A086975 -id:A086975 - cp's OEIS frontend

A086974 Numbers of the form p^2 * q * r with distinct primes p, q and r such that p^2 < q*r.

60, 84, 90, 126, 132, 140, 156, 198, 204, 220, 228, 234, 260, 276, 306, 308, 315, 340, 342, 348, 364, 372, 380, 414, 444, 460, 476, 492, 495, 516, 522, 532, 558, 564, 572, 580, 585, 620, 636, 644, 650, 666, 693, 708, 732, 738, 740, 748, 765, 774, 804

Offset: 1

Reinhard Zumkeller, Jul 25 2003

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A002033, A086975.

q[n_] := Module[{f = SortBy[FactorInteger[n], Last]}, f[[;;, 2]] == {1, 1, 2} && f[[1, 1]] * f[[2, 1]] > f[[3, 1]]^2]; Select[Range[1000], q] (* Amiram Eldar, Sep 15 2024 *)

A002033(a(n)-1) = 44. - Juri-Stepan Gerasimov, Sep 26 2009

Incorrect comment removed by Charles R Greathouse IV, Mar 22 2010

A353423 For even n, a(n) = -Sum_{d|n, dA064989(n)), with a(1) = 1.

Original entry on oeis.org

1, -1, -1, 0, -1, -1, -1, 0, 0, -1, -1, -2, -1, -1, -1, 0, -1, 0, -1, -2, -1, -1, -1, -8, 0, -1, 0, -2, -1, -5, -1, 0, -1, -1, -1, 0, -1, -1, -1, -8, -1, -5, -1, -2, -2, -1, -1, -96, 0, 0, -1, -2, -1, 0, -1, -8, -1, -1, -1, -70, -1, -1, -2, 0, -1, -5, -1, -2, -1, -5, -1, 0, -1, -1, 0, -2, -1, -5, -1, -96, 0, -1, -1, -70

Offset: 1

Antti Karttunen, Apr 21 2022

sign

Apparently, for all i, j >= 1, A077462(i) = A077462(j) => a(i) = a(j).

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences computed from indices in prime factorization

Cf. A003961, A064989, A077462, A348717.
Cf. A070003 (positions of 0's), A167171 (positions of -1's), A096156 (positions of -2's), A007304 (positions of -5's), A086975 (positions of -70's), all these are so far conjectural. Also a subsequence of A178739 seems to give the positions of -96's.
Cf. also A353454, A353457, A353458, A353467 for similar recurrences.

A000265(n) = (n>>valuation(n,2));
A064989(n) = { my(f=factor(A000265(n))); for(i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f); };
memoA353423 = Map();
A353423(n) = if(1==n,1,my(v); if(mapisdefined(memoA353423,n,&v), v, if(!(n%2), v = -sumdiv(n,d,if(dA353423(n/2)*A353423(d),0)), v = A353423(A064989(n))); mapput(memoA353423,n,v); (v)));

a(p) = -1 for all primes p.

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

A369209 Numbers whose number of divisors has the largest prime factor 3.

Original entry on oeis.org

4, 9, 12, 18, 20, 25, 28, 32, 36, 44, 45, 49, 50, 52, 60, 63, 68, 72, 75, 76, 84, 90, 92, 96, 98, 99, 100, 108, 116, 117, 121, 124, 126, 132, 140, 147, 148, 150, 153, 156, 160, 164, 169, 171, 172, 175, 180, 188, 196, 198, 200, 204, 207, 212, 220, 224, 225, 228

Offset: 1

Amiram Eldar, Jan 16 2024

Subsequence of A059269 and first differs from it at n = 36: A059269(136) = 44 has 15 = 3 * 5 divisors and thus is not a term of this sequence.
Numbers k such that A000005(k) is in A065119.
Numbers k such that A071188(k) = 3.
Equals the complement of A354181, without the terms of A036537 (i.e., complement(A354181) \ A036537).
The asymptotic density of this sequence is Product_{p prime} (1-1/p) * (Sum_{k>=1} 1/p^(A003586(k)-1)) - A327839 = 0.26087647470200496716... .

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000005, A003586, A006530, A036537, A065119, A336595, A071188, A211337, A211338, A327839, A354181.
Subsequence of A013929 and A059269.
Subsequences: A001248, A030627, A050997, A054753, A062503, A067259, A079395, A085986, A085987, A086975, A095990, A096156, A138032, A162143, A179643, A179645.

gpf[n_] := FactorInteger[n][[-1, 1]]; Select[Range[300], gpf[DivisorSigma[0, #]] == 3 &]

gpf(n) = if(n == 1, 1, vecmax(factor(n)[, 1]));
is(n) = gpf(numdiv(n)) == 3;

cp's OEIS Frontend

A086974 Numbers of the form p^2 * q * r with distinct primes p, q and r such that p^2 < q*r.

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A353423 For even n, a(n) = -Sum_{d|n, dA064989(n)), with a(1) = 1.

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A369209 Numbers whose number of divisors has the largest prime factor 3.

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