A378665 -id:A378665 - cp's OEIS frontend

A378738 Primitively abundant numbers k for which A378665(k) > A378664(k).

66, 102, 174, 186, 222, 246, 258, 282, 318, 354, 366, 402, 426, 438, 474, 498, 534, 582, 606, 618, 642, 654, 678, 748, 762, 786, 822, 834, 894, 906, 942, 978, 1002, 1038, 1074, 1086, 1146, 1158, 1182, 1194, 1266, 1338, 1362, 1374, 1398, 1434, 1446, 1506, 1542, 1578, 1614, 1626, 1662, 1686, 1698, 1758, 1842, 1866

Offset: 1

Antti Karttunen, Dec 07 2024

nonn

Subsequence of A378737: 1496 is its first term that does not occur here.
Equal to primitively abundant numbers k such that A032742(k) > A378664(k), because for primitively abundant numbers the greatest non-abundant divisor is the largest proper divisor, A378665(k) = A032742(k).
Question: What is the asymptotic density of these numbers among A091191? Does it tend to 1?
Conjecture: A001222(a(n)) = 3 <=> 3|a(n).

			Examples given in A378737 for 66, 748, 1866, and 1870 all work also here, because those four numbers are all in A091191.

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

Intersection of A091191 and A378737.

Cf. A001222, A032742, A294930, A337372, A341612, A341614, A378664, A378665, A378735, A378736, A378739 [= A378664(a(n))], A378741 (subsequence), A378742 (subsequence after its initial term).

A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A341612(n) = ((sigma(n)<=(2*n))&&((2*n)<A003961(n)));
A378664(n) = { fordiv(n,d,if(A341612(n/d), return(n/d))); (1); };
A032742(n) = if(1==n,n,n/vecmin(factor(n)[,1]));
is_A091191(n) = if(sigma(n)<=2*n, 0, fordiv(n,d,if(d2*d, return(0))); (1));
is_A378738(n) = (is_A091191(n) && (A378664(n)!=A032742(n)));

{k such that A294930(k) = 1 and A032742(k) > A378664(k)}.

A378737 Abundant numbers k for which A378665(k) > A378664(k).

Original entry on oeis.org

66, 102, 174, 186, 222, 246, 258, 282, 318, 354, 366, 402, 426, 438, 474, 498, 534, 582, 606, 618, 642, 654, 678, 748, 762, 786, 822, 834, 894, 906, 942, 978, 1002, 1038, 1074, 1086, 1146, 1158, 1182, 1194, 1266, 1338, 1362, 1374, 1398, 1434, 1446, 1496, 1506, 1542, 1578, 1614, 1626, 1662, 1686, 1698, 1758, 1842

Offset: 1

Antti Karttunen, Dec 06 2024

nonn

For most of these numbers k, A378664(k) = 6. Note that A003961(6) = A003961(2*3) = 3*5 = 15 > 2*6, while sigma(6) = 12, making 6 non-abundant. Note that there seems to be only a finite amount (namely 13, see A337372) of such "stopper semiprimes" that would prevent of A378736 obtaining value 1.

Other possible values that A378664 obtains on these numbers are for example 68, 136, 170, 256, 290, 370, 410, 430, 470, 530, 646, 682, 754, 1276, 1292, 1364, 1508, 1628, 1804, 1892, 2068, 2332, 4756, 6844, 10846, 15334, etc. See A378740, which contains some of these.

			66 is a term as A378665(66) = 33, but A378664(66) = 6.
748 is a term as A378665(748) = 374, but A378664(748) = 68.
1866 is a term as A378665(1866) = 933, but A378664(1866) = 6.
1870 is a term as A378665(1870) = 935, but A378664(1870) = 170.

Subsequence of A005101.
Cf. A378738 (subsequence).
Cf. A337372, A378664, A378665, A378735, A378736, A378740.

A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A294935(n) = (sigma(n)<=(2*n));
A341612(n) = ((sigma(n)<=(2*n))&&((2*n)<A003961(n)));
A378664(n) = { fordiv(n,d,if(A341612(n/d), return(n/d))); (1); };
A378665(n) = { fordiv(n,d,if(A294935(n/d), return(n/d))); (1); };
is_A378737(n) = (!A294935(n) && (A378664(n)!=A378665(n)));

{A005101(i) for such indices i where A378735(i) > A378736(i)}.

