),"b293215.txt");

A283475 a(n) = A019565(A005187(n)).

Original entry on oeis.org

1, 2, 6, 5, 30, 7, 21, 42, 210, 11, 33, 66, 165, 330, 154, 231, 2310, 13, 39, 78, 195, 390, 182, 273, 1365, 2730, 286, 429, 1430, 2145, 1001, 2002, 30030, 17, 51, 102, 255, 510, 238, 357, 1785, 3570, 374, 561, 1870, 2805, 1309, 2618, 19635, 39270, 442, 663, 2210, 3315, 1547, 3094, 15470, 23205, 2431, 4862, 12155
Offset: 0

Views

Author

Antti Karttunen, Mar 15 2017

Keywords

Links

Crossrefs

Cf. A000040, A000051, A001221, A001222, A004198, A005117, A005187, A019565, A046523, A097248, A108951, A280700, A280705, A283477, A283478.
Cf. A283476 (same sequence sorted into ascending order).

Programs

  • Mathematica
    Map[Times @@ Prime@ Flatten@ Position[#, 1] &@ Reverse@ IntegerDigits[#, 2] &, Table[2 n - DigitCount[2 n, 2, 1], {n, 0, 60}]] (* Michael De Vlieger, Mar 16 2017 *)
  • Scheme
    (define (A283475 n) (A019565 (A005187 n)))

Formula

a(n) = A019565(A005187(n)).
a(n) = A097248(A283477(n)) = A097248(A108951(A019565(n))) = A283478(A019565(n)).
Other identities:
If A004198(x,y) = 0, then a(x+y) = A097248(a(x)*a(y)).
For all n >= 1, a(A000051(n)) = A000040(n+2).
For all n >= 0, A001221(a(n)) = A001222(a(n)) = A280700(n).
For all n >= 0, A046523(a(n)) = A280705(n).

A293221 a(n) = Product_{d|n, dA019565(A289813(d)); a product obtained from the 1-digits present in ternary expansions of proper divisors of n.

Original entry on oeis.org

1, 2, 2, 2, 2, 6, 2, 12, 6, 6, 2, 36, 2, 4, 18, 12, 2, 30, 2, 360, 12, 10, 2, 540, 6, 60, 30, 360, 2, 900, 2, 120, 30, 10, 12, 2700, 2, 4, 180, 360, 2, 540, 2, 360, 450, 6, 2, 5400, 4, 120, 30, 360, 2, 210, 30, 5040, 12, 14, 2, 1701000, 2, 84, 180, 2520, 180, 1260, 2, 840, 18, 12600, 2, 94500, 2, 140, 180, 840, 20, 18900, 2, 756000, 210, 210, 2, 23814000, 30
Offset: 1

Views

Author

Antti Karttunen, Oct 03 2017

Keywords

Links

Crossrefs

Cf. A019565, A289813, A293214, A293222, A293223 (restricted growth sequence transform), A293226.
Cf. also A290091.

Programs

  • PARI
    A019565(n) = {my(j,v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ This function from M. F. Hasler
A289813(n) = { my (d=digits(n, 3)); fromdigits(vector(#d, i, if (d[i]==1, 1, 0)), 2); } \\ From Remy Sigrist
A293221(n) = { my(m=1); fordiv(n,d,if(d < n,m *= A019565(A289813(d)))); m; };

Formula

a(n) = Product_{d|n, dA019565(A289813(d)).
For all n >= 0, a(3^n) = A002110(n).

A293222 a(n) = Product_{d|n, dA019565(A289814(d)); a product obtained from the 2-digits present in ternary expansions of proper divisors of n.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 2, 1, 4, 1, 6, 1, 6, 2, 12, 1, 6, 1, 4, 3, 4, 1, 36, 2, 2, 1, 12, 1, 36, 1, 36, 2, 12, 6, 30, 1, 10, 1, 240, 1, 180, 1, 20, 6, 20, 1, 1620, 3, 60, 6, 60, 1, 30, 4, 72, 5, 4, 1, 360, 1, 2, 15, 72, 2, 180, 1, 36, 10, 144, 1, 2700, 1, 2, 90, 20, 6, 180, 1, 720, 1, 4, 1, 540, 12, 6, 2, 720, 1, 900, 3, 100, 1, 20, 10, 16200, 1, 60, 6
Offset: 1

Views

Author

Antti Karttunen, Oct 03 2017

Keywords

Links

Crossrefs

Cf. A019565, A289814, A293221, A293224 (restricted growth sequence transform), A293226.

Programs

  • PARI
    A019565(n) = {my(j,v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ This function from M. F. Hasler
A289814(n) = { my (d=digits(n, 3)); fromdigits(vector(#d, i, if (d[i]==2, 1, 0)), 2); } \\ From Remy Sigrist
A293222(n) = { my(m=1); fordiv(n,d,if(d < n,m *= A019565(A289814(d)))); m; };

Formula

a(n) = Product_{d|n, dA019565(A289814(d)).

A339809 a(n) = A019565(n) - 1.

Original entry on oeis.org

0, 1, 2, 5, 4, 9, 14, 29, 6, 13, 20, 41, 34, 69, 104, 209, 10, 21, 32, 65, 54, 109, 164, 329, 76, 153, 230, 461, 384, 769, 1154, 2309, 12, 25, 38, 77, 64, 129, 194, 389, 90, 181, 272, 545, 454, 909, 1364, 2729, 142, 285, 428, 857, 714, 1429, 2144, 4289, 1000, 2001, 3002, 6005, 5004, 10009, 15014, 30029, 16, 33, 50
Offset: 0

Views

Author

Antti Karttunen, Dec 18 2020

Keywords

Links

Crossrefs

Cf. A019565, A339810, A339811, A339812.
Cf. also A339820, A339821.

Programs

  • PARI
    A019565(n) = { my(m=1, p=1); while(n>0, p = nextprime(1+p); if(n%2, m *= p); n >>= 1); (m); };
A339809(n) = (A019565(n)-1);

A339821 a(n) = phi(A019565(2n)), where phi is Euler totient function.

Original entry on oeis.org

1, 2, 4, 8, 6, 12, 24, 48, 10, 20, 40, 80, 60, 120, 240, 480, 12, 24, 48, 96, 72, 144, 288, 576, 120, 240, 480, 960, 720, 1440, 2880, 5760, 16, 32, 64, 128, 96, 192, 384, 768, 160, 320, 640, 1280, 960, 1920, 3840, 7680, 192, 384, 768, 1536, 1152, 2304, 4608, 9216, 1920, 3840, 7680, 15360, 11520, 23040, 46080, 92160
Offset: 0

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Author

Antti Karttunen, Dec 18 2020

Keywords

Crossrefs

Bisection of A339820.
Cf. A000010, A003961, A003972, A006093, A019565, A339822 (2-adic valuation).
Cf. also A324651.

Programs

  • PARI
    A019565(n) = { my(m=1, p=1); while(n>0, p = nextprime(1+p); if(n%2, m *= p); n >>= 1); (m); };
A339821(n) = eulerphi(A019565(n+n));
  • PARI
    A339821(n) = { my(m=1, p=2); while(n>0, p = nextprime(1+p); if(n%2, m *= (p-1)); n >>= 1); (m); };

Formula

If 4n = 2^e1 + 2^e2 + ... + 2^ek [e1 ... ek distinct], then a(n) = A006093(e1) * A006093(e2) * ... * A006093(ek).
a(n) = A339820(2n) = A000010(A019565(2n)) = A000010(A019565(2n+1)).
a(n) = A003972(A019565(n)) = A000010(A003961(A019565(n))).
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