A337384 -id:A337384 - 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

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));

A378752 a(n) = 2*sigma(n) - sigma(A003961(n)), where A003961 is fully multiplicative with a(prime(i)) = prime(i+1).

Original entry on oeis.org

1, 2, 2, 1, 4, 0, 4, -10, -5, 4, 10, -22, 10, 0, 0, -59, 16, -46, 16, -20, -8, 16, 18, -120, 5, 12, -76, -44, 28, -48, 26, -238, 12, 28, 0, -221, 34, 24, 4, -140, 40, -96, 40, -14, -92, 24, 42, -478, -19, -42, 24, -38, 48, -384, 32, -240, 16, 52, 58, -288, 56, 40, -164, -839, 24, -48, 64, -8, 12, -96, 70, -850, 68, 60

Offset: 1

Antti Karttunen, Dec 09 2024

sign

Cf A000203, A003961, A003973.

Cf. A337381 (positions of terms <= 0), A337382 (of terms > 0), A337384 (positions of 0's),

A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A378752(n) = ((2*sigma(n)) - sigma(A003961(n)));

a(n) = 2*A000203(n) - A003973(n).

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).

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

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A378752 a(n) = 2*sigma(n) - sigma(A003961(n)), where A003961 is fully multiplicative with a(prime(i)) = prime(i+1).

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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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Examples

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Programs

Mathematica

PARI

Scheme

Scheme

Extensions

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

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Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A378752 a(n) = 2*sigma(n) - sigma(A003961(n)), where A003961 is fully multiplicative with a(prime(i)) = prime(i+1).

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Programs

PARI

Formula

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).