A378664 Greatest divisor d of n such that sigma(d) <= 2*d < A003961(d), or 1 if no such divisor exists, where A003961 is fully multiplicative with a(p) = nextprime(p).

Original entry on oeis.org

1, 1, 1, 4, 1, 6, 1, 8, 9, 10, 1, 6, 1, 14, 15, 16, 1, 9, 1, 10, 21, 1, 1, 8, 1, 1, 27, 28, 1, 15, 1, 32, 1, 1, 35, 9, 1, 1, 39, 10, 1, 21, 1, 44, 45, 1, 1, 16, 49, 50, 1, 52, 1, 27, 1, 28, 57, 1, 1, 15, 1, 1, 63, 64, 1, 6, 1, 68, 69, 35, 1, 9, 1, 1, 75, 76, 1, 39, 1, 16, 81, 1, 1, 28, 1, 1, 1, 44, 1, 45, 91, 92

Offset: 1

Antti Karttunen, Dec 06 2024

nonn

Largest term of {1} U A341614 that divides n.

Cf. A000203, A003961, A337372, A341612, A341614, A378662, A378665, A378736 [= a(A005101(n))], A378737, A378738, A378739, A378740, A378741, A378742.
Positions of fixed points (where a(n)=n) is given by {1} U A341614.
Cf. A246281 (positions of 1's), A246282 (of terms > 0), A005101 (of terms that are neither 1 nor fixed points).

Table[If[Length[#] == 0, 1, Max[#]] &@ Select[Divisors[n], DivisorSigma[1, #] <= 2 # < (Times @@ Map[Power @@ # &, FactorInteger[#] /. {p_, e_} /; e > 0 :> {Prime[PrimePi[p] + 1], e}] - Boole[# == 1]) &], {n, 92}] (* Michael De Vlieger, Dec 06 2024 *)

A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A341612(n) = ((sigma(n)<=(2*n))&&((2*n)<A003961(n)));
A378664(n) = { fordiv(n,d,if(A341612(n/d), return(n/d))); (1); };

a(n) <= A378665(n).

A378735 Greatest non-abundant divisor of the n-th abundant number.

Original entry on oeis.org

6, 9, 10, 8, 15, 9, 10, 21, 16, 27, 28, 15, 33, 35, 9, 39, 16, 28, 44, 45, 32, 50, 51, 52, 27, 28, 57, 15, 63, 44, 69, 35, 16, 75, 52, 32, 81, 28, 87, 44, 45, 93, 64, 98, 99, 50, 68, 52, 105, 27, 110, 111, 32, 76, 117, 16, 123, 63, 129, 130, 44, 135, 136, 92, 35, 141, 32, 147, 75, 152, 153, 154, 52, 159, 64, 81, 165

Offset: 1

Antti Karttunen, Dec 06 2024

nonn

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

Cf. A005101, A294935, A378665.

Cf. also A378736 and the points of differences A378737.

A294935(n) = (sigma(n)<=(2*n));
A378665(n) = { fordiv(n,d,if(A294935(n/d), return(n/d))); (1); };
k=0; n=0; while(k<105, n++; if(!A294935(n), k++; print1(A378665(n),", ")));

a(n) = A378665(A005101(n)).

a(n) >= A378736(n).

A378660 Smallest divisor d of n for which n/d is non-abundant, i.e. sigma(n/d) <= 2*(n/d).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 3, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 4, 1, 1, 1, 4, 1, 2, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 4, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 8, 1, 1, 1, 1, 1, 2, 1, 5, 1, 1, 1, 3, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 3, 1, 1, 1, 2, 1, 2, 1, 2, 1

Offset: 1

Antti Karttunen, Dec 06 2024

nonn

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

Cf. A000203, A263837, A294935, A378665.

A294935(n) = (sigma(n)<=(2*n));
A378660(n) = { fordiv(n,d,if(A294935(n/d), return(d))); (n); };

a(n) = n / A378665(n).

cp's OEIS Frontend

A378738 Primitively abundant numbers k for which A378665(k) > A378664(k).

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A378737 Abundant numbers k for which A378665(k) > A378664(k).

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A378664 Greatest divisor d of n such that sigma(d) <= 2*d < A003961(d), or 1 if no such divisor exists, where A003961 is fully multiplicative with a(p) = nextprime(p).

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A378735 Greatest non-abundant divisor of the n-th abundant number.

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A378660 Smallest divisor d of n for which n/d is non-abundant, i.e. sigma(n/d) <= 2*(n/d).

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