A326134 -id:A326134 - cp's OEIS frontend

A246282 Numbers k for which A003961(k) > 2k; numbers n such that if n = Product_{k >= 1} (p_k)^(c_k), then Product_{k >= 1} (p_{k+1})^(c_k) > 2n, where p_k indicates the k-th prime, A000040(k).

4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 24, 27, 28, 30, 32, 35, 36, 39, 40, 42, 44, 45, 48, 49, 50, 52, 54, 56, 57, 60, 63, 64, 66, 68, 69, 70, 72, 75, 76, 78, 80, 81, 84, 88, 90, 91, 92, 96, 98, 99, 100, 102, 104, 105, 108, 110, 112, 114, 116, 117, 120, 124, 125, 126, 128, 130, 132, 135, 136, 138, 140, 144

Offset: 1

Antti Karttunen, Aug 24 2014

Numbers n such that A003961(n) > 2*n.
Numbers n such that A048673(n) > n.
The sequence grows as:
        a(10) = 18
       a(100) = 192
      a(1000) = 1830
     a(10000) = 18636
    a(100000) = 187350
   a(1000000) = 1865226
  a(10000000) = 18654333
and the powers of 10 occur at:
        a(5) = 10
       a(53) = 100
      a(536) = 1000
     a(5423) = 10000
    a(53290) = 100000
   a(535797) = 1000000
  a(5361886) = 10000000
suggesting that the ratio a(n)/n is converging to an constant and an arbitrary natural number is slightly more likely to be in this sequence than in the complement A246281. See also comments at A246351 and compare to quite a different ratio present in the "inverse" case A246362.
From Antti Karttunen, Aug 27 2020: (Start)
Any perfect number, including all odd perfect numbers (if such numbers exist), must occur in this sequence. See A286385 and A326042 for the reason why.
Like abundancy index (ratio A000203(n)/n), also ratio A003961(n)/n is multiplicative and always > 1 for all n > 1. Thus if the number has a proper divisor that is in this sequence, then the number itself also is. See A337372 for terms included here, but with no proper divisor in this sequence. (End)
For k >= 2, if m * A130789(k) is a term then m * A130789(k-1) is a term. - Peter Munn, Sep 01 2025
Could be called "primeshift-abundant numbers", in analogy with A005101. - Antti Karttunen, Sep 01 2025

			3 = p_2 (3 is the second prime, A000040(2)) is not a member, because p_3 = 5 (5 is the next prime after 3, A000040(3)) and 5/3 < 2.
4 = 2*2 = p_1 * p_1 is a member, as p_2 * p_2 = 3*3 = 9, and 9/4 > 2.
33 = 3*11 = p_2 * p_5 is not a member, as p_3 * p_6 = 5*13 = 65, and 65/33 < 2.
35 = 5*7 = p_3 * p_4 is a member, as p_4 * p_5 = 7*11 = 77, and 77/35 > 2.

Complement: A246281.
Setwise difference of A246352 and A048674.
Cf. A000040, A003961, A048673, A130789, A246362, A252742 (characteristic function), A286385, A326042, A337345.
Positions of positive terms in A252748 and in A337345.
Union of A337372 (primitive terms), A341610 (non-primitive terms).
Subsequences: A000396, (A005101 U A341614), (A023196 U A341615), A326134, A337373, A337374, A337378, A337381, A337384, A337386, A337543, A341611.
Cf. also A275717, A275718.

Select[Range[144], 2 # < Times @@ Map[#1^#2 & @@ # &, FactorInteger[#] /. {p_, e_} /; e > 0 :> {Prime[PrimePi@ p + 1], e}] - Boole[# == 1] &] (* 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); }; \\ From A003961
isA246282(n) = (A003961(n) > (n+n));
n = 0; i = 0; while(i < 10000, n++; if(isA246282(n), i++; write("b246282.txt", i, " ", n)));

;; With Antti Karttunen's IntSeq-library.
(define A246282 (MATCHING-POS 1 1 (lambda (n) (> (A003961 n) (* 2 n)))))

;; With Antti Karttunen's IntSeq-library.
(define A246282 (MATCHING-POS 1 1 (lambda (n) (> (A048673 n) n))))

A new shorter version of name prepended by Antti Karttunen, Aug 27 2020

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

Original entry on oeis.org

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)}.

