A329605 -id:A329605 - cp's OEIS frontend

A329614 Smallest prime factor of the number of divisors of A108951(n).

1, 2, 2, 3, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 5, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 7, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 5, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2

Offset: 1

Antti Karttunen, Nov 17 2019

nonn

Differs from A071187 for the first time at n=324, where a(324) = 5, while A071187(324) = 3. The positions of the differences are listed at A329613.

			324 = 18^2 = 2^2 * 3^4, thus A108951(324) = 2^2 * (2*3)^4 = 2^6 * 3^4 = 5184, which has (6+1)*(4+1) = 7 * 5 = 35 divisors, thus a(324) = A020639(35) = 5.

Cf. A020639, A034386, A071187, A108951, A329605, A329608, A329612, A329613.

Array[FactorInteger[DivisorSigma[0, #]][[1, 1]] &@ Apply[Times, Map[#1^#2 & @@ # &, FactorInteger[#] /. {p_, e_} /; e > 0 :> {Times @@ Prime@ Range@ PrimePi@ p, e}]] &, 105] (* Michael De Vlieger, Nov 18 2019 *)

A034386(n) = prod(i=1, primepi(n), prime(i));
A108951(n) = { my(f=factor(n)); prod(i=1, #f~, A034386(f[i, 1])^f[i, 2]) };  \\ From A108951
A071187(n) = if(1==n, n, my(f = factor(numdiv(n))); vecmin(f[, 1]));
A329614(n) = A071187(A108951(n));

a(n) = A071187(A108951(n)).

a(n) = A020639(A329605(n)).

A329378 Least common multiple of exponents of prime factors of A108951(n), where A108951 is fully multiplicative with a(prime(i)) = prime(i)# = Product_{i=1..i} A000040(i).

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 3, 1, 2, 2, 4, 1, 6, 1, 3, 2, 2, 1, 4, 2, 2, 3, 3, 1, 6, 1, 5, 2, 2, 2, 4, 1, 2, 2, 4, 1, 6, 1, 3, 3, 2, 1, 5, 2, 6, 2, 3, 1, 12, 2, 4, 2, 2, 1, 4, 1, 2, 3, 6, 2, 6, 1, 3, 2, 6, 1, 10, 1, 2, 6, 3, 2, 6, 1, 5, 4, 2, 1, 4, 2, 2, 2, 4, 1, 12, 2, 3, 2, 2, 2, 6, 1, 6, 3, 4, 1, 6, 1, 4, 6

Offset: 1

Antti Karttunen, Nov 17 2019

nonn

Cf. A019565, A034386, A072411, A108951, A284002, A329382, A329600, A329605.

Differs from related A329617 for the first time at n=36.

A034386(n) = prod(i=1, primepi(n), prime(i));
A072411(n) = lcm(factor(n)[, 2]); \\ From A072411
A108951(n) = { my(f=factor(n)); prod(i=1, #f~, A034386(f[i, 1])^f[i, 2]) };  \\ From A108951
A329378(n) = A072411(A108951(n));

a(n) = A072411(A108951(n)) = A072411(A329600(n)).
a(n) <= A329617(n) <= A329382(n) <= A329605(n).
a(A019565(n)) = A284002(n).

A329617 Product of distinct exponents of prime factors of A108951(n), where A108951 is fully multiplicative with a(prime(i)) = prime(i)# = Product_{i=1..i} A000040(i).

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 3, 1, 2, 2, 4, 1, 6, 1, 3, 2, 2, 1, 4, 2, 2, 3, 3, 1, 6, 1, 5, 2, 2, 2, 8, 1, 2, 2, 4, 1, 6, 1, 3, 3, 2, 1, 5, 2, 6, 2, 3, 1, 12, 2, 4, 2, 2, 1, 8, 1, 2, 3, 6, 2, 6, 1, 3, 2, 6, 1, 10, 1, 2, 6, 3, 2, 6, 1, 5, 4, 2, 1, 8, 2, 2, 2, 4, 1, 12, 2, 3, 2, 2, 2, 6, 1, 6, 3, 8, 1, 6, 1, 4, 6

Offset: 1

Antti Karttunen, Nov 17 2019

nonn

Cf. A034386, A108951, A290107, A329600, A329605.

Differs from related A329378 for the first time at n=36. See also A329382.

