A378662 -id:A378662 - cp's OEIS frontend

A337372 Primitively primeshift-abundant numbers: Numbers that are included in A246282 (k with A003961(k) > 2k), but none of whose proper divisors are.

4, 6, 9, 10, 14, 15, 21, 35, 39, 49, 57, 69, 91, 125, 242, 275, 286, 325, 338, 363, 418, 425, 442, 475, 494, 506, 561, 575, 598, 646, 682, 715, 722, 725, 754, 775, 782, 806, 845, 847, 867, 874, 925, 957, 962, 1023, 1025, 1045, 1054, 1058, 1066, 1075, 1105, 1118, 1175, 1178, 1221, 1222, 1235, 1265, 1309, 1325, 1334, 1353

Offset: 1

Antti Karttunen, Aug 27 2020

nonn

Numbers k whose only divisor in A246282 is k itself, i.e., A003961(k) > 2k, but for none of the proper divisors d|k, dA003961(d) > 2d.
Question: Do the odd terms in A326134 all occur here? Answer is yes, if the following conjecture holds: This is a subsequence of A263837, nonabundant numbers. In other words, we claim that any abundant number k (A005101) has A337345(k) > 1 and thus is a term of A341610. (The conjecture indeed holds. See the proof below).
From Antti Karttunen, Dec 06 2024: (Start)
Observation 1: The thirteen initial terms (4, 6, 9, ..., 69, 91) are only semiprimes in A246282, all other semiprimes being in A246281 (but none in A341610), and there seems to be only 678 terms m with A001222(m) = 3, from a(14) = 125 to the last one of them, a(2691) = 519963. There are more than 150000 terms m with A001222(m) = 4. In general, there should be only a finite number of terms m for any given k = A001222(m). Compare for example with A287728.
Observation 2: The intersection with A005101 (and thus also with A091191) is empty, which then implies the claims made in the sequences A378662, A378664, from which further follows that there are no 1's present in any of these sequences: A378658, A378736, A378740.
(End)
Proof of the latter observation by Jianing Song, Dec 11 2024: (Start)
Let's write p' for the next prime after the prime p. Also, write Q(n) = A003961(n)/sigma(n) which is multiplicative.
Proposition: For n > 1 not being a prime nor twice a prime, n has a factor p such that Q(n) > p'/p.
This implies that if n is abundant [including any primitively abundant n in A091191], then n has a factor p such that A003961(n/p)/(n/p) = (A003961(n)/n)/(p'/p) > sigma(n)/n [which is > 2 because n is abundant], so n/p is in A246282, meaning that n cannot be in this sequence.
Proof. We see that 1 <= Q(p) <= Q(p^2) <= ..., which implies that if n verifies the proposition, then every multiple of n also verifies it. Since n = p^2 > 4 and n = 8 verify the proposition, it suffices to consider the case where n = pq is the product of two distinct odd primes. Suppose WLOG that p < q, so q >= p', then using q/(q+1) >= p'/(p'+1) we have
Q(n) = p'q'/((p+1)(q+1)) >= p'^2*q'/(q(p+1)(p'+1)) > (p'^2-1)*q'/(q(p+1)(p'+1)) = (p'-1)/(p+1) * q'/q >= q'/q.
(End)

			14 = 2*7 is in the sequence as setting every prime to the next larger prime gives 3*11 = 33 > 28 = 2*14. Doing so for any proper divisor d of 14 gives a number < 2 * d. - _David A. Corneth_, Dec 07 2024

Antti Karttunen, Table of n, a(n) for n = 1..20000
David A. Corneth, PARI programs
Index entries for sequences related to prime indices in the factorization of n.
Index entries for primes, gaps between.

Setwise difference A246282 \ A341610.
Cf. A000101, A003961, A246281, A252742, A337346, A378745, A378746.
Cf. A378658, A378662, A378664, A378736, A378740.
Positions of ones in A337345 and in A341609 (characteristic function).
Subsequence of A263837 and thus also of A341614.
Cf. also A005101, A091191, A326134.
Cf. also A337543.

Block[{a = {}, b = {}}, Do[If[2 i < Times @@ Map[#1^#2 & @@ # &, FactorInteger[i] /. {p_, e_} /; e > 0 :> {Prime[PrimePi@ p + 1], e}] - Boole[i == 1], AppendTo[a, i]; If[IntersectingQ[Most@ Divisors[i], a], AppendTo[b, i]]], {i, 1400}]; Complement[a, b]] (* Michael De Vlieger, Feb 22 2021 *)

A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A252742(n) = (A003961(n) > (2*n));
A337346(n) = sumdiv(n,d,(dA252742(d));
isA337372(n) = ((1==A252742(n))&&(0==A337346(n)));

PARI
```
is_A337372 = A341609;
```
PARI
```
\\ See Corneth link
```

{k: 1==A337345(k)}.