A326057 a(n) = gcd(A003961(n)-2n, A003961(n)-sigma(n)), where A003961(n) is fully multiplicative function with a(prime(k)) = prime(k+1).

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 3, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 3, 3, 1, 1, 1, 1, 3, 1, 1, 1, 43, 1, 3, 5, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 3, 3, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 3, 19, 1, 1, 1, 1, 3, 5, 1, 1, 1, 1, 3, 3, 5, 7, 1, 1, 3, 1, 1, 1, 1, 1, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 5, 1, 1, 1, 1, 3, 3, 1, 1, 1, 1, 3, 3, 1, 1

Offset: 1

Antti Karttunen, Jun 06 2019

nonn

Terms a(n) larger than 1 and equal to A252748(n) occur at n = 6, 28, 69, 91, 496, ..., see A326134. See also A349753.

Records 1, 3, 43, 45, 2005, 79243, ... occur at n = 1, 6, 28, 360, 496, 8128, ...

Cf. A000203, A000360, A003961, A033879, A033880, A252748, A286385, A326134.

Cf. also A009194, A325385, A325813, A325975, A326046, A326047, A326048, A326056, A326060, A326062, A349753.

Array[GCD[#3 - #1, #3 - #2] & @@ {2 #, DivisorSigma[1, #], Times @@ Map[#1^#2 & @@ # &, FactorInteger[#] /. {p_, e_} /; e > 0 :> {Prime[PrimePi@ p + 1], e}] - Boole[# == 1]} &, 78] (* 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); \\ From A003961
A252748(n) = (A003961(n) - (2*n));
A286385(n) = (A003961(n) - sigma(n));
A326057(n) = gcd(A252748(n), A286385(n));

a(n) = gcd(A252748(n), A286385(n)) = gcd(A003961(n) - 2n, A003961(n) - A000203(n)).

a(n) = gcd(A252748(n), A033879(n)) = gcd(A286385(n), A033879(n)). [Also A033880 can be used] - Antti Karttunen, May 06 2024

A349169 Numbers k such that k * gcd(sigma(k), A003961(k)) is equal to the odd part of {sigma(k) * gcd(k, A003961(k))}, where A003961 shifts the prime factorization one step towards larger primes, and sigma is the sum of divisors function.

Original entry on oeis.org

1, 15, 105, 3003, 3465, 13923, 45045, 264537, 459459, 745875, 1541475, 5221125, 8729721, 10790325, 14171625, 29288025, 34563375, 57034575, 71430975, 99201375, 109643625, 144729585, 205016175, 255835125, 295708875, 356080725, 399242025, 419159475, 449323875, 928602675, 939495375, 1083656925, 1941623775, 1962350685, 2083228875

Offset: 1

Antti Karttunen, Nov 10 2021

Numbers k such that A348990(k) [= k/gcd(k, A003961(k))] is equal to A348992(k), which is the odd part of A349162(k), thus all terms must be odd, as A348990 preserves the parity of its argument.
Equally, numbers k for which gcd(A064987(k), A191002(k)) is equal to A000265(gcd(A064987(k), A341529(k))).
Also odd numbers k for which A348993(k) = A319627(k).
Odd terms of A336702 are given by the intersection of this sequence and A349174.
Conjectures:
  (1) After 1, all terms are multiples of 3. (Why?)
  (2) After 1, all terms are in A104210, in other words, for all n > 1, gcd(a(n), A003961(a(n))) > 1. Note that if we encountered a term k with gcd(k, A003961(k)) = 1, then we would have discovered an odd multiperfect number.
  (3) Apart from 1, 15, 105, 3003, 13923, 264537, all other terms are abundant. [These apparently are also the only terms that are not Zumkeller, A083207. Note added Dec 05 2024]
  (4) After 1, all terms are in A248150. (Cf. also A386430).
  (5) After 1, all terms are in A348748.
  (6) Apart from 1, there are no common terms with A349753.
Note: If any of the last four conjectures could be proved, it would refute the existence of odd perfect numbers at once. Note that it seems that gcd(sigma(k), A003961(k)) < k, for all k except these four: 1, 2, 20, 160.
Questions:
  (1) For any term x here, can 2*x be in A349745? (Partial answer: at least x should be in A191218 and should not be a multiple of 3). Would this then imply that x is an odd perfect number? (Which could explain the points (1) and (4) in above, assuming the nonexistence of opn's).

