Displaying 1-10 of 145 results found.
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0, 1, 1, 2, 1, 1, 1, 3, 2, 7, 1, 4, 1, 1, 8, 4, 1, 1, 1, 8, 2, 1, 1, 1, 2, 1, 3, 32, 1, 1, 1, 5, 2, 1, 12, 6, 1, 1, 8, 1, 1, 1, 1, 4, 1, 1, 1, 2, 2, 1, 4, 8, 1, 1, 8, 1, 2, 1, 1, 2, 1, 1, 17, 6, 6, 1, 1, 8, 2, 1, 1, 1, 1, 1, 1, 16, 6, 1, 1, 2, 4, 1, 1, 2, 2, 1, 8, 1, 1, 1, 20, 4, 2, 1, 12, 1, 1, 1, 1, 14, 1, 1, 1, 1, 1
PROG
(PARI)
A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, 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)); };
a(n) = 1 if A276085(n) is a multiple of 3, otherwise 0, where A276085 is the primorial base log-function.
+20
12
1, 0, 0, 0, 1, 1, 1, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 1, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 1, 1, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1
COMMENTS
a(n) = 1 iff n is of the form 2^i * 3^j * k, with k in A007310 [i.e., gcd(k, 6) = 1], and i == j (mod 3).
FORMULA
a(n) = [ A276085(n) == 0 (mod 3)], where [ ] is the Iverson bracket. Sum_{k=1..n} a(k) ~ (43/91) * n. - Amiram Eldar, May 29 2024
MATHEMATICA
a[n_] := If[Divisible[Differences[IntegerExponent[n, {2, 3}]][[1]], 3], 1, 0]; Array[a, 100] (* Amiram Eldar, May 29 2024 *)
PROG
(PARI)
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)); };
CROSSREFS
Characteristic function of A339746.
COMMENTS
Intersection with A048103 gives the fixed points (1, 2, 10, 15, 5005, ...) of A327859. Question: Does that set preclude nonsquarefree numbers? Certainly it does not contain any multiples of 9. See also comments in A328110. If k == 2 (mod 4), then both A003415(k) and A276085(k) are odd, and the latter is of the form 4m+1 (if k has an odd number of prime factors), or of the form 4m+3 (if k has an even number of prime factors). Therefore, for k of the form 4m+2 to be included in this sequence, a necessary condition is that it must be either in the intersection of A026424 and A358772 (like, for example, 2 is) or in A369668 (the intersection of A028260 and A358774), like for example, 10 is. If k is odd, then A276085(k) is even, and for A003415(k) to be even with k odd, then k has to be in A046337 (odd numbers with an even number of prime factors, counted with multiplicity). But A276085( A046337(n)) == 0 (mod 4) for all n, so also A003415(k) has to be a multiple of 4, so k has to be in A360110 (itself a subsequence of A369002), like for example k=15 and k=5005 are. If it exists, a(7) > 2^19.
EXAMPLE
As 5005 = 5*7*11*13, A003415(5*7*11*13) = (5*7*11) + (5*7*13) + (5*11*13) + (7*11*13) = 2556 = 2^2 * 3^2 * 71 = A276085(5005) = A002110(2) + A002110(3) + A002110(4) + A002110(5) [as 5, 7, 11 and 13 are prime(3) .. prime(6)], therefore 5005 is included in this sequence.
PROG
(PARI)
A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, 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))); };
a(n) = gcd( A001414(n), A276085(n)), where A001414 is the sum of prime factors with repetition, and A276085 is the primorial base log-function.
+20
11
0, 1, 1, 2, 1, 1, 1, 3, 2, 7, 1, 1, 1, 1, 8, 4, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 1, 1, 1, 5, 2, 1, 12, 2, 1, 1, 8, 1, 1, 3, 1, 1, 1, 1, 1, 1, 2, 1, 4, 17, 1, 1, 8, 1, 2, 1, 1, 2, 1, 1, 1, 6, 6, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 6, 9, 1, 1, 4, 1, 1, 2, 2, 1, 8, 1, 1, 1, 20, 1, 2, 1, 12, 1, 1, 1, 1, 14, 1, 1, 1, 1, 1
COMMENTS
As A001414 and A276085 are both fully additive sequences, all sequences that give the positions of multiples of some k > 1 in this sequence are closed under multiplication: For example, A373373, which gives the indices of multiples of 3.
PROG
(PARI)
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)); };
CROSSREFS
Cf. A345452 (positions of even terms), A373373 (positions of multiples of 3).
a(n) = 1 if both A003415(n) and A276085(n) are multiples of 3, otherwise 0, where A003415 is the arithmetic derivative and A276085 is the primorial base log-function.
+20
9
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, 1, 0, 0, 0, 0, 0, 0, 0, 1, 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, 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
COMMENTS
Question: Does this sequence have an asymptotic mean?
PROG
(PARI)
A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, 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)); };
CROSSREFS
Characteristic function of A373144.