A337345 Number of divisors d of n for which A003961(d) > 2*d, where A003961 is fully multiplicative with a(p) = nextprime(p).

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 0, 2, 1, 1, 0, 3, 0, 1, 1, 3, 0, 3, 0, 3, 1, 0, 0, 5, 0, 0, 2, 3, 0, 4, 0, 4, 0, 0, 1, 6, 0, 0, 1, 5, 0, 4, 0, 2, 3, 0, 0, 7, 1, 2, 0, 2, 0, 5, 0, 5, 1, 0, 0, 8, 0, 0, 3, 5, 0, 2, 0, 2, 1, 4, 0, 9, 0, 0, 2, 2, 0, 3, 0, 7, 3, 0, 0, 8, 0, 0, 0, 4, 0, 8, 1, 2, 0, 0, 0, 9, 0, 3, 2, 5, 0, 2, 0, 4, 4

Offset: 1

Antti Karttunen, Aug 27 2020

nonn

Number of terms of A246282 that divide n.

Number of divisors d of n for which A048673(d) > d.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences computed from indices in prime factorization

Inverse Möbius transform of A252742.
Cf. A003961, A048673, A246282, A337346, A337372 (positions of ones), A341609 (their characteristic function), A341610 (positions of terms > 1), A378658 [= a(A091191(n))], A378662, A378663.
Cf. also A080224, A337541, A341620.

A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A337345(n) = sumdiv(n,d,A003961(d)>(d+d));

a(n) = Sum_{d|n} A252742(d).
a(n) = A337346(n) + A252742(n).
From Antti Karttunen, Dec 10 2024: (Start)
a(n) = 1 <=> A341609(n) = 1.
a(n) = A378662(n) + A080224(n) = A378663(n) + A341620(n).
(End)

A341614 Numbers k such that sigma(k) <= 2k < A003961(k).

Original entry on oeis.org

4, 6, 8, 9, 10, 14, 15, 16, 21, 27, 28, 32, 35, 39, 44, 45, 49, 50, 52, 57, 63, 64, 68, 69, 75, 76, 81, 91, 92, 98, 99, 105, 110, 116, 117, 124, 125, 128, 130, 135, 136, 147, 148, 152, 153, 154, 164, 165, 170, 171, 172, 175, 182, 184, 188, 189, 190, 195, 207, 212, 225, 230, 231, 232, 236, 238, 242, 243, 244, 245, 248, 250, 255, 256

Offset: 1

Antti Karttunen, Feb 22 2021

Index entries for sequences where odd perfect numbers must occur, if they exist at all

Intersection of A263837 and A246282 (nonabundant numbers in A246282).
Union of A000396 and A341615.
Union of A337372 and A341611 (see also A341610).
Cf. A341612 (characteristic function), A326134 (a subsequence).
Cf. also A378662, A378664.

Select[Range[256], DivisorSigma[1, #] <= 2 # <
Times @@ Map[#1^#2 & @@ # &, FactorInteger[#] /. {p_, e_} /; e > 0 :> {Prime[PrimePi@ p + 1], e}] &] (* Michael De Vlieger, Feb 22 2021 *)

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).

A378736 Greatest divisor d of the n-th abundant number such that sigma(d) <= 2*d < A003961(d).

Original entry on oeis.org

6, 9, 10, 8, 15, 9, 10, 21, 16, 27, 28, 15, 6, 35, 9, 39, 16, 28, 44, 45, 32, 50, 6, 52, 27, 28, 57, 15, 63, 44, 69, 35, 16, 75, 52, 32, 81, 28, 6, 44, 45, 6, 64, 98, 99, 50, 68, 52, 105, 27, 110, 6, 32, 76, 117, 16, 6, 63, 6, 130, 44, 135, 136, 92, 35, 6, 32, 147, 75, 152, 153, 154, 52, 6, 64, 81, 165, 28, 170, 171

Offset: 1

Antti Karttunen, Dec 06 2024

nonn

There are no 1's in this sequence. See A378662, A378664 and A337372 for a proof.

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

Cf. A003961, A005101, A294935, A341612, A378664, A378735, A378737.

Cf. also A337372.

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); };
k=0; n=0; while(k<20000, n++; if(!A294935(n), k++; print1(A378664(n),", ")));

a(n) = A378664(A005101(n)).

a(n) <= A378735(n).

