A374212 -id:A374212 - cp's OEIS frontend

A374211 Lexicographically earliest infinite sequence such that for all i, j >= 1, a(i) = a(j) => f(i) = f(j), with f(1) = 1, and for n > 1, f(n) = [A278226(A328768(n)), A374212(n), A374213(n)], where A328768 is the first primorial based variant of the arithmetic derivative, and A374212 and A374213 are its 2- and 3-adic valuations.

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

Offset: 1

Antti Karttunen, Jun 30 2024

nonn

Restricted growth sequence transform of the function f given in the definition.
For all i, j >= 1:
  A305900(i) = A305900(j) => a(i) = a(j),
  a(i) = a(j) => A152822(i) = A152822(j),
  a(i) = a(j) => A373982(i) = A373982(j) => A328771(i) = A328771(j),
  a(i) = a(j) => A373991(i) = A373991(j).

Antti Karttunen, Table of n, a(n) for n = 1..100000
Index entries for sequences related to primorial numbers

Cf. A002110, A278226, A328768, A374212, A374213.
Cf. A305900, A152822, A328771, A373982, A373991.
Cf. also A374131.

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; };
A002110(n) = prod(i=1,n,prime(i));
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A278226(n) = A046523(A276086(n));
A328768(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]*A002110(primepi(f[i,1])-1)/f[i, 1]));
Aux374211(n) = if(1==n, n, my(u=A328768(n)); [A278226(u), valuation(u, 2), valuation(u, 3)]);
v374211 = rgs_transform(vector(up_to, n, Aux374211(n)));
A374211(n) = v374211[n];

A328768 The first primorial based variant of arithmetic derivative: a(prime(i)) = A002110(i-1), where prime(i) = A000040(i), a(uv) = a(u)v + u*a(v), with a(0) = a(1) = 0.

Original entry on oeis.org

0, 0, 1, 2, 4, 6, 7, 30, 12, 12, 17, 210, 20, 2310, 67, 28, 32, 30030, 33, 510510, 44, 104, 431, 9699690, 52, 60, 4633, 54, 148, 223092870, 71, 6469693230, 80, 652, 60077, 192, 84, 200560490130, 1021039, 6956, 108, 7420738134810, 229, 304250263527210, 884, 114, 19399403, 13082761331670030, 128, 420, 145, 90124, 9292, 614889782588491410, 135, 1116, 324

Offset: 0

Antti Karttunen, Oct 28 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 0..1001
Index entries for sequences related to primorial numbers

Cf. A002110, A108951, A276085, A276150, A328771, A373982, A374211.
Cf. A042965 (indices of even terms), A016825 (of odd terms), A152822 (antiparity of terms), A373992 (indices of multiples of 3), A374212 (2-adic valuation), A374213 (3-adic valuation), A374123 [a(n) mod 360].
Cf. A374031 [gcd(a(n), A276085(n))], A374116 [gcd(a(n), A328845(n))].
For variants of the same formula, see A003415, A258851, A328769, A328845, A328846, A371192.

A002110(n) = prod(i=1,n,prime(i));
A328768(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]*A002110(primepi(f[i,1])-1)/f[i, 1]));

A328768(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]*A002110(primepi(f[i, 1]))/(f[i, 1]^2)));

a(n) = n * Sum e_j * A276085(p_j)/p_j for n = Product p_j^e_j, where for primes p, A276085(p) = A002110(A000720(p)-1).
a(n) = n * Sum e_j * (p_j)#/(p_j^2) for n = Product p_j^e_j with (p_j)# = A034386(p_j).
For all n >= 0, A276150(a(n)) = A328771(n).

A374213 The 3-adic valuation of A328768(n), where A328768 is the first primorial based variant of the arithmetic derivative.

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 0, 3, 0, 1, 0, 1, 0, 0, 0, 1, 1, 1, 0, 0, 3, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 3, 2, 4, 0, 0, 1, 0, 1, 0, 1, 1, 2, 0, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 2, 0, 1, 0, 3, 0, 1, 0, 1, 0, 0, 1, 1, 1, 2, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 0, 1, 4

Offset: 2

Antti Karttunen, Jun 30 2024

nonn

Antti Karttunen, Table of n, a(n) for n = 2..100000

Cf. A007949, A328768, A373991, A373992 (after its two initial terms, gives the indices of nonzero terms in this sequence).

