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Search: a083345 -id:a083345
Displaying 1-10 of 61 results found. page 1 2 3 4 5 6 7
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A373363 a(n) = gcd(A001414(n), A083345(n)), where A001414 is the sum of prime factors with repetition, and A083345 is the numerator of the sum of the inverses of prime factors with repetition. +20
10
0, 1, 1, 1, 1, 5, 1, 3, 2, 7, 1, 1, 1, 9, 8, 2, 1, 1, 1, 3, 10, 13, 1, 1, 2, 15, 1, 1, 1, 1, 1, 5, 14, 19, 12, 5, 1, 21, 16, 1, 1, 1, 1, 3, 1, 25, 1, 1, 2, 3, 20, 1, 1, 1, 16, 1, 22, 31, 1, 1, 1, 33, 1, 3, 18, 1, 1, 3, 26, 1, 1, 1, 1, 39, 1, 1, 18, 1, 1, 1, 4, 43, 1, 1, 22, 45, 32, 1, 1, 1, 20, 3, 34, 49, 24, 1, 1, 1, 1, 7 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,6
LINKS
Antti Karttunen, Table of n, a(n) for n = 1..65537
PROG
(PARI)
A001414(n) = ((n=factor(n))[, 1]~*n[, 2]); \\ From A001414.
A083345(n) = { my(f=factor(n)); numerator(vecsum(vector(#f~, i, f[i, 2]/f[i, 1]))); };
A373363(n) = gcd(A001414(n), A083345(n));
CROSSREFS
Cf. A001414, A083345.
Cf. A345452 (positions of even terms), A353374 (their characteristic function).
Cf. also A082299, A373362.
KEYWORD
nonn
AUTHOR
Antti Karttunen, Jun 02 2024
STATUS
approved
A369644 Positions of multiples of 3 in A083345. +20
8
1, 8, 14, 20, 26, 35, 38, 44, 50, 54, 62, 64, 65, 68, 74, 77, 86, 92, 95, 110, 112, 116, 119, 122, 125, 134, 135, 143, 146, 155, 158, 160, 161, 164, 170, 185, 188, 194, 196, 203, 206, 208, 209, 212, 215, 218, 221, 230, 236, 242, 254, 275, 278, 280, 284, 287, 290, 297, 299, 302, 304, 305, 314, 323, 326, 329, 332, 335 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
LINKS
Table of n, a(n) for n=1..68.
PROG
(PARI) \\ See A369643.
CROSSREFS
Cf. A083345, A369643 (characteristic function).
Subsequence of A327863.
KEYWORD
nonn
AUTHOR
Antti Karttunen, Jan 28 2024
STATUS
approved
A373475 Numbers k such that A001414(k) and A083345(k) are both multiples of 3, where A001414 is fully additive with a(p) = p, and A083345 is the numerator of the fully additive function with a(p) = 1/p. +20
8
1, 8, 14, 20, 26, 35, 38, 44, 50, 62, 64, 65, 68, 74, 77, 86, 92, 95, 110, 112, 116, 119, 122, 125, 134, 143, 146, 155, 158, 160, 161, 164, 170, 185, 188, 194, 196, 203, 206, 208, 209, 212, 215, 218, 221, 230, 236, 242, 254, 275, 278, 280, 284, 287, 290, 299, 302, 304, 305, 314, 323, 326, 329, 332, 335, 341, 343 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
If k is a term, then 3^9 * k is also a term. See A373476.
A369659 is a subsequence of this sequence, giving the terms that are not multiples of 3. This follows because A083345(n) = n' / gcd(n',n) and from the following lemma: When k is not a multiple of 3, then either sopfr(k) [= A001414(k)] and k' [= A003415(k)] are both multiples of 3, or both are non-multiples of 3.
Proof of the lemma: As k is not a multiple of 3, all its prime factors p, q, r, s, t, u, v, w, ... (not necessarily all distinct) are either of the form 3m+1 or 3m-1. Let's first eliminate from k all triplets of primes that are of the same type modulo 3, either -1 or +1, (marked now as p, q, r) as they do not affect the divisibility by 3 of either the sopfr(k) or k'. In the case of the arithmetic derivative this is because we have k' = (pqr)' * (k/pqr) + (k/pqr)' * pqr, and as we know that the first summand is a multiple of 3 (because (pqr)' is), therefore the divisibility of the whole expression by 3 depends only on whether (k/pqr)' is a multiple of 3, as certainly pqr is not a multiple of 3.