Cf. A000203, A003961, A007949, A064989, A083207, A104210, A161942, A191002, A191218, A228058, A319627, A322361, A326134, A329963, A342671, A348748, A348990, A348992, A348993, A349162, A349164, A349174.
Cf. also A349745, A349746, A349753.
Subsequence of A349749, and of A386430.

Select[Range[10^6], #1/GCD[#1, #3] == #2/(2^IntegerExponent[#2, 2]*GCD[#2, #3]) & @@ {#, DivisorSigma[1, #], Times @@ Map[NextPrime[#1]^#2 & @@ # &, FactorInteger[#]]} &] (* Michael De Vlieger, Nov 11 2021 *)

A000265(n) = (n >> valuation(n, 2));
A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
isA349169(n) = { my(s=sigma(n),u=A003961(n)); (n*gcd(s,u) == A000265(s)*gcd(n,u)); }; \\ (Program simplified Nov 30 2021)

For all n >= 1, A007949(A000203(a(n))) = A007949(a(n)). [sigma preserves the 3-adic valuation of the terms of this sequence] - Antti Karttunen, Nov 29 2021

Name changed and comment section rewritten by Antti Karttunen, Nov 29 2021

A378980 Numbers k such that (A003961(k)-2*k) divides (A003961(k)-sigma(k)), where A003961 is fully multiplicative with a(prime(i)) = prime(i+1), and sigma is the sum of divisors function.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 10, 25, 26, 28, 33, 46, 55, 57, 69, 91, 93, 496, 1034, 1054, 1558, 2211, 2626, 4825, 8128, 11222, 12046, 12639, 28225, 32043, 68727, 89575, 970225, 1392386, 2245557, 8550146, 12371554, 16322559, 22799825, 33550336, 48980427, 51326726, 55037217, 60406599, 68258725, 142901438, 325422273, 342534446

Offset: 1

Antti Karttunen, Dec 12 2024

nonn

Numbers k such that A252748(k) divides A286385(k).

Conjecture: Apart from a(5)=6, this is a subsequence of A319630, i.e., for all terms k<>6, gcd(k, A003961(k)) = 1. See also A372562, A372566.

Amiram Eldar, Table of n, a(n) for n = 1..69 (terms below 10^11; terms 1..60 from Antti Karttunen)
Index entries for sequences related to prime indices in the factorization of n.
Index entries for sequences related to sigma(n).

Cf. A000203, A003961, A252748, A286385, A319630, A372562, A372566.
Positions of 0's in A378981.
Subsequence of A263837.
Subsequences: A000396, A048674, A348514, A326134, A349753 (odd terms of this sequence).
Cf. also A378983.

f1[p_, e_] := (p^(e + 1) - 1)/(p - 1); f2[p_, e_] := NextPrime[p]^e; q[k_] := Module[{fct = FactorInteger[k], m, s}, s = Times @@ f1 @@@ fct; m = Times @@ f2 @@@ fct; Divisible[m - s, m - 2*k]]; q[1] = True; Select[Range[10^5], q] (* Amiram Eldar, Dec 19 2024 *)

A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A378981(n) = { my(u=A003961(n)); ((u-sigma(n))%((2*n)-u)); };
isA378980(n) = !A378981(n);

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 *)

A348514 Numbers k for which A003961(k) = 2k+1, where A003961 shifts the prime factorization of n one step towards larger primes.

Original entry on oeis.org

4, 10, 57, 1054, 2626, 68727, 12371554, 1673018314, 10475647197, 11154517557, 27594844918, 630178495917, 7239182861878

Offset: 1

Antti Karttunen, Oct 29 2021

Numbers k such that A064216(1+k) = k.
It seems that after 4, all other terms are squarefree. See conjecture in A348511.
a(9)..a(13) <= 10475647197, 11154517557, 27594844918, 630178495917, 7239182861878, which are also terms. - David A. Corneth, Oct 30 2021

Index entries for sequences computed from indices in prime factorization

Fixed points of map A108228. (Compare to A048674).
Positions of ones in A252748.
Subsequence of the following sequences: A246282, A319630, A348511, A378980 (see also A379216), A387411, A387414.
Cf. A000230, A003961, A064216, A326134.

f[p_, e_] := NextPrime[p]^e; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; Select[Range[10^5], s[#] == 2*# + 1 &] (* Amiram Eldar, Oct 30 2021 *)

a(9)-a(11) verified by Amiram Eldar, Nov 01 2021

a(12)-a(13) verified by Martin Ehrenstein, Nov 08 2021

A349753 Odd numbers k for which A003961(k)-2k divides A003961(k)-sigma(k), where A003961 shifts the prime factorization one step towards larger primes, and sigma is the sum of divisors function.