Cf. also A374133, A374212.

A002110(n) = prod(i=1,n,prime(i));
A328768(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]*A002110(primepi(f[i,1])-1)/f[i, 1]));
A374213(n) = valuation(A328768(n), 3);

a(n) = A007949(A328768(n)).

A374202 The 2-adic valuation of A328845(n), where A328845 is a Fibonacci-based variant of the arithmetic derivative.

Original entry on oeis.org

0, 1, 2, 0, 0, 0, 2, 2, 0, 0, 2, 0, 0, 0, 5, 0, 0, 0, 3, 0, 0, 0, 2, 1, 0, 1, 4, 0, 0, 0, 4, 0, 0, 2, 2, 0, 0, 0, 2, 0, 0, 0, 4, 0, 0, 0, 7, 1, 0, 0, 3, 0, 0, 2, 2, 0, 0, 0, 5, 0, 0, 0, 6, 1, 0, 0, 3, 0, 0, 0, 2, 0, 0, 3, 5, 1, 0, 0, 4, 3, 0, 0, 3, 1, 0, 0, 2, 0, 0, 3, 5, 0, 0, 3, 4, 0, 0, 0, 2, 0, 0, 0, 2, 1, 0, 0, 2

Offset: 2

Antti Karttunen, Jul 01 2024

nonn

Antti Karttunen, Table of n, a(n) for n = 2..100000

Cf. A007814, A328845, A374045, A374046 (after its 2 initial terms, gives the indices of nonzero terms in this sequence), A374047 (indices of 0's).

Cf. also A374132, A374212, A374203, A374205.

A374202[n_] := IntegerExponent[n*Total[MapApply[#2*Fibonacci[#]/# &, FactorInteger[n]]], 2];
Array[A374202, 100, 2] (* Paolo Xausa, Dec 16 2024 *)

A328845(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]*fibonacci(f[i,1])/f[i, 1]));
A374202(n) = valuation(A328845(n), 2);

a(n) = A007814(A328845(n)).

A374206 The 2-adic valuation of A113177(n), where A113177 is fully additive with a(p) = Fibonacci(p).

Original entry on oeis.org

0, 1, 1, 0, 0, 0, 0, 2, 1, 0, 2, 0, 1, 0, 2, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 1, 0, 0, 3, 0, 0, 0, 1, 1, 1, 0, 1, 0, 3, 0, 4, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 1, 4, 0, 1, 0, 0, 0, 1, 0, 1, 1, 2, 0, 0, 0, 0, 0, 0, 0, 1, 2, 0, 1, 2, 0, 0, 3, 1, 0, 0, 1, 1, 0, 2, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 2, 0, 6, 0, 2, 2, 1, 0, 3

Offset: 2

Antti Karttunen, Jul 01 2024

nonn

Antti Karttunen, Table of n, a(n) for n = 2..100000

Cf. A007814, A113177, A373586 (gives the indices of nonzero terms here, after its initial 1), A373587 (gives the indices of 0's).

Cf. also A374132, A374212, A374202, A374207, A374208.

A113177(n) = if(n<=1, 0, my(f=factor(n)); sum(i=1, #f~, f[i, 2]*fibonacci(f[i,1])));
A374206(n) = valuation(A113177(n), 2);

a(n) = A007814(A113177(n)).

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A328768 The first primorial based variant of arithmetic derivative: a(prime(i)) = A002110(i-1), where prime(i) = A000040(i), a(u*v) = a(u)*v + u*a(v), with a(0) = a(1) = 0.

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A374213 The 3-adic valuation of A328768(n), where A328768 is the first primorial based variant of the arithmetic derivative.

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A374202 The 2-adic valuation of A328845(n), where A328845 is a Fibonacci-based variant of the arithmetic derivative.

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A374206 The 2-adic valuation of A113177(n), where A113177 is fully additive with a(p) = Fibonacci(p).

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A328768 The first primorial based variant of arithmetic derivative: a(prime(i)) = A002110(i-1), where prime(i) = A000040(i), a(uv) = a(u)v + u*a(v), with a(0) = a(1) = 0.