What will remain after such elimination process has been completed as far as possible, must be either 1, or of the form p*q (p and q of different types), or p*q*r*s (with two primes of one type, and two primes of the other type), in which cases both sopfr(k) and k' are multiples of 3, or then alternatively, what remains must be of the form p*q (p and q of the same type), or p*q*r (with two primes of one type and the third of the other type), both cases which indicate that both sopfr(k) and k' are non-multiples of 3.
LINKS
Antti Karttunen, Table of n, a(n) for n = 1..33911
Index entries for sequences which agree for a long time but are different
FORMULA
a(n) = A373476(n) / 3^9.
EXAMPLE
110 = 2*5*11 is a term of this sequence because 2+5+11 = 18 is a multiple of 3, and also 2*5 + 2*11 + 5*11 = 87 is a multiple of 3.
54 (= A369644(10)) is NOT a term of this sequence, because A001414(54) = 11 is not a multiple of 3, although A083345(54) = 3 is.
19683 = 3^9 is a term of this sequence, because both A001414(19683) = 9*3 = 27 and A083345(19683) = A003415(3^9)/gcd(3^9, A003415(3^9)) = 3, are multiples of 3.
PROG
(PARI) isA373475 = A373474;
CROSSREFS
Cf. A001414, A003415, A083345, A373474 (characteristic function).
Positions of multiples of 3 in A373363.
Intersection of A289142 and A369644.
Subsequence of A373478.
Disjoint union of A369659 and A373476.
Differs from A369659 for the first time at n=4186, where a(4186) = A373476(1) = 19683, a term not present in A369659, as it is the first multiple of 3 in this sequence.
KEYWORD
nonn
AUTHOR
Antti Karttunen, Jun 06 2024
STATUS
approved
A373151 Lexicographically earliest infinite sequence such that a(i) = a(j) => A083345(i) = A083345(j) and A373145(i) = A373145(j), for all i, j >= 1. +20
7
1, 2, 2, 3, 2, 4, 2, 5, 6, 7, 2, 8, 2, 9, 10, 11, 2, 12, 2, 13, 14, 15, 2, 16, 6, 17, 18, 19, 2, 20, 2, 21, 22, 23, 24, 25, 2, 26, 27, 28, 2, 29, 2, 30, 15, 31, 2, 32, 6, 9, 33, 34, 2, 35, 27, 36, 37, 20, 2, 38, 2, 39, 40, 41, 42, 43, 2, 44, 45, 46, 2, 15, 2, 47, 16, 48, 42, 49, 2, 50, 8, 51, 2, 52, 37, 53, 54, 55, 2, 29, 56, 57, 58, 59, 60, 28, 2, 16, 31, 61 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
Restricted growth sequence transform of the ordered pair [A083345(n), A373145(n)].
For all i, j >= 1:
A373150(i) = A373150(j) => a(i) = a(j),
a(i) = a(j) => A369001(i) = A369001(j),
a(i) = a(j) => A369004(i) = A369004(j),
a(i) = a(j) => A373143(i) = A373143(j).
LINKS
Antti Karttunen, Table of n, a(n) for n = 1..65537
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; };
A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A083345(n) = { my(f=factor(n)); numerator(vecsum(vector(#f~, i, f[i, 2]/f[i, 1]))); };
A002110(n) = prod(i=1, n, prime(i));
A276085(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*A002110(primepi(f[k, 1])-1)); };
A373145(n) = gcd(A003415(n), A276085(n));
Aux373151(n) = [A083345(n), A373145(n)];
v373151 = rgs_transform(vector(up_to, n, Aux373151(n)));
A373151(n) = v373151[n];
CROSSREFS
Cf. A003415, A002110, A083345, A276085, A373145.
Cf. A369001, A369004, A373143, A373150.
