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Search: a373592 -id:a373592
Displaying 1-6 of 6 results found. page 1
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A373595 Lexicographically earliest infinite sequence such that for all i, j >= 1, a(i) = a(j) => f(i) = f(j), where f(n<=3) = n, f(p) = 0 for primes p > 3, and for composite n, f(n) = [A007949(n), A373591(n), A373592(n)]. +20
3
1, 2, 3, 4, 5, 6, 5, 7, 8, 4, 5, 9, 5, 10, 6, 11, 5, 12, 5, 7, 13, 4, 5, 14, 4, 10, 15, 16, 5, 9, 5, 17, 6, 4, 10, 18, 5, 10, 13, 11, 5, 19, 5, 7, 12, 4, 5, 20, 21, 7, 6, 16, 5, 22, 4, 23, 13, 4, 5, 14, 5, 10, 24, 25, 10, 9, 5, 7, 6, 16, 5, 26, 5, 10, 9, 16, 10, 19, 5, 17, 27, 4, 5, 28, 4, 10, 6, 11, 5, 18, 21, 7, 13, 4, 10, 29, 5, 30, 12, 11, 5, 9, 5, 23, 19 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
Restricted growth sequence transform of the function f given in the definition.
For all i, j > 1:
A305900(i) = A305900(j) => A373594(i) = A373594(j) => a(i) = a(j),
A373593(i) = A373593(j) => a(i) = a(j),
a(i) = a(j) => b(i) = b(j), where b can be (but is not limited to) any of the sequences listed at the crossrefs-section, under "some of the matched sequences".
LINKS
Antti Karttunen, Table of n, a(n) for n = 1..100000
PROG
(PARI)
up_to = 100000;
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; };
A007949(n) = valuation(n, 3);
A373591(n) = sum(i=1, #n=factor(n)~, (1==n[1, i]%3)*n[2, i]);
A373592(n) = sum(i=1, #n=factor(n)~, (2==n[1, i]%3)*n[2, i]);
Aux373595(n) = if(n<=3, n, if(isprime(n), 0, [A007949(n), A373591(n), A373592(n)]));
v373595 = rgs_transform(vector(up_to, n, Aux373595(n)));
A373595(n) = v373595[n];
CROSSREFS
Cf. A007949, A373591, A373592.
Cf. A305900, A373593, A373594.
Some of the matched sequences (see comments): A001222, A359430, A369643, A369658, A373371, A373383, A373474, A373491, A373493, A373585, A373588, A373596.
KEYWORD
nonn
AUTHOR
Antti Karttunen, Jun 13 2024
STATUS
approved
A373594 Lexicographically earliest infinite sequence such that for all i, j >= 1, a(i) = a(j) => f(i) = f(j), where f(n<=3) = n, f(p) = 0 for primes p > 3, and for composite n, f(n) = [A007814(n), A065339(n), A083025(n), A373591(n), A373592(n)]. +20
2
1, 2, 3, 4, 5, 6, 5, 7, 8, 9, 5, 10, 5, 11, 12, 13, 5, 14, 5, 15, 16, 17, 5, 18, 19, 20, 21, 22, 5, 23, 5, 24, 25, 9, 26, 27, 5, 11, 28, 29, 5, 30, 5, 31, 32, 17, 5, 33, 34, 35, 12, 36, 5, 37, 38, 39, 16, 9, 5, 40, 5, 11, 41, 42, 43, 44, 5, 15, 25, 45, 5, 46, 5, 20, 47, 22, 48, 49, 5, 50, 51, 9, 5, 52, 19, 11, 12, 53, 5, 54, 55, 31, 16, 17, 26, 56, 5, 57, 58 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
Restricted growth sequence transform of the function f given in the definition.
Note that for composite n, f(n) can be defined in general as a quintuple vector [v(n), w(n), x(n), y(n), z(n)], where v, w, x, y and z are any five of these six sequences: A007814, A007949, A065339, A083025, A373591, A373592. This follows because A007814(n) + A065339(n) + A083025(n) = A007949(n) + A373591(n) + A373592(n) = A001222(n), so the omitted sixth element can be always worked out from the remaining five.
