A246281 -id:A246281 - cp's OEIS frontend

A003961 Completely multiplicative with a(prime(k)) = prime(k+1).

1, 3, 5, 9, 7, 15, 11, 27, 25, 21, 13, 45, 17, 33, 35, 81, 19, 75, 23, 63, 55, 39, 29, 135, 49, 51, 125, 99, 31, 105, 37, 243, 65, 57, 77, 225, 41, 69, 85, 189, 43, 165, 47, 117, 175, 87, 53, 405, 121, 147, 95, 153, 59, 375, 91, 297, 115, 93, 61, 315, 67, 111, 275, 729, 119

Offset: 1

Marc LeBrun

Meyers (see Guy reference) conjectures that for all r >= 1, the least odd number not in the set {a(i): i < prime(r)} is prime(r+1). - N. J. A. Sloane, Jan 08 2021
Meyers' conjecture would be refuted if and only if for some r there were such a large gap between prime(r) and prime(r+1) that there existed a composite c for which prime(r) < c < a(c) < prime(r+1), in which case (by Bertrand's postulate) c would necessarily be a term of A246281. - Antti Karttunen, Mar 29 2021
a(n) is odd for all n and for each odd m there exists a k with a(k) = m (see A064216). a(n) > n for n > 1: bijection between the odd and all numbers. - Reinhard Zumkeller, Sep 26 2001
a(n) and n have the same number of distinct primes with (A001222) and without multiplicity (A001221). - Michel Marcus, Jun 13 2014
From Antti Karttunen, Nov 01 2019: (Start)
More generally, a(n) has the same prime signature as n, A046523(a(n)) = A046523(n). Also A246277(a(n)) = A246277(n) and A287170(a(n)) = A287170(n).
Many permutations and other sequences that employ prime factorization of n to encode either polynomials, partitions (via Heinz numbers) or multisets in general can be easily defined by using this sequence as one of their constituent functions. See the last line in the Crossrefs section for examples.
(End)

			a(12) = a(2^2 * 3) = a(prime(1)^2 * prime(2)) = prime(2)^2 * prime(3) = 3^2 * 5 = 45.
a(A002110(n)) = A002110(n + 1) / 2.

Richard K. Guy, editor, Problems From Western Number Theory Conferences, Labor Day, 1983, Problem 367 (Proposed by Leroy F. Meyers, The Ohio State U.).

Indranil Ghosh, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
Index entries for sequences computed from indices in prime factorization.
Index entries for sequences related to Heinz numbers.

See A045965 for another version.
Row 1 of table A242378 (which gives the "k-th powers" of this sequence), row 3 of A297845 and of A306697. See also arrays A066117, A246278, A255483, A308503, A329050.
Cf. A064989 (a left inverse), A064216, A000040, A002110, A000265, A027746, A046523, A048673 (= (a(n)+1)/2), A108228 (= (a(n)-1)/2), A191002 (= a(n)*n), A252748 (= a(n)-2n), A286385 (= a(n)-sigma(n)), A283980 (= a(n)*A006519(n)), A341529 (= a(n)*sigma(n)), A326042, A049084, A001221, A001222, A122111, A225546, A260443, A245606, A244319, A246269 (= A065338(a(n))), A322361 (= gcd(n, a(n))), A305293.
Cf. A191555, A252738.
Cf. A249734, A249735 (bisections).
Cf. A246261 (a(n) is of the form 4k+1), A246263 (of the form 4k+3), A246271, A246272, A246259, A246281 (n such that a(n) < 2n), A246282 (n such that a(n) > 2n), A252742.
Cf. A275717 (a(n) > a(n-1)), A275718 (a(n) < a(n-1)).
Cf. A003972 (Möbius transform), A003973 (Inverse Möbius transform), A318321.
Cf. A300841, A305421, A322991, A250469, A269379 for analogous shift-operators in other factorization and quasi-factorization systems.
Cf. also following permutations and other sequences that can be defined with the help of this sequence: A005940, A163511, A122111, A260443, A206296, A265408, A265750, A275733, A275735, A297845, A091202 & A091203, A250245 & A250246, A302023 & A302024, A302025 & A302026.
A version for partition numbers is A003964, strict A357853.
A permutation of A005408.
Applying the same transformation again gives A357852.
Other multiplicative sequences: A064988, A357977, A357978, A357980, A357983.
A056239 adds up prime indices, row-sums