Numbers k such that both A003415(k) and A276085(k) are multiples of 3, where A003415 is the arithmetic derivative and A276085 is the primorial base log-function.
+20
9
1, 8, 27, 35, 36, 64, 65, 77, 95, 119, 125, 135, 143, 155, 161, 162, 180, 185, 189, 203, 209, 215, 216, 221, 252, 275, 280, 287, 288, 297, 299, 305, 323, 329, 335, 341, 343, 351, 365, 371, 377, 395, 396, 407, 413, 425, 437, 459, 468, 473, 485, 497, 512, 513, 515, 520, 527, 533, 545, 551, 575, 581, 605, 611, 612, 616
COMMENTS
This is a multiplicative semigroup; if m and n are in the sequence then so is m*n.
EXAMPLE
65 is present as A003415(65) = 18 = 3*6 and A276085(65) = 2316 = 3*772.
CROSSREFS
Positions of multiples of 3 in A373145.
Numbers k such that A276085(k) is a multiple of 8, where A276085 is the primorial base log-function.
+20
8
1, 15, 20, 21, 28, 39, 51, 52, 55, 57, 68, 76, 77, 81, 87, 93, 108, 115, 116, 124, 141, 143, 144, 161, 183, 185, 187, 188, 192, 201, 205, 209, 215, 219, 225, 237, 244, 256, 259, 265, 267, 268, 287, 291, 292, 295, 297, 299, 300, 301, 303, 309, 315, 316, 319, 327, 339, 341, 355, 356, 371, 381, 388, 391, 396, 400, 404
COMMENTS
Because A276085 is completely additive, this is a multiplicative semigroup; if m and n are in the sequence then so is m*n. The terms should be the integers in a multiplicative subgroup of the positive rationals. Denoting the k-th prime by p_k, a set of generators for this subgroup might be the union of {20, 81} with an infinite set constituted as follows: if p_k == 3 (mod 4) then p_k * p_{k+1} is in the set, if p_k == 1 (mod 4) then p_k^3 * p_{k+1} is in the set. - Peter Munn, Jul 15 2024
a(n) is -1, 0, or 1 such that a(n) == A276085(n) (mod 3), where A276085 is the primorial base log-function.
+20
8
0, 1, -1, -1, 0, 0, 0, 0, 1, 1, 0, 1, 0, 1, -1, 1, 0, -1, 0, -1, -1, 1, 0, -1, 0, 1, 0, -1, 0, 0, 0, -1, -1, 1, 0, 0, 0, 1, -1, 0, 0, 0, 0, -1, 1, 1, 0, 0, 0, 1, -1, -1, 0, 1, 0, 0, -1, 1, 0, 1, 0, 1, 1, 0, 0, 0, 0, -1, -1, 1, 0, 1, 0, 1, -1, -1, 0, 0, 0, 1, -1, 1, 0, 1, 0, 1, -1, 0, 0, -1, 0, -1, -1, 1, 0, 1, 0, 1, 1, -1, 0, 0, 0, 0, -1
COMMENTS
Completely additive modulo 3.
PROG
(PARI)
A002110(n) = prod(i=1, n, prime(i));
A373153(n) = { my(f = factor(n), u); u=sum(k=1, #f~, f[k, 2]* A002110(primepi(f[k, 1])-1))%3; if(2==u, -1, u); };
(PARI) A373153(n) = { my(u=(valuation(n, 2)-valuation(n, 3))%3); if(2==u, -1, u); }; \\ Antti Karttunen, Jun 01 2024
For all such terms k in A143293 (partial sums of primorials) for which A129251(k) = 0, the term A276085(k) is included here.
+20
7
2, 4, 2312, 3217644767340672907899084554132
COMMENTS
The next terms are quite big and can be found in the b-file. Note the nonmonotonic order, a(8) < a(5), a(6) and a(7).
PROG
(PARI)
A002110(n) = prod(i=1, n, prime(i));
A143293(n) = if(n==0, 1, my(P=1, s=1); forprime(p=2, prime(n), s+=P*=p); (s)); \\ From A143293.
A276085(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]* A002110(primepi(f[k, 1])-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; };
1, 3, 9, 6, 39, 18, 249, 9, 39, 78, 2559, 36, 32589, 498, 234, 18, 543099, 78, 10242789, 156, 1494, 5118, 233335659, 57, 996, 65178, 258, 996, 6703028889, 405, 207263519019, 42, 15354, 1086198, 6612, 156, 7628001653829, 20485578, 195534, 249, 311878265181039, 2559, 13394639596851069, 10236, 1245, 466671318, 628284422185342479
PROG
(PARI)
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1, 1]==2), f[1, 2] = 0); for (i=1, #f~, f[i, 1] = precprime(f[i, 1]-1)); factorback(f)};
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)); };
CROSSREFS
Cf. A002110, A064989, A108951, A276085, A276086, A324886, A319626, A346096, A346105, A346106, A346109.
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