For all i, j > 1:
A305900(i) = A305900(j) => a(i) = a(j) => A373595(i) = A373595(j).
LINKS
Antti Karttunen, Table of n, a(n) for n = 1..100000
PROG
(PARI)
up_to = 100000;
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; };
A007814(n) = valuation(n, 2);
A065339(n) = sum(i=1, #n=factor(n)~, (3==n[1, i]%4)*n[2, i]);
A083025(n) = sum(i=1, #n=factor(n)~, (1==n[1, i]%4)*n[2, i]);
A373591(n) = sum(i=1, #n=factor(n)~, (1==n[1, i]%3)*n[2, i]);
A373592(n) = sum(i=1, #n=factor(n)~, (2==n[1, i]%3)*n[2, i]);
Aux373594(n) = if(n<=3, n, if(isprime(n), 0, [A007814(n), A083025(n), A065339(n), A373591(n), A373592(n)]));
v373594 = rgs_transform(vector(up_to, n, Aux373594(n)));
A373594(n) = v373594[n];
CROSSREFS
Cf. A007814, A007949, A065339, A083025, A373591, A373592.
Cf. also A305900, A373595.
KEYWORD
nonn
AUTHOR
Antti Karttunen, Jun 13 2024
STATUS
approved
A359430 a(n) = 1 if the arithmetic derivative of n is a multiple of 3, otherwise 0. +10
13
1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 1, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET
0
LINKS
Antti Karttunen, Table of n, a(n) for n = 0..100000
Index entries for characteristic functions
FORMULA
a(n) = [A003415(n) == 0 (mod 3)], where [ ] is the Iverson bracket.
From Antti Karttunen, Jun 13 2024: (Start)
a(n) = A267142(n) + A369658(n) = A267142(n) + A011655(n)*A373371(n).
For n > 0, a(n) = [n == 0 (mod 9)] + [n != 0 (mod 3)]*[A373591(n) == A373592(n) (mod 3)].
a(n) = [0 == A373253(n)] = 1 - (A373254(n) + A373256(n)).
a(n) >= A369643(n).
a(n) >= A373143(n).
a(n) >= A370118(n).
(End)
PROG
(PARI)
A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A359430(n) = !(A003415(n)%3);
(PARI) A359430(n) = if(!n, 1, if(!(n%3), !(n%9), my(f = factor(n), m1=0, m2=0); for(i=1, #f~, if(1==(f[i, 1]%3), m1 += f[i, 2], m2 += f[i, 2])); 0==((m1-m2)%3))); \\ Antti Karttunen, Jun 13 2024
CROSSREFS
Characteristic function of A327863.
Cf. A003415, A011655, A267142, A369658, A373253, A373254, A373256, A373371, A373591, A373592.
Cf. also A369643, A369653 [= a(A276086(n))], A370118, A370122 [= a(A003415(n))], A373143.
KEYWORD
nonn
AUTHOR
Antti Karttunen, Jan 02 2023
STATUS
approved
A369658 a(n) = 1 if n is not multiple of 3, but its arithmetic derivative is, otherwise 0. +10
11
0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET
0
COMMENTS
Conjecture: the asymptotic mean of this sequence is (2/3)*(1/3) = 2/9. Compare to the comment at A369653, but consider also the four lowermost rows of the table given at A369252 (and further generalizations to various number of primes), and also A007352, A096629, and how they affect such probabilities.
Sum_{i=1..10^n} a(i), for n = 1..10 gives: 2, 18, 201, 2110, 21484, 216973, 2181521, 21896827, 219541804, 2199637607. - Antti Karttunen, Jun 17 2024
LINKS
Antti Karttunen, Table of n, a(n) for n = 0..100000
Eric Weisstein's World of Mathematics, Chebyshev Bias
Index entries for characteristic functions
Index entries for sequences which agree for a long time but are different
FORMULA
a(n) = A011655(n) * A079978(A003415(n)) = A011655(n) * A359430(n).
a(n) <= A369643(n) <= A359430(n).
