A117366 -id:A117366 - cp's OEIS frontend

A331289 a(n) = A329348(n) - A001222(n).

1, 0, 0, 0, 0, 0, 0, -2, -1, 0, 0, 1, 0, 0, 4, -2, 0, -1, 0, 1, 4, 0, 0, -1, 0, 0, -2, 1, 0, 2, 0, -4, 4, 0, 6, 0, 0, 0, 4, -3, 0, -2, 0, 1, -2, 0, 0, -4, -1, 1, 4, 1, 0, -2, 2, 4, 4, 0, 0, -1, 0, 0, 0, -4, 11, 9, 0, 1, 4, 2, 0, -2, 0, 0, 2, 1, 0, 9, 0, -3, -3, 0, 0, -2, 9, 0, 4, 4, 0, -2, 4, 1, 4, 0, 5, -4, 0, -1, 7, -3, 0, 9, 0, 4, 1

Offset: 1

Antti Karttunen, Jan 15 2020

Cf. A001222, A117366, A329348, A329619, A331188, A331290.

A002110(n) = prod(i=1, n, prime(i));
A331289(n) = if(1==n, 1, my(f=factor(n), p=nextprime(1+vecmax(f[, 1]))); (prod(i=1, #f~, A002110(primepi(f[i, 1]))^(f[i, 2]-(#f~==i)))%p)-bigomega(n));

a(n) = A329348(n) - A001222(n).

a(n) = (A331188(n) mod A117366(n)) - A001222(n).

A331290 a(n) = gcd(A001222(n), A329348(n)).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Jan 15 2020

Records occur at n = 1, 4, 36, 112, 352, 1088, 2016, 2688, 8064, 63360, ...

Cf. A001222, A002110, A117366, A329348, A331188, A331289, A331291.

Cf. also A329618, A329621.

A002110(n) = prod(i=1, n, prime(i));
A331290(n) = if(1==n, 1, my(f=factor(n), p=nextprime(1+vecmax(f[, 1]))); gcd((prod(i=1, #f~, A002110(primepi(f[i, 1]))^(f[i, 2]-(#f~==i)))%p),bigomega(n)));

a(n) = gcd(A001222(n), A329348(n)) = gcd(A001222(n), A331188(n) mod A117366(n)).

a(n) = gcd(A001222(n), A331289(n)).

A117364 a(n) = largest prime less than the largest prime dividing n (or 1 if there is no such prime).

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 5, 1, 2, 3, 7, 2, 11, 5, 3, 1, 13, 2, 17, 3, 5, 7, 19, 2, 3, 11, 2, 5, 23, 3, 29, 1, 7, 13, 5, 2, 31, 17, 11, 3, 37, 5, 41, 7, 3, 19, 43, 2, 5, 3, 13, 11, 47, 2, 7, 5, 17, 23, 53, 3, 59, 29, 5, 1, 11, 7, 61, 13, 19, 5, 67, 2, 71, 31, 3, 17, 7, 11, 73, 3, 2, 37, 79, 5, 13, 41, 23

Offset: 1

Leroy Quet, Mar 10 2006

nonn

a(n) = 1 if and only if n is a power of 2 (including 1).
a(n/3) = 2 iff n/3 is A003586: 3-smooth numbers: numbers of the form 2^i*3^j with i, j >= 0.
a(n/5) = 3 iff n/5 is A051037: 5-smooth numbers: i.e. numbers whose prime divisors are all <= 5, etc.

			5 is the largest prime dividing 10. So a(10) is the largest prime < 5, which is 3.

Cf. A117365, A117366, A117367.

PrevPrime[n_] := Block[{k = n - 1}, While[ ! PrimeQ[k], k-- ]; k]; f[n_] := Block[{k = PrevPrime@ FactorInteger[Max[2, n]][[ -1, 1]]}, If[k > 1, k, 1]]; Array[f, 87] (* Robert G. Wilson v *)

More terms from Robert G. Wilson v, May 01 2006

A331291 a(n) = A001222(n) * A329348(n).

Original entry on oeis.org

1, 1, 1, 4, 1, 4, 1, 3, 2, 4, 1, 12, 1, 4, 12, 8, 1, 6, 1, 12, 12, 4, 1, 12, 4, 4, 3, 12, 1, 15, 1, 5, 12, 4, 16, 16, 1, 4, 12, 4, 1, 3, 1, 12, 3, 4, 1, 5, 2, 12, 12, 12, 1, 8, 8, 32, 12, 4, 1, 12, 1, 4, 9, 12, 26, 36, 1, 12, 12, 15, 1, 15, 1, 4, 15, 12, 4, 36, 1, 10, 4, 4, 1, 8, 22, 4, 12, 32, 1, 8, 12, 12, 12, 4

Offset: 1

Antti Karttunen, Jan 15 2020

Records occur at n = 1, 4, 12, 30, 35, 56, 66, 132, 145, 208, 276, 279, 469, ...

Cf. A001222, A117366, A329348, A331188, A331289, A331290.

A002110(n) = prod(i=1, n, prime(i));
A331291(n) = if(1==n, 1, my(f=factor(n), p=nextprime(1+vecmax(f[, 1]))); ((prod(i=1, #f~, A002110(primepi(f[i, 1]))^(f[i, 2]-(#f~==i)))%p)*bigomega(n)));

a(n) = A001222(n) * A329348(n) = A001222(n) * (A331188(n) mod A117366(n)).

a(p) = 1 for all primes p.

cp's OEIS Frontend

A331289 a(n) = A329348(n) - A001222(n).

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A331290 a(n) = gcd(A001222(n), A329348(n)).

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A117364 a(n) = largest prime less than the largest prime dividing n (or 1 if there is no such prime).

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A331291 a(n) = A001222(n) * A329348(n).

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