A353304 -id:A353304 - cp's OEIS frontend

A329609 Numbers k such that A156552(k) is a multiple of 3.

1, 4, 9, 10, 16, 21, 22, 25, 30, 34, 36, 39, 40, 46, 49, 55, 57, 62, 64, 66, 81, 82, 84, 85, 87, 88, 90, 91, 94, 100, 102, 105, 111, 115, 118, 120, 121, 129, 133, 134, 136, 138, 144, 146, 154, 155, 156, 159, 160, 166, 169, 183, 184, 186, 187, 189, 194, 195, 196, 198, 203, 205, 206, 213, 218, 220, 225, 228, 235, 237, 238, 246

Offset: 1

Antti Karttunen, Nov 21 2019

nonn

Not a multiplicative semigroup. For example, although 10 and 21 are present, 210 is missing. Compare to A332820. - Antti Karttunen, Jan 17 2023

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A000290 (subsequence), A156552, A329603, A329604, A332812.
Positions of zeros in A329903, of nonzeros in A341353, of ones in A353269 (characteristic function), A353418 (Dirichlet inverse of char.fun), A359836.
Sequence A332449 sorted into ascending order.
Cf. also A353303, A353304 and A332820.

A156552(n) = {my(f = factor(n), p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res}; \\ From A156552
isA329609(n) = !(A156552(n)%3);

A353303 Number of factorizations of n into factors k > 1 for which A156552(k) is a multiple of three.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Apr 10 2022

nonn

Number of factorizations of n into terms of A329609 that are larger than one.

			Divisors of 16 are [1, 2, 4, 8, 16]. When we apply A156552 to them, we obtain [0, 1, 3, 7, 15], of which only 0, 3 and 15 are multiples of three, therefore only factorizations 1*16 and 4*4 of 16 are counted, therefore a(16) = 2.
792 has 24 divisors in total, but only d = [1, 4, 9, 22, 36, 66, 88, 198, 264, 792] are such that A156552(d) is a multiple of 3. When using them, the following five factorizations are possible: 792 = 4*198 = 9*88 = 22*36 = 4*9*22, therefore a(792) = 5.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences computed from indices in prime factorization

Cf. A003961, A156552, A329609, A348717, A353269, A353304 [= a(n^2)].

A156552(n) = { my(f = factor(n), p, p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res };
A353269(n) = (!(A156552(n)%3));
A353303(n, m=n) = if(1==n, 1, my(s=0); fordiv(n, d, if((d>1)&&(d<=m)&&A353269(d), s += A353303(n/d, d))); (s));

a(n) = 0 iff A353269(n) = 0.

a(n) = a(A003961(n)) = a(A348717(n)), for all n >= 1.

A353362 Number of divisors d of n for which A156552(d) is a multiple of 3.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Apr 15 2022

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences computed from indices in prime factorization

Inverse Möbius transform of A353269.
Cf. A000005, A003961, A156552, A348717, A353303, A353304, A353361.
Cf. also A353352.
Differs from A353332 for the first time at n=30, where a(30) = 3, while A353332(30) = 2.

A156552(n) = { my(f = factor(n), p, p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res };
A353269(n) = (!(A156552(n)%3));
A353362(n) = sumdiv(n,d,A353269(d));

a(n) = Sum_{d|n} A353269(d).
a(n) = A000005(n) - A353361(n).
a(p) = 1 for all primes p.
a(n) = a(A003961(n)) = a(A348717(n)), for all n >= 1.

A353334 Number of factorizations of the square of n into factors k > 1 for which both A001222(k) and A056239(k) are even.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 3, 2, 3, 1, 4, 1, 2, 2, 5, 1, 4, 1, 6, 3, 3, 1, 7, 2, 2, 3, 4, 1, 7, 1, 7, 2, 3, 2, 9, 1, 2, 3, 12, 1, 7, 1, 6, 4, 3, 1, 12, 2, 6, 2, 4, 1, 7, 3, 7, 3, 2, 1, 17, 1, 3, 6, 11, 2, 7, 1, 6, 2, 7, 1, 16, 1, 2, 4, 4, 2, 7, 1, 21, 5, 3, 1, 16, 3, 2, 3, 12, 1, 16, 3, 6, 2, 3, 2, 19, 1, 4, 4, 16, 1, 7

Offset: 1

Antti Karttunen, Apr 14 2022

nonn

Number of factorizations of n^2 into terms of A340784 that are larger than one.

Antti Karttunen, Table of n, a(n) for n = 1..10080
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences computed from indices in prime factorization

Cf. A000290, A001222, A003961, A056239, A340784, A348717, A353331, A353333.

Differs from A353304 for the first time at n=30, where a(30) = 7, while A353304(30) = 8.

