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Lexicographically earliest such sequence a that a(i) = a(j) => f(i) = f(j) for all i, j, where f(n) = -1 if n is an odd prime, and f(n) = A285330(n) for all other numbers.
+20
7
1, 2, 3, 4, 3, 5, 3, 6, 5, 7, 3, 8, 3, 9, 8, 10, 3, 11, 3, 12, 12, 13, 3, 14, 7, 15, 9, 16, 3, 17, 3, 18, 14, 19, 11, 20, 3, 21, 22, 23, 3, 24, 3, 25, 26, 27, 3, 28, 17, 29, 30, 31, 3, 32, 23, 33, 34, 35, 3, 36, 3, 37, 38, 39, 28, 40, 3, 22, 41, 42, 3, 43, 3, 44, 45, 46, 20, 47, 3, 48, 49, 50, 3, 51, 52, 53, 54, 55, 3, 56, 29, 57, 58, 59, 60, 61, 3, 62, 15
PROG
(PARI)
up_to = 65537;
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; };
A048675(n) = my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; \\ From A048675
A285328(n) = { my(r); if((n > 1 && !bitand(n, (n-1))), (n/2), r= A007947(n); if(r==n, 1, n = n-r; while( A007947(n) <> r, n = n-r); n)); };
A322807aux(n) = if((n%2)&&isprime(n), -1, A285330(n)); v322807 = rgs_transform(vector(up_to, n, A322807aux(n)));
Lexicographically earliest such sequence a that a(i) = a(j) => A285330(i) = A285330(j) for all i, j.
+20
4
1, 2, 3, 3, 4, 5, 6, 4, 5, 7, 8, 9, 10, 11, 9, 6, 12, 13, 14, 15, 15, 16, 17, 18, 7, 19, 11, 20, 21, 22, 23, 8, 18, 24, 13, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 22, 38, 39, 40, 41, 42, 29, 43, 44, 45, 46, 47, 48, 49, 50, 10, 37, 51, 52, 28, 53, 54, 55, 56, 57, 58, 59, 60, 25, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 38
COMMENTS
Restricted growth sequence transform of A285330.
PROG
(PARI)
up_to = 65537;
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; };
A048675(n) = my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; \\ From A048675
A285328(n) = { my(r); if((n > 1 && !bitand(n, (n-1))), (n/2), r= A007947(n); if(r==n, 1, n = n-r; while( A007947(n) <> r, n = n-r); n)); };
v322806 = rgs_transform(vector(up_to, n, A285330(n)));
0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 1, 2, 2, 2, 2, 1, 2, 2, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 2, 3, 2, 1, 2, 2, 2, 3, 2, 1, 3, 1, 2, 2, 1, 2, 1, 1, 2, 3, 1, 1, 2, 1, 2, 2, 2, 2, 2, 1, 2, 1, 2, 1, 3, 2, 2, 2, 2, 1, 3, 2, 2, 3, 3, 3, 2, 1, 2, 2, 2, 1, 1, 1, 2, 2
PROG
(PARI)
A048675(n) = my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; \\ From A048675
A285328(n) = { my(r); if((n > 1 && !bitand(n, (n-1))), (n/2), r= A007947(n); if(r==n, 1, n = n-r; while( A007947(n) <> r, n = n-r); n)); };
1, 2, 2, 2, 2, 3, 2, 2, 3, 4, 2, 3, 2, 4, 3, 2, 2, 3, 2, 4, 4, 4, 2, 4, 4, 4, 4, 5, 2, 5, 2, 2, 4, 4, 3, 3, 2, 4, 4, 4, 2, 6, 2, 6, 7, 4, 2, 4, 5, 4, 4, 6, 2, 3, 4, 5, 4, 4, 2, 7, 2, 4, 8, 2, 4, 6, 2, 4, 4, 6, 2, 9, 2, 4, 10, 6, 3, 6, 2, 6, 9, 4, 2, 8, 4, 4, 4, 6, 2, 7, 4, 11, 4, 4, 4, 4, 2, 5, 4, 4, 2, 6, 2, 6, 5
PROG
(PARI)
up_to = 65537;
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; };
A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t };
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); }; \\ From A046523
A048675(n) = my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; \\ From A048675
A285328(n) = { my(r); if((n > 1 && !bitand(n, (n-1))), (n/2), r= A007947(n); if(r==n, 1, n = n-r; while( A007947(n) <> r, n = n-r); n)); };
0, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 3, 1, 3, 1, 1, 2, 2, 2, 2, 1, 2, 2, 2, 1, 3, 1, 3, 4, 2, 1, 2, 3, 2, 2, 3, 1, 2, 2, 3, 2, 2, 1, 4, 1, 2, 3, 1, 2, 3, 1, 2, 2, 3, 1, 4, 1, 2, 4, 3, 2, 3, 1, 3, 4, 2, 1, 3, 2, 2, 2, 3, 1, 4, 2, 4, 2, 2, 2, 2, 1, 3, 2, 2, 1, 3, 1, 3, 3
MATHEMATICA
Table[DigitCount[#, 2, 1] &@ Which[n == 1, 0, MoebiusMu@ n != 0, Total@ Map[#2*2^(PrimePi@ #1 - 1) & @@ # &, FactorInteger[n]], True, With[{r = DivisorSum[n, EulerPhi[#] Abs@ MoebiusMu[#] &]}, SelectFirst[Range[n - 2, 2, -1], DivisorSum[#, EulerPhi[#] Abs@ MoebiusMu[#] &] == r &]]], {n, 105}] (* Michael De Vlieger, Dec 31 2018 *)
PROG
(PARI)
A048675(n) = my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; \\ From A048675
A285328(n) = { my(r); if((n > 1 && !bitand(n, (n-1))), (n/2), r= A007947(n); if(r==n, 1, n = n-r; while( A007947(n) <> r, n = n-r); n)); };
\\ Or just as:
A322862(n) = if(issquarefree(n), omega(n), hammingweight( A285328(n)));
A binary representation of the primes that divide a number, shown in decimal.
+10
50
0, 1, 2, 1, 4, 3, 8, 1, 2, 5, 16, 3, 32, 9, 6, 1, 64, 3, 128, 5, 10, 17, 256, 3, 4, 33, 2, 9, 512, 7, 1024, 1, 18, 65, 12, 3, 2048, 129, 34, 5, 4096, 11, 8192, 17, 6, 257, 16384, 3, 8, 5, 66, 33, 32768, 3, 20, 9, 130, 513, 65536, 7, 131072, 1025, 10, 1, 36, 19, 262144, 65, 258
COMMENTS
The binary representation of a(n) shows which prime numbers divide n, but not the multiplicities. a(2)=1, a(3)=10, a(4)=1, a(5)=100, a(6)=11, a(10)=101, a(30)=111, etc.
For n > 1, a(n) gives the (one-based) index of the column where n is located in array A285321. A008479 gives the other index. - Antti Karttunen, Apr 17 2017 For all n > 1 for which the value of A285331(n) is well-defined, we have A285331(a(n)) <= floor( A285331(n)/2), because then n is included in the binary tree A285332 and a(n) is one of its ancestors (in that tree), and thus must be at least one step nearer to its root than n itself. Conjecture: Starting at any n and iterating the map n -> a(n), we will always reach 0 (see A288569). This conjecture is equivalent to the conjecture that at any n that is neither a prime nor a power of two, we will eventually hit a prime number (which then becomes a power of two in the next iteration). If this conjecture is false then sequence A285332 cannot be a permutation of natural numbers. On the other hand, if the conjecture is true, then A285332 must be a permutation of natural numbers, because all primes and powers of 2 occur in definite positions in that tree. This conjecture also implies the conjectures made in A019565 and A285320 that essentially claim that there are neither finite nor infinite cycles in A019565. If there are any 2-cycles in this sequence, then both terms of the cycle should be present in A286611 and the larger one should be present in A286612. (End)
Binary rank of the distinct prime indices of n, where the binary rank of an integer partition y is given by Sum_i 2^(y_i-1). For all prime indices (with multiplicity) we have A048675. - Gus Wiseman, May 25 2024
FORMULA
Additive with a(p^e) = 2^(i-1) where p is the i-th prime. - Vladeta Jovovic, Oct 29 2003 (End)
a(n^2) = a(n).
