A374115 -id:A374115

Search: a374115 -id:a374115

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A374112

a(n) = gcd(A113177(n), A276085(n)), where A113177 and A276085 are fully additive with a(p) = Fibonacci(p) and a(p) = p#/p, respectively.

+10
7

0, 1, 2, 2, 1, 3, 1, 3, 4, 1, 1, 4, 1, 1, 1, 4, 1, 5, 1, 1, 1, 1, 1, 5, 2, 1, 6, 1, 1, 1, 1, 5, 1, 1, 18, 6, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 6, 2, 1, 1, 1, 1, 7, 2, 1, 1, 1, 1, 1, 1, 1, 17, 6, 2, 1, 1, 1, 1, 1, 1, 7, 1, 1, 2, 1, 6, 1, 1, 1, 8, 1, 1, 17, 6, 1, 1, 1, 1, 1, 6, 1, 1, 1, 2, 7, 1, 1, 1, 2, 1, 1, 1, 1, 2

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..65537

PROG

(PARI)

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

A374112(n) = gcd(A113177(n), A276085(n));

CROSSREFS

Cf. A113177, A276085, A374113, A374114 (indices of even terms), A374115 (of odd terms).

Cf. also A374116.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Jun 29 2024

STATUS

approved

A374114

Numbers k such that A113177(k) and A276085(k) are both even, where A113177 and A276085 are fully additive with a(p) = Fibonacci(p) and a(p) = p#/p, respectively.

+10
6

1, 3, 4, 9, 12, 16, 25, 27, 35, 36, 48, 49, 55, 64, 65, 75, 77, 81, 85, 91, 95, 100, 105, 108, 115, 119, 121, 133, 140, 143, 144, 145, 147, 155, 161, 165, 169, 185, 187, 192, 195, 196, 203, 205, 209, 215, 217, 220, 221, 225, 231, 235, 243, 247, 253, 255, 256, 259, 260, 265, 273, 285, 287, 289, 295, 299, 300, 301, 305

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Numbers whose 2-adic valuation (A007814) is even, and the number of the prime factors (with multiplicity, A001222) and the 3-adic valuation (A007949) have the same parity.

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..12000

PROG

(PARI) isA374114 = A374113;

CROSSREFS

Intersection of A003159 and A373586.

Indices of even terms in A374112.

Cf. A001222, A007814, A007949, A113177, A276085, A374113 (characteristic function), A374115 (complement).

KEYWORD

nonn

AUTHOR

Antti Karttunen, Jun 29 2024

STATUS

approved

A374113

a(n) = 1 if A113177(n) and A276085(n) are both even, otherwise 0, where A113177 and A276085 are fully additive with a(p) = Fibonacci(p) and a(p) = p#/p, respectively.

+10
5

1, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1

(list; graph; refs; listen; history; text; internal format)

OFFSET

COMMENTS

a(n) = 1 if the 2-adic valuation of n is even, and the number of its prime factors (with multiplicity, A001222) and its 3-adic valuation (A007949) have the same parity, otherwise 0.

LINKS

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

Index entries for characteristic functions

FORMULA

a(n) = A035263(n) * A373585(n).

a(n) = A059841(A374112(n)).

PROG

(PARI)

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

A374113(n) = (!(A113177(n)%2) && !(A276085(n)%2));

(PARI) A374113(n) = (!(valuation(n, 2)%2) && !((bigomega(n)-valuation(n, 3))%2));

CROSSREFS

Characteristic function of A374114, whose complement A374115 gives the indices of 0's.

Cf. A001222, A007949, A035263, A113177, A059841, A276085, A373585, A374112.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Jun 29 2024

STATUS

approved

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