A056239(n) = { my(f); if(1==n, 0, f=factor(n); sum(i=1, #f~, f[i, 2] * primepi(f[i, 1]))); }
A353331(n) = ((!(bigomega(n)%2)) && (!(A056239(n)%2)));
A353333(n, m=n) = if(1==n, 1, my(s=0); fordiv(n, d, if((d>1) && (d<=m) && A353331(d), s += A353333(n/d, d))); (s));
A353334(n) = A353333(n^2);

a(n) = A353333(A000290(n)).
a(n) = a(A003961(n)) = a(A348717(n)), for all n >= 1.
a(p) = 1 for all primes p.

A353378 Number of ways to write the square of n as a product of the terms of A345452 larger than 1; a(1) = 1 by convention (an empty product).

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 4, 1, 2, 3, 5, 1, 4, 1, 4, 3, 2, 1, 7, 2, 2, 3, 4, 1, 7, 1, 7, 3, 2, 3, 9, 1, 2, 3, 7, 1, 7, 1, 4, 6, 2, 1, 12, 2, 4, 3, 4, 1, 7, 3, 7, 3, 2, 1, 16, 1, 2, 6, 11, 3, 7, 1, 4, 3, 7, 1, 16, 1, 2, 6, 4, 3, 7, 1, 12, 5, 2, 1, 16, 3, 2, 3, 7, 1, 17, 3, 4, 3, 2, 3, 19, 1, 4, 6, 9, 1, 7

Offset: 1

Antti Karttunen, Apr 17 2022

nonn

Number of factorizations of n^2 into factors k > 1 for which there is an even number of primes (when counted with multiplicity, A001222) in their prime factorization, and the 2-adic valuation of k (A007814) is also even.

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

Cf. A000290, A001222, A007814, A345452, A353374, A353377.

Cf. also A353304, A353334, A353338, A353359.

A353374(n) = (!(bigomega(n)%2) && !(valuation(n, 2)%2));
A353377(n, m=n) = if(1==n, 1, my(s=0); fordiv(n, d, if((d>1)&&(d<=m)&&A353374(d), s += A353377(n/d, d))); (s));
A353378(n) = A353377(n^2);

a(n) = A353377(A000290(n)).

For all n >= 1, a(n) <= A353338(n).

A353359 Number of ways to write the cube of n as a product of the terms of A332820 larger than 1; a(1) = 1 by convention (an empty product).

Original entry on oeis.org

1, 1, 1, 2, 1, 4, 1, 3, 2, 3, 1, 9, 1, 4, 4, 5, 1, 9, 1, 8, 3, 3, 1, 19, 2, 4, 3, 9, 1, 26, 1, 7, 4, 3, 4, 29, 1, 4, 3, 16, 1, 26, 1, 8, 9, 3, 1, 36, 2, 8, 4, 9, 1, 19, 3, 19, 3, 4, 1, 89, 1, 3, 8, 11, 4, 26, 1, 8, 4, 26, 1, 67, 1, 4, 9, 9, 4, 26, 1, 31, 5, 3, 1, 91, 3, 4, 3, 16, 1, 91, 3, 8, 4, 3, 4, 64, 1, 9, 9

Offset: 1

Antti Karttunen, Apr 16 2022

nonn

Number of factorizations of n^3 into factors k > 1 for which A048675(k) is a multiple of three.

Antti Karttunen, Table of n, a(n) for n = 1..16415
Index entries for sequences computed from indices in prime factorization

Cf. A000578, A003961, A048675, A332820, A348717, A353350, A353353.

Cf. also A353304, A353334, A353378.

A048675(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; };
A353350(n) = (0==(A048675(n)%3));
A353353(n, m=n) = if(1==n, 1, my(s=0); fordiv(n, d, if((d>1)&&(d<=m)&&A353350(d), s += A353353(n/d, d))); (s));
A353359(n) = A353353(n^3);

a(n) = A353353(A000578(n)).
a(n) = a(A003961(n)) = a(A348717(n)), for all n >= 1.
a(p) = 1 for all primes p.

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A329609 Numbers k such that A156552(k) is a multiple of 3.

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A353303 Number of factorizations of n into factors k > 1 for which A156552(k) is a multiple of three.

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A353362 Number of divisors d of n for which A156552(d) is a multiple of 3.

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A353334 Number of factorizations of the square of n into factors k > 1 for which both A001222(k) and A056239(k) are even.

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A353378 Number of ways to write the square of n as a product of the terms of A345452 larger than 1; a(1) = 1 by convention (an empty product).

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A353359 Number of ways to write the cube of n as a product of the terms of A332820 larger than 1; a(1) = 1 by convention (an empty product).

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