(End)
EXAMPLE
a(38) = 129 because 38 = 2*19 = prime(1)*prime(8) and 129 = 2^0 + 2^7 (in binary 10000001).
a(140) = 13, binary 1101 because 140 is divisible by the first, third and fourth primes and 2^(1-1) + 2^(3-1) + 2^(4-1) = 13.
MATHEMATICA
a[n_] := Total[ 2^(PrimePi /@ FactorInteger[n][[All, 1]] - 1)]; a[1] = 0; Table[a[n], {n, 1, 69}] (* Jean-François Alcover, Dec 12 2011 *)
PROG
(Haskell)
a087207 = sum . map ((2 ^) . (subtract 1) . a049084) . a027748_row
(PARI) a(n) = {if (n==1, 0, my(f=factor(n), v = []); forprime(p=2, vecmax(f[, 1]), v = concat(v, vecsearch(f[, 1], p)!=0); ); fromdigits(Vecrev(v), 2)); } \\ Michel Marcus, Jun 05 2017 (PARI) A087207(n)=vecsum(apply(p->1<<primepi(p-1), factor(n)[, 1])) \\ Significantly faster than using sum(...). - M. F. Hasler, Jun 23 2017 (Python)
from sympy import factorint, primepi
def a(n):
return sum(2**primepi(i - 1) for i in factorint(n))
(Scheme)
CROSSREFS
Cf. A000040, A000120, A001221, A005117, A008479, A019565, A055396, A285320, A285321, A285329, A285330, A285332. Positions of particular values are: A000079\{1} (1), A000244\{1} (2), A033845 (3), A000351\{1} (4), A033846 (5), A033849 (6), A143207 (7), A000420\{1} (8), A033847 (9), A033850 (10), A033851 (12), A147576 (14), A147571 (15), A001020\{1} (16), A033848 (17). A048675 gives binary rank of prime indices.
Cf. A000720, A005940, A018819, A023506, A071814, A225620, A277319, A277905, A304818, A372689, A372890.
AUTHOR
Mitch Cervinka (puritan(AT)planetkc.com), Oct 26 2003
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
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
FORMULA
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.]
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 *)
PROG
(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
for(n=0, 4095, write("b285332.txt", n, " ", A285332(n))); (Scheme, with memoization-macro definec)
(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 Wudef 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))
CROSSREFS
Cf. A007947, A019565, A033845, A048675, A065642, A087207, A109162, A285319, A285320, A285328, A285330, A285333, A286542, A286543.
0, 1, 1, 9, 1, 5, 1, 15, 6, 7, 1, 25, 1, 9, 150, 225, 1, 21, 1, 375, 10, 13, 1, 625, 10, 15, 3750, 5625, 1, 31, 1, 21, 14, 19, 126, 35, 1, 21, 210, 315, 1, 41, 1, 525, 39, 25, 1, 875, 14, 45, 5250, 7875, 1, 4375, 8750, 13125, 22, 31, 1, 49, 1, 33, 51, 441, 18, 61, 1, 735, 26, 59, 1, 1225, 1, 39, 55, 11025, 18, 71, 1, 18375, 36750, 43, 1, 30625, 22, 45
COMMENTS
After zero, sequence contains only terms of A048103.