KEYWORD
nonn
AUTHOR
Antti Karttunen, May 27 2024
STATUS
approved
A373474 a(n) = 1 if A001414(n) and A083345(n) are both multiples of 3, otherwise 0, where A001414 is fully additive with a(p) = p, and A083345 is the numerator of the fully additive function with a(p) = 1/p. +20
7
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
1
COMMENTS
Differs from A369658 for the first time at n=19683, 157464, 275562, 393660, ..., see A373476.
LINKS
Antti Karttunen, Table of n, a(n) for n = 1..100000
Index entries for characteristic functions
Index entries for sequences which agree for a long time but are different
FORMULA
a(n) = [A373363(n) == 0 (mod 3)], where [ ] is the Iverson bracket.
a(3^9 * n) = a(n).
PROG
(PARI)
A001414(n) = ((n=factor(n))[, 1]~*n[, 2]); \\ From A001414.
A083345(n) = { my(f=factor(n)); numerator(vecsum(vector(#f~, i, f[i, 2]/f[i, 1]))); };
A373474(n) = (!(A001414(n)%3) && !(A083345(n)%3));
CROSSREFS
Characteristic function of A373475.
Cf. A001414, A083345, A373363, A373476 [k where a(k) != A369658(k)].
KEYWORD
nonn
AUTHOR
Antti Karttunen, Jun 06 2024
STATUS
approved
A369069 Dirichlet convolution of Liouville's lambda (A008836) with A083345, where A083345(n) = n' / gcd(n,n'), and n' stands for the arithmetic derivative of n, A003415. +20
6
0, 1, 1, 0, 1, 3, 1, 3, 1, 5, 1, 0, 1, 7, 6, -1, 1, 2, 1, 0, 8, 11, 1, 8, 1, 13, 0, 0, 1, 14, 1, 6, 12, 17, 10, 0, 1, 19, 14, 14, 1, 20, 1, 0, 5, 23, 1, -3, 1, 2, 18, 0, 1, 0, 14, 20, 20, 29, 1, 0, 1, 31, 7, -3, 16, 32, 1, 0, 24, 34, 1, 5, 1, 37, 3, 0, 16, 38, 1, -5, 4, 41, 1, 0, 20, 43, 30, 32, 1, 9, 18, 0, 32, 47 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,6
COMMENTS
In contrast to A347395, this sequence has also nonpositive values after the initial term. It seems that A342090 gives most of the positions of nonpositive terms here, apart from k = 0, 729, 1458, 3645, 5103, 5832, 7290, ..., etc.
LINKS
Antti Karttunen, Table of n, a(n) for n = 1..16384
FORMULA
a(n) = Sum_{d|n} A008836(n/d) * A083345(d).
PROG
(PARI)
A008836(n) = ((-1)^bigomega(n));
A083345(n) = { my(f=factor(n)); numerator(vecsum(vector(#f~, i, f[i, 2]/f[i, 1]))); };
A369069(n) = sumdiv(n, d, A008836(n/d)*A083345(d));
CROSSREFS
Cf. A003415, A008836, A083345, A342090.
Cf. also A347395, A369068.
KEYWORD
sign
AUTHOR
Antti Karttunen, Jan 16 2024
STATUS
approved
A373491 a(n) = 1 if A059975(n) and A083345(n) are both multiples of 3, otherwise 0, where A059975 is fully additive with a(p) = p-1, and A083345 is the numerator of the fully additive function with a(p) = 1/p. +20
6
1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET
1
LINKS
Antti Karttunen, Table of n, a(n) for n = 1..100000
Index entries for characteristic functions
FORMULA
a(n) = [A373377(n) == 0 (mod 3)], where [ ] is the Iverson bracket.
a(n) <= A373493(n).
PROG
(PARI)
A059975(n) = {my(f = factor(n)); sum(i = 1, #f~, f[i, 2]*(f[i, 1] - 1)); };
A083345(n) = { my(f=factor(n)); numerator(vecsum(vector(#f~, i, f[i, 2]/f[i, 1]))); };
A373491(n) = (!(A059975(n)%3) && !(A083345(n)%3));
CROSSREFS
Characteristic function of A373492.
Cf. A059975, A083345, A373377, A373493.
