A374133 -id:A374133 - cp's OEIS frontend

A374131 Lexicographically earliest infinite sequence such that for all i, j >= 1, a(i) = a(j) => f(i) = f(j), where f(1) = 1, and for n > 1, f(n) = [A083345(n), A374132(n), A374133(n)], where A083345 is the numerator of the fully additive function with a(p) = 1/p, and A374132 and A374133 are the 2- and 3-adic valuations of A276085, which is fully additive with a(p) = p#/p.

1, 2, 3, 3, 4, 5, 4, 6, 7, 8, 4, 9, 4, 10, 11, 7, 4, 8, 4, 12, 13, 14, 4, 15, 16, 17, 4, 18, 4, 19, 4, 20, 21, 22, 23, 24, 4, 25, 26, 27, 4, 28, 4, 29, 30, 31, 4, 32, 16, 10, 33, 34, 4, 35, 36, 37, 38, 39, 4, 40, 4, 41, 42, 43, 44, 45, 4, 46, 47, 48, 4, 14, 4, 49, 50, 33, 51, 52, 4, 50, 53, 54, 4, 55, 56, 57, 58, 59, 4, 60, 61, 62, 63, 64, 65, 66, 4, 15, 67, 68

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) => A035263(i) = A035263(j),
  a(i) = a(j) => A369001(i) = A369001(j),
  a(i) = a(j) => A369004(i) = A369004(j),
  a(i) = a(j) => A372573(i) = A372573(j),
  a(i) = a(j) => A373137(i) = A373137(j),
  a(i) = a(j) => A373258(i) = A373258(j),
  a(i) = a(j) => A373483(i) = A373483(j).

Antti Karttunen, Table of n, a(n) for n = 1..100000

Cf. A083345, A374132, A374133.
Cf. A035263, A305900, A369001, A369004, A372573, A373137, A373258, A373483.
Cf. also A373151.

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; };
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))); };
Aux374131(n) = if(1==n, n, my(u=A276085(n)); [A083345(n), valuation(u, 2), valuation(u, 3)]);
v374131 = rgs_transform(vector(up_to, n, Aux374131(n)));
A374131(n) = v374131[n];

A374205 The 5-adic valuation of A328845(n), where A328845 is a Fibonacci-based variant of the arithmetic derivative.

Original entry on oeis.org

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

Offset: 2

Antti Karttunen, Jul 01 2024

nonn

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

Cf. A000045, A112765, A328845.

Cf. also A374133, A374213, A374202.

A374205[n_] := IntegerExponent[n*Total[MapApply[#2*Fibonacci[#]/# &, FactorInteger[n]]], 5];
Array[A374205, 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]));
A374205(n) = valuation(A328845(n), 5);

a(n) = A112765(A328845(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)).

A374132 The 2-adic valuation of A276085(n), where A276085 is the primorial base log-function.

Original entry on oeis.org

0, 1, 1, 1, 0, 1, 0, 2, 0, 1, 2, 1, 0, 3, 2, 1, 0, 1, 3, 5, 0, 1, 0, 2, 0, 1, 5, 1, 0, 1, 0, 2, 0, 2, 1, 1, 0, 3, 0, 1, 0, 1, 2, 1, 0, 1, 1, 2, 0, 4, 3, 1, 0, 3, 0, 4, 0, 1, 1, 1, 0, 1, 1, 2, 0, 1, 4, 2, 0, 1, 0, 1, 0, 1, 4, 4, 0, 1, 1, 3, 0, 1, 1, 2, 0, 3, 0, 1, 0, 2, 2, 4, 0, 2, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 3

Offset: 2

Antti Karttunen, Jun 30 2024

nonn

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

Cf. A007814, A276085.
Cf. A036554 (indices of 0's), A003159 (of nonzero terms), A373142 (of 1's), A373267 (of 2's), A369002 (of terms >= 2), A373138 (of terms >= 3).
Cf. also A374133.

A276085(n) = { my(f=factor(n)); sum(k=1, #f~, f[k, 2]*prod(i=1,primepi(f[k, 1]-1),prime(i))); };
A374132(n) = valuation(A276085(n),2);

a(n) = A007814(A276085(n)).

A374207 The 3-adic valuation of A113177(n), where A113177 is fully additive with a(p) = Fibonacci(p).

Original entry on oeis.org

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

Offset: 2

Antti Karttunen, Jul 01 2024

nonn

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

Cf. A007949, A113177, A374051, A374052 (indices of nonzero terms).

Cf. also A374133, A374213, A374203, A374206, A374208.

A113177(n) = if(n<=1, 0, my(f=factor(n)); sum(i=1, #f~, f[i, 2]*fibonacci(f[i,1])));
A374207(n) = valuation(A113177(n), 3);

a(n) = A007949(A113177(n)).

A374203 The 3-adic valuation of A328845(n), where A328845 is a Fibonacci-based variant of the arithmetic derivative.

Original entry on oeis.org

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

Offset: 2

Antti Karttunen, Jul 01 2024

nonn

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

Cf. A007949, A328845, A374121, A374122 (after its 2 initial terms, gives the indices of nonzero terms in this sequence).

Cf. also A374133, A374213, A374202, A374205.

A374203[n_] := IntegerExponent[n*Total[MapApply[#2*Fibonacci[#]/# &, FactorInteger[n]]], 3];
Array[A374203, 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]));
A374203(n) = valuation(A328845(n), 3);

a(n) = A007949(A328845(n)).

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A374205 The 5-adic valuation of A328845(n), where A328845 is a Fibonacci-based variant of the arithmetic derivative.

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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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A374132 The 2-adic valuation of A276085(n), where A276085 is the primorial base log-function.

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A374207 The 3-adic valuation of A113177(n), where A113177 is fully additive with a(p) = Fibonacci(p).

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A374203 The 3-adic valuation of A328845(n), where A328845 is a Fibonacci-based variant of the arithmetic derivative.

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