PROG
(PARI)
A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A129251(n) = { my(f = factor(n)); sum(k=1, #f~, (f[k, 2]>=f[k, 1])); };
A276086(n) = { my(i=0, m=1, pr=1, nextpr); while((n>0), i=i+1; nextpr = prime(i)*pr; if((n%nextpr), m*=(prime(i)^((n%nextpr)/pr)); n-=(n%nextpr)); pr=nextpr); m; };
CROSSREFS
Cf. A003415, A048103, A129251, A276085, A276086, A327859, A327928, A327929, A327964, A327965, A328097, A328098, A328099, A328113 (rgs-transform).
0, 1, 2, 2, 3, 4, 4, 3, 6, 4, 9, 6, 5, 8, 8, 4, 15, 8, 12, 5, 14, 10, 27, 8, 10, 6, 25, 12, 7, 16, 16, 5, 210, 16, 45, 10, 35, 16, 18, 5, 105, 16, 28, 11, 462, 28, 81, 10, 21, 12, 20, 7, 154, 26, 125, 16, 30, 8, 49, 24, 11, 32, 32, 6, 10659, 212, 420, 17, 910, 46, 75, 10, 78, 36, 175, 20, 33, 20, 24, 6, 3094, 106, 315, 18, 385, 32, 56, 17, 780045
COMMENTS
Following A285332, also this sequence can be represented in a form of a binary tree: 0
|
...................1...................
2 2
3......../ \........4 4......../ \........3
/ \ / \ / \ / \
/ \ / \ / \ / \
/ \ / \ / \ / \
6 4 9 6 5 8 8 4
15 8 12 5 14 10 27 8 10 6 25 12 7 16 16 5
etc.
FORMULA
For all n >= 1, a(2n) = A285332(n). a(2^n) = A109162(1+n). [The left edge of the tree.] a( A000225(n)) = n. [The right edge of tree.]
PROG
(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
A048675(n) = my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; \\ Michel Marcus, Oct 10 2016
Lexicographically earliest such sequence a that a(i) = a(j) => f(i) = f(j) for all i, j, where f(n) = -1 if n is an odd prime, and f(n) = floor(n/2) for all other numbers.
+10
6
1, 2, 3, 4, 3, 5, 3, 6, 6, 7, 3, 8, 3, 9, 9, 10, 3, 11, 3, 12, 12, 13, 3, 14, 14, 15, 15, 16, 3, 17, 3, 18, 18, 19, 19, 20, 3, 21, 21, 22, 3, 23, 3, 24, 24, 25, 3, 26, 26, 27, 27, 28, 3, 29, 29, 30, 30, 31, 3, 32, 3, 33, 33, 34, 34, 35, 3, 36, 36, 37, 3, 38, 3, 39, 39, 40, 40, 41, 3, 42, 42, 43, 3, 44, 44, 45, 45, 46, 3, 47, 47, 48, 48, 49, 49, 50, 3, 51, 51, 52, 3, 53, 3, 54, 54
COMMENTS
This sequence is a restricted growth sequence transform of a function f which is defined as f(n) = A004526(n), unless n is an odd prime, in which case f(n) = -1, which is a constant not in range of A004526. See the Crossrefs section for a list of similar sequences. For all i, j:
The shifted version of this filter, A323161, has a remarkable ability to find many sequences related to primes and prime chains. - Antti Karttunen, Jan 06 2019
PROG
(PARI)
up_to = 65537;
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; };
A322809aux(n) = if((n>2)&&isprime(n), -1, (n>>1));
v322809 = rgs_transform(vector(up_to, n, A322809aux(n)));
CROSSREFS
A list of few similarly constructed sequences follows, where each sequence is an rgs-transform of such function f, for which the value of f(n) is the n-th term of the sequence whose A-number follows after a parenthesis, unless n is of the form ..., in which case f(n) is given a constant value outside of the range of that sequence:
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