For n >= 1, a(n) <= A373474(n). - Antti Karttunen, Jun 07 2024
For n >= 1, a(n) = A011655(n) * A373371(n) = A011655(n) * [A373591(n) == A373592(n) (mod 3)]. - Antti Karttunen, Jun 13 2024
PROG
(PARI)
A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A369658(n) = ((n%3)&&(0==(A003415(n)%3)));
(PARI) A369658(n) = if(n<2, n, if(!(n%3), 0, my(f = factor(n), m1=0, m2=0); for(i=1, #f~, if(1==(f[i, 1]%3), m1 += f[i, 2], if(2==(f[i, 1]%3), m2 += f[i, 2]))); 0==((m1-m2)%3))); \\ Antti Karttunen, Jun 16 2024
CROSSREFS
Characteristic function of A369659.
Cf. A003415, A011655, A079978, A359430, A369653, A373371, A373591, A373592.
Cf. A007352, A096629, A369252.
Differs from related A369643 for the first time at n=54, where a(54) = 0, while A369643(54) = 1.
Differs from related A373474 for the first time at n=19683, where a(19683) = 0, while A373474(19683) = 1.
Cf. also A353557, A360109, A369968, for cases k = 2, 4, 5 of the characteristic functions for nonmultiples of k whose arithmetic derivative is multiple of k.
KEYWORD
nonn
AUTHOR
Antti Karttunen, Feb 10 2024
STATUS
approved
A373371 a(n) = 1 if the sum of prime factors with repetition is a multiple of 3, otherwise 0. +10
10
1, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET
1
COMMENTS
a(n) = 1 if the multiplicities of prime factors of the forms 3m+1 (A002476) and 3m-1 (A003627) are equal modulo 3, otherwise 0. - Antti Karttunen, Jun 13 2024
LINKS
Antti Karttunen, Table of n, a(n) for n = 1..100000
Index entries for characteristic functions
FORMULA
a(n) = [A001414(n) == 0 (mod 3)], where [ ] is the Iverson bracket.
From Antti Karttunen, Jun 13 2024: (Start)
a(n) = [A373591(n) == A373592(n) (mod 3)].
a(n) = a(n/A038500(n)) = A359430(n/A038500(n)) = A369658(n/A038500(n)).
(End)
PROG
(PARI)
A001414(n) = ((n=factor(n))[, 1]~*n[, 2]); \\ From A001414.
A373371(n) = !(A001414(n)%3);
(PARI) A373371(n) = { my(f = factor(n), m1=0, m2=0); for(i = 1, #f~, if(1==(f[i, 1]%3), m1 += f[i, 2], if(2==(f[i, 1]%3), m2 += f[i, 2]))); 0==((m1-m2)%3); }; \\ Antti Karttunen, Jun 13 2024
CROSSREFS
Characteristic function of A289142.
Cf. A001414, A002476, A003627, A038500, A373591, A373592.
Cf. also A359430, A369658, A373372.
KEYWORD
nonn
AUTHOR
Antti Karttunen, Jun 02 2024
STATUS
approved
A373591 Number of primes congruent to 1 modulo 3 dividing n (with multiplicity). +10
8
0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 2, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 1, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 2, 0, 1, 0, 1, 0, 1, 2, 0, 0, 0, 0, 1, 1, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,49
LINKS
Antti Karttunen, Table of n, a(n) for n = 1..100000
FORMULA
a(n) = A001222(A248909(n)).
a(n) = A001222(n) - (A007949(n)+A373592(n)).
Totally additive with a(3) = 0, a(p) = 1 if p == 1 (mod 3), and a(p) = 0 if p == 2 (mod 3). - Amiram Eldar, Jun 17 2024
MATHEMATICA
f[p_, e_] := If[Mod[p, 3] == 1, e, 0]; f[3, e_] := 0; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jun 17 2024 *)
PROG
(PARI) A373591(n) = sum(i=1, #n=factor(n)~, (1==n[1, i]%3)*n[2, i]); \\ After code in A083025
CROSSREFS
Cf. A001222, A007949, A248909, A373592.
Cf. also A065339, A083025.
KEYWORD
nonn
AUTHOR
Antti Karttunen, Jun 13 2024
STATUS
approved
page 1

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Last modified August 29 11:15 EDT 2024. Contains 375512 sequences. (Running on oeis4.)