A378738 -id:A378738 - cp's OEIS frontend

A378741 Terms k of A378738 for which A378664(k) obtains novel values.

66, 748, 1870, 3190, 3230, 3410, 3770, 4070, 4510, 4730, 5170, 5830, 52316, 75284, 90596, 152830, 182410, 202130, 211990, 216070, 225910, 231710, 236930, 257890, 261290, 270470, 290870, 300730, 310930, 314470, 330310, 378430, 395890, 434830, 449570, 460790, 473770, 476410, 489830, 512710, 517010, 523270, 530090

Offset: 1

Antti Karttunen, Dec 07 2024

nonn

a(n) is a multiple of A378740(n), by definition.

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

Cf. A378740 [= A378664(a(n))].

Subsequence of A378738, which is a subsequence of A091191.

\\ Uses program from A378738:
memoA378740 = Map();
k=0; n=0; while(k<20000, n++; if(is_A378738(n), t=A378664(n); if(!mapisdefined(memoA378740, t), mapput(memoA378740, t, n); k++; print1(n, ", "))));

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).

A378740 Distinct values of A378664(k) in the order of appearance, when k ranges over those primitively abundant numbers k for which A378664(k) is less than the largest proper divisor of k.

Original entry on oeis.org

6, 68, 170, 290, 646, 682, 754, 370, 410, 430, 470, 530, 4756, 6844, 8236, 30566, 10730, 11890, 12470, 43214, 45182, 46342, 47386, 15170, 15370, 54094, 17110, 17690, 62186, 62894, 19430, 75686, 79178, 39530, 89914, 41890, 43070, 95282, 97966, 46610, 103402, 47570, 106018, 107602, 107666, 48970, 109798, 111386, 51830

Offset: 1

Antti Karttunen, Dec 07 2024

nonn

Term 1 does not occur in this sequence, thus all terms are in A341614, and also in A246282. See A378658, A378662, A378664, A378736 and A337372 for a proof.

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

Cf. A246282, A337372, A378658, A378662, A378664, A378736, A378738, A378739, A378741 (corresponding numbers of A091191).

\\ Uses program from A378738:
memoA378740 = Map();
k=0; n=0; while(k<200, n++; if(is_A378738(n), t=A378664(n); if(!mapisdefined(memoA378740, t), mapput(memoA378740, t, n); k++; print1(t, ", "); write("b378740.txt", k, " ", t))));

a(n) = A378664(A378741(n)), a(n)| A378741(n).

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)}.

A378739 A378664(k) computed for those primitively abundant numbers k for which A378664(k) is less than the largest proper divisor of k.

Original entry on oeis.org

6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 68, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 170, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 290, 646, 6, 6, 6, 6, 682, 6

Offset: 1

Antti Karttunen, Dec 07 2024

nonn

Of the initial 20000 terms, 19803 are 6's.

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

Cf. A091191, A378664, A378738, A378740 (distinct terms in the order of appearance), A378741.

a(n) = A378664(A378738(n)).

A378742 Primitively abundant numbers k for which A378664(k) = 6, where A378664 is the greatest divisor d of n such that sigma(d) <= 2*d < A003961(d), or 1 if no such divisor exists.

Original entry on oeis.org

12, 66, 102, 174, 186, 222, 246, 258, 282, 318, 354, 366, 402, 426, 438, 474, 498, 534, 582, 606, 618, 642, 654, 678, 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

Apparently all the terms are of the form 6*p, where p is any prime except one of the 3, 5, 7, 13, 19, 23.

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

Cf. A378664.
After the initial term, a subsequence of A378738.
Subsequence of A008588 and of A091191.

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); };
is_A091191(n) = if(sigma(n)<=2*n, 0, fordiv(n,d,if(d2*d, return(0))); (1));
is_A378742(n) = (is_A091191(n) && (A378664(n)==6));

cp's OEIS Frontend

A378741 Terms k of A378738 for which A378664(k) obtains novel values.

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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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A378740 Distinct values of A378664(k) in the order of appearance, when k ranges over those primitively abundant numbers k for which A378664(k) is less than the largest proper divisor of k.

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

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A378739 A378664(k) computed for those primitively abundant numbers k for which A378664(k) is less than the largest proper divisor of k.

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A378742 Primitively abundant numbers k for which A378664(k) = 6, where A378664 is the greatest divisor d of n such that sigma(d) <= 2*d < A003961(d), or 1 if no such divisor exists.

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