A034386(n) = prod(i=1, primepi(n), prime(i));
A108951(n) = { my(f=factor(n)); prod(i=1, #f~, A034386(f[i, 1])^f[i, 2]) };  \\ From A108951
A290107(n) = factorback(vecsort((factor(n)[, 2]), , 8));
A329617(n) = A290107(A108951(n));

a(n) = A290107(A108951(n)) = A290107(A329600(n)).

A329378(n) <= a(n) <= A329382(n) <= A329605(n).

A329612 a(n) = gcd(d(n), d(A108951(n))), where d(n) gives the number of divisors of n, A000005(n), and A108951 is fully multiplicative with a(prime(i)) = prime(i)# = prime(1) * ... * prime(i).

Original entry on oeis.org

1, 2, 2, 3, 2, 2, 2, 4, 3, 4, 2, 2, 2, 4, 2, 5, 2, 6, 2, 2, 4, 4, 2, 2, 3, 4, 4, 2, 2, 8, 2, 6, 4, 4, 2, 3, 2, 4, 4, 4, 2, 8, 2, 2, 2, 4, 2, 2, 3, 6, 4, 2, 2, 4, 4, 8, 4, 4, 2, 6, 2, 4, 2, 7, 4, 8, 2, 2, 4, 8, 2, 6, 2, 4, 6, 2, 2, 8, 2, 2, 5, 4, 2, 12, 4, 4, 4, 8, 2, 4, 4, 2, 4, 4, 4, 2, 2, 6, 2, 9, 2, 8, 2, 8, 8

Offset: 1

Antti Karttunen, Nov 18 2019

nonn

Cf. A000005, A034386, A108951, A329614.

A034386(n) = prod(i=1, primepi(n), prime(i));
A108951(n) = { my(f=factor(n)); prod(i=1, #f~, A034386(f[i, 1])^f[i, 2]) };  \\ From A108951
A329612(n) = gcd(numdiv(n),numdiv(A108951(n)));

a(n) = gcd(A000005(n),A329605(n)) = gcd(A000005(n),A000005(A108951(n))).

A331285 a(n) is the index of the first occurrence of n in A331284.

Original entry on oeis.org

1, 8, 27, 108, 180, 396, 672, 1056, 1372, 1760, 2352, 3087, 3696, 3744, 4896, 5733, 7497, 9724, 11907, 13600, 15200, 18513, 19773, 23940, 24752, 28917, 32319, 33534, 42282, 45472, 47500, 52668, 55890, 59976, 66048, 74240, 77792, 81144, 86944, 100035, 105248, 109368, 122825, 127908, 134368, 144648, 156325, 168948, 175770

Offset: 1

Antti Karttunen, Jan 14 2020

nonn

Also positions of records in A331284.
a(n) is the least k such that in range 1 .. k of A329605 (equally: in A329606[1..k]) there can be found exactly n occurrences of some term. In A329605 these "champion terms" are A329605(a(n)): 1, 4, 16, 24, 48, 192, 96, 192, 384, 288, 288, 576, 576, 576, 1152, 2304, 4608, 9216, 576, 2304, ..., that appear all to be 3-smooth numbers (in A003586).
For example, a(5)=180 and 48 is the first term in A329605 to occur for five times, as A329605(22) = A329605(42) = A329605(75) = A329605(112) = A329605(180) = 48.

Cf. A331284 (a left inverse).

Cf. A003586, A329605, A329606.

A331284(a(n)) = n for all n >= 1.

cp's OEIS Frontend

A329614 Smallest prime factor of the number of divisors of A108951(n).

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A329378 Least common multiple of exponents of prime factors of A108951(n), where A108951 is fully multiplicative with a(prime(i)) = prime(i)# = Product_{i=1..i} A000040(i).

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A329617 Product of distinct exponents of prime factors of A108951(n), where A108951 is fully multiplicative with a(prime(i)) = prime(i)# = Product_{i=1..i} A000040(i).

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A329612 a(n) = gcd(d(n), d(A108951(n))), where d(n) gives the number of divisors of n, A000005(n), and A108951 is fully multiplicative with a(prime(i)) = prime(i)# = prime(1) * ... * prime(i).

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A331285 a(n) is the index of the first occurrence of n in A331284.

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