A337543 -id:A337543 - 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)}.

A337381 Numbers k for which A003973(k) >= 2*sigma(k).

Original entry on oeis.org

6, 8, 9, 12, 14, 15, 16, 18, 20, 21, 24, 27, 28, 30, 32, 35, 36, 40, 42, 44, 45, 48, 49, 50, 52, 54, 56, 60, 63, 64, 66, 68, 70, 72, 75, 76, 78, 80, 81, 84, 88, 90, 92, 96, 98, 99, 100, 102, 104, 105, 108, 110, 112, 114, 117, 120, 124, 125, 126, 128, 130, 132, 135, 136, 138, 140, 144, 147, 148, 150, 152, 153, 154

Offset: 1

Antti Karttunen, Aug 27 2020

nonn

Note that A003973(n) >= sigma(n) for all n. See A336852.
Like the abundancy index (ratio A000203(n)/n), and ratio A003961(n)/n, the ratio A003973(n)/sigma(n) is also multiplicative and > 1 for all n > 1. Thus if the number has a proper divisor that is in this sequence, then the number itself is also. See A337543 for those terms included here, but which have no proper divisor in this sequence. - Antti Karttunen, Aug 31 2020
All terms are in A246282 because A341528(n) < A341529(n) for all n > 1. - Antti Karttunen, Feb 22 2021

Cf. A000203, A003961, A003973, A336852, A337541, A337542, A341528, A341529.
Cf. A337382 (complement), A337383 (characteristic function).
Subsequences: A337378, A337384, A337386, A337543 (primitive terms).
Subsequence of A246282.

A003973(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); sigma(factorback(f)); };
isA337381(n) = (A003973(n)>=2*sigma(n));

A337479 Primitive elements of A337386: numbers k for which sigma(A003961(k)) >= 2*A003961(k), but none of the proper divisors of k satisfy the same condition.

Original entry on oeis.org

120, 180, 300, 420, 504, 630, 660, 780, 924, 990, 1020, 1050, 1092, 1140, 1170, 1380, 1470, 1650, 1740, 1860, 2220, 2310, 2460, 2580, 2730, 2820, 2856, 3168, 3180, 3192, 3432, 3540, 3570, 3660, 3864, 3990, 4020, 4260, 4284, 4290, 4380, 4488, 4590, 4740, 4752, 4788, 4830

Offset: 1

Antti Karttunen, Sep 01 2020

nonn

Equivalently, numbers k such that A003961(k) is in A006039, i.e., numbers that become an (odd) primitive nondeficient number when prime-shifted once.

Conjecture: every positive integer is either a (possibly trivial) multiple of a sequence term or divides infinitely many terms of this sequence. - Peter Munn, Sep 24 2020

Cf. A000203, A003961, A003973, A006039, A337386, A337539, A337690.

Cf. also A071395, A091191, A337372, A337543.

Block[{f}, f[1] = 1; f[n_] := Times @@ Map[#1^#2 & @@ # &, FactorInteger[n] /. {p_, e_} /; e > 0 :> {Prime[PrimePi@ p + 1], e}]; Select[Range[5000], And[DivisorSigma[1, Last[#]] >= 2 Last[#], NoneTrue[Most[#], DivisorSigma[1, #] >= 2 # &]] &@ Map[f, Divisors@ #] &] ] (* Michael De Vlieger, Oct 05 2020 *)

A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
isA337386(n) = { my(x=A003961(n)); (sigma(x)>=2*x); };
isA337479(n) = (1==sumdiv(n,d,isA337386(d)));

For all n >= 1, A337690(a(n)) = A337539(n).

A337541 Number of divisors d of n for which sigma(A003961(d)) >= 2*sigma(d), where sigma is the sum of divisors, and A003961(x) shifts the prime factorization of x one step towards larger primes.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Aug 31 2020

nonn

Number of terms of A337381 that divide n.

Inverse Möbius transform of A337383.
Cf. A000203, A003961, A003973, A337381, A337542, A337543 (positions of ones).
Cf. also A337345.

A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A337541(n) = sumdiv(n,d,sigma(A003961(d))>=2*sigma(d));

a(n) = Sum_{d|n} A337383(d).

a(n) = A337542(n) + A337383(n).

A337542 Number of proper divisors d of n for which sigma(A003961(d)) >= 2*sigma(d), where sigma is the sum of divisors, and A003961(x) shifts the prime factorization of x one step towards larger primes.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Aug 31 2020

nonn

Number of terms of A337381 less than n that divide n.

Cf. A000203, A003961, A337381, A337383, A337541, A337543.

Cf. also A337346.

A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A337542(n) = sumdiv(n,d,(dA003961(d))>=2*sigma(d));

a(n) = Sum_{d|n, dA337383(d).

a(n) = A337541(n) - A337383(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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A337381 Numbers k for which A003973(k) >= 2*sigma(k).

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A337479 Primitive elements of A337386: numbers k for which sigma(A003961(k)) >= 2*A003961(k), but none of the proper divisors of k satisfy the same condition.

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A337541 Number of divisors d of n for which sigma(A003961(d)) >= 2*sigma(d), where sigma is the sum of divisors, and A003961(x) shifts the prime factorization of x one step towards larger primes.

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A337542 Number of proper divisors d of n for which sigma(A003961(d)) >= 2*sigma(d), where sigma is the sum of divisors, and A003961(x) shifts the prime factorization of x 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).