Original entry on oeis.org

1, 3, 7, 25, 33, 55, 57, 69, 91, 93, 2211, 4825, 12639, 28225, 32043, 68727, 89575, 970225, 2245557, 16322559, 22799825, 48980427, 55037217, 60406599, 68258725, 325422273, 414690595, 569173299, 794579511, 10056372275, 10475647197, 10759889913, 11154517557

Offset: 1

Antti Karttunen, Dec 01 2021

nonn

Numbers k for which A326057(k) = gcd(A003961(k)-2k, A003961(k)-sigma(k)) is equal to abs(A252748(k)) = |A003961(k)-2k|.
The odd terms of A326134 form a subsequence of this sequence. Unlike in A326134, here we don't constrain the value of A252748(k) = A003961(k)-2k, thus allowing also values <= +1. Because of that, the odd terms of A048674 and A348514 are all included here, for example 57 and 68727 that occur in A348514, and 1, 3, 25, 33, 93, 970225, 325422273, 414690595 that occur in A048674.
Conjecture (1): This is a subsequence of A319630, in other words, for all terms k, gcd(k, A003961(k)) = 1.
Conjecture (2): Apart from 1, there are no common terms with A349169, which would imply that no odd perfect numbers exist.
None of the 36 initial terms is Zumkeller, in A083207, because all are deficient (in A005100). See also A337372. - Antti Karttunen, Dec 05 2024

Amiram Eldar, Table of n, a(n) for n = 1..36 (terms below 10^11)
Index entries for sequences where any odd perfect numbers must occur
Index entries for sequences computed from indices in prime factorization
Index entries for sequences related to sigma(n)

Cf. A000203, A003961, A005100, A083207, A252748, A286385, A319630, A326057, A326134, A337372, A349169.
Cf. A048674, A348514.
Subsequence of A378980 (its odd terms).

f[p_, e_] := NextPrime[p]^e; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; q[n_] := Divisible[(sn = s[n]) - DivisorSigma[1, n], sn - 2*n]; Select[Range[1, 10^6, 2], q] (* Amiram Eldar, Dec 04 2021 *)

A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
isA349753(n) = if(!(n%2), 0, my(s = A003961(n), t = (s-(2*n)), u = s-sigma(n)); !(u%t));

A341615 Numbers k for which sigma(k) < 2k < A003961(k).

Original entry on oeis.org

4, 8, 9, 10, 14, 15, 16, 21, 27, 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

nonn

Deficient numbers in A246282.

Not all odd terms are in A337372.

Intersection of A005100 and A246282. Subsequence of A341614.
Cf. A341613 (characteristic function).
Cf. A000203, A003961, A252748, A286385, A326057, A326134, A337372.

A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
isA341615(n) = A341613(n);

cp's OEIS Frontend

A246282 Numbers k for which A003961(k) > 2*k; numbers n such that if n = Product_{k >= 1} (p_k)^(c_k), then Product_{k >= 1} (p_{k+1})^(c_k) > 2*n, where p_k indicates the k-th prime, A000040(k).

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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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A326057 a(n) = gcd(A003961(n)-2n, A003961(n)-sigma(n)), where A003961(n) is fully multiplicative function with a(prime(k)) = prime(k+1).

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A349169 Numbers k such that k * gcd(sigma(k), A003961(k)) is equal to the odd part of {sigma(k) * gcd(k, A003961(k))}, where A003961 shifts the prime factorization one step towards larger primes, and sigma is the sum of divisors function.

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A378980 Numbers k such that (A003961(k)-2*k) divides (A003961(k)-sigma(k)), where A003961 is fully multiplicative with a(prime(i)) = prime(i+1), and sigma is the sum of divisors function.

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

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A348514 Numbers k for which A003961(k) = 2k+1, where A003961 shifts the prime factorization of n one step towards larger primes.

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A246282 Numbers k for which A003961(k) > 2k; numbers n such that if n = Product_{k >= 1} (p_k)^(c_k), then Product_{k >= 1} (p_{k+1})^(c_k) > 2n, where p_k indicates the k-th prime, A000040(k).