KEYWORD
nonn
AUTHOR
Antti Karttunen, Jun 10 2024
STATUS
approved
A373483 a(n) = 1 if A083345(n) and A276085(n) are both multiples of 3, otherwise 0, where A276085 is fully additive with a(p) = p#/p, and A083345 is the numerator of the fully additive function with a(p) = 1/p. +20
5
1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET
1
LINKS
Antti Karttunen, Table of n, a(n) for n = 1..100000
Index entries for characteristic functions
FORMULA
a(n) = A369643(n) * A372573(n).
a(n) = [A373485(n) == 0 (mod 3)], where [ ] is the Iverson bracket.
a(n) <= A373143(n).
PROG
(PARI)
A083345(n) = { my(f=factor(n)); numerator(vecsum(vector(#f~, i, f[i, 2]/f[i, 1]))); };
A276085(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*prod(i=1, primepi(f[k, 1]-1), prime(i))); };
A373483(n) = (!(A083345(n)%3) && !(A276085(n)%3));
CROSSREFS
Characteristic function of A373484.
Cf. A083345, A276085, A369643, A372573, A373143, A373485.
KEYWORD
nonn
AUTHOR
Antti Karttunen, Jun 09 2024
STATUS
approved
A373485 a(n) = gcd(A083345(n), A276085(n)), where A276085 is fully additive with a(p) = p#/p, and A083345 is the numerator of the fully additive function with a(p) = 1/p. +20
5
0, 1, 1, 1, 1, 1, 1, 3, 2, 7, 1, 4, 1, 1, 8, 2, 1, 1, 1, 2, 2, 1, 1, 1, 2, 1, 1, 8, 1, 1, 1, 5, 2, 1, 12, 1, 1, 1, 8, 1, 1, 1, 1, 4, 1, 1, 1, 1, 2, 1, 4, 2, 1, 1, 8, 1, 2, 1, 1, 1, 1, 1, 17, 3, 6, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 4, 6, 1, 1, 1, 4, 1, 1, 1, 2, 1, 8, 1, 1, 1, 20, 4, 2, 1, 12, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,8
COMMENTS
For all n >= 1, A373145(n) is a multiple of a(n).
For all i, j: A373151(i) = A373151(j) => a(i) = a(j) => A373483(i) = A373483(j).
LINKS
Antti Karttunen, Table of n, a(n) for n = 1..100000
PROG
(PARI)
A083345(n) = { my(f=factor(n)); numerator(vecsum(vector(#f~, i, f[i, 2]/f[i, 1]))); };
A276085(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*prod(i=1, primepi(f[k, 1]-1), prime(i))); };
A373485(n) = gcd(A083345(n), A276085(n));
CROSSREFS
Cf. A083345, A276085, A373145, A373151.
Cf. A369002 (positions of even terms), A369003 (of odd terms), A373483, A373484 (of multiples of 3).
KEYWORD
nonn
AUTHOR
Antti Karttunen, Jun 09 2024
STATUS
approved
A373492 Numbers k for which A059975(k) and A083345(k) are both multiples of 3, where A059975 is fully additive with a(p) = p-1, and A083345 is the numerator of the fully additive function with a(p) = 1/p. +20
5
1, 8, 20, 44, 50, 64, 68, 92, 110, 116, 125, 160, 164, 170, 188, 212, 230, 236, 242, 275, 284, 290, 332, 343, 352, 356, 374, 400, 404, 410, 425, 428, 452, 470, 506, 512, 524, 530, 544, 548, 575, 578, 590, 596, 605, 637, 638, 668, 692, 710, 716, 725, 736, 764, 782, 788, 830, 880, 890, 902, 908, 928, 931, 932, 935 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
A multiplicative semigroup: if m and n are in the sequence, then so is m*n.
LINKS
Antti Karttunen, Table of n, a(n) for n = 1..10000
PROG
(PARI) isA373492 = A373491;
CROSSREFS
Cf. A059975, A083345, A373491 (characteristic function).
Positions of multiples of 3 in A373377.
Intersection of A373385 and A369644.
Subsequence of A373494.
KEYWORD
nonn
AUTHOR
Antti Karttunen, Jun 10 2024
STATUS
approved
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Last modified August 18 19:26 EDT 2024. Contains 375273 sequences. (Running on oeis4.)