Unnecessary escape-clause removed from the definition by Antti Karttunen, Dec 12 2024

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).

A378658 a(n) = A337345(A091191(n)), where A337345 is the number of divisors d of n for which A003961(d) > 2*d, and A091191 lists the primitive abundant numbers.

Original entry on oeis.org

3, 3, 3, 4, 4, 5, 2, 4, 3, 4, 2, 4, 3, 3, 2, 2, 6, 2, 2, 2, 6, 2, 6, 6, 2, 2, 7, 2, 6, 2, 2, 2, 6, 2, 6, 2, 6, 2, 5, 5, 2, 2, 2, 2, 6, 5, 2, 2, 4, 2, 2, 6, 2, 2, 5, 6, 2, 2, 2, 12, 2, 8, 2, 6, 2, 2, 2, 2, 6, 2, 2, 8, 2, 6, 2, 8, 6, 2, 2, 2, 8, 2, 6, 2, 2, 6, 8, 2, 2, 13, 2, 2, 2, 6, 2, 2, 8, 2, 6, 2, 2, 2, 4, 6

Offset: 1

Antti Karttunen, Dec 05 2024

For all n, a(n) > 1. This follows from a proof given in A337372. See also A378662.

Among the initial 10 million terms, there are 7835064 2's.

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

Cf. A000203, A003961, A080224, A091191, A337345, A337372, A378662.

s = Select[Range[2^11], DivisorSigma[1, #] > 2 # && Times @@ Boole@ Map[DivisorSigma[1, #] <= 2 # &, Most@ Divisors@ #] == 1 &];
Map[Length@ Select[Divisors[#], 2 # < (Times @@ Map[Power @@ # &, FactorInteger[#] /. {p_, e_} /; e > 0 :> {Prime[PrimePi[p] + 1], e}] - Boole[# == 1]) &] &, s] (* 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); };
A337345(n) = sumdiv(n,d,A003961(d)>(2*d));
is_A091191(n) = if(sigma(n)<=2*n, 0, fordiv(n,d,if(d2*d, return(0))); (1));
k=0; n=0; while(k<100000, n++; if(is_A091191(n), k++; print1(A337345(n), ", "); write("b378658.txt", k, " ", A337345(n))));

{A337345(k) for k such that A080224(k) = 1}.

a(n) = 1+A378662(A091191(n)).

A378663 Number of divisors d of n such that sigma(d) < 2*d < A003961(d), where A003961 is fully multiplicative with a(p) = nextprime(p).

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 2, 1, 1, 0, 1, 0, 1, 1, 3, 0, 1, 0, 2, 1, 0, 0, 2, 0, 0, 2, 2, 0, 2, 0, 4, 0, 0, 1, 2, 0, 0, 1, 3, 0, 2, 0, 2, 3, 0, 0, 3, 1, 2, 0, 2, 0, 2, 0, 3, 1, 0, 0, 3, 0, 0, 3, 5, 0, 0, 0, 2, 1, 3, 0, 3, 0, 0, 2, 2, 0, 1, 0, 4, 3, 0, 0, 3, 0, 0, 0, 3, 0, 4, 1, 2, 0, 0, 0, 4, 0, 3, 2, 3, 0, 0, 0, 3, 4

Offset: 1

Antti Karttunen, Dec 06 2024

nonn

Number of terms of A341615 that divide n.

Inverse Möbius transform of A341613.

Cf. A000203, A003961, A080225, A337345, A341615, A341620, A378662.

A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A341613(n) = ((sigma(n)<(2*n))&&((2*n)<A003961(n)));
A378663(n) = sumdiv(n,d,A341613(d));

a(n) = Sum_{d|n} A341613(d).
a(n) = A337345(n) - A341620(n).
a(n) = A378662(n) - A080225(n).

cp's OEIS Frontend

A337372 Primitively primeshift-abundant numbers: Numbers that are included in A246282 (k with A003961(k) > 2k), but none of whose proper divisors are.

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A337345 Number of divisors d of n for which A003961(d) > 2*d, where A003961 is fully multiplicative with a(p) = nextprime(p).

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A341614 Numbers k such that sigma(k) <= 2k < A003961(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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A378736 Greatest divisor d of the n-th abundant number such that sigma(d) <= 2*d < A003961(d).

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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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A378658 a(n) = A337345(A091191(n)), where A337345 is the number of divisors d of n for which A003961(d) > 2*d, and A091191 lists the primitive abundant numbers.

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A378663 Number of divisors d of n such that sigma(d) < 2*d < A003961(d), where A003961 is fully multiplicative with a(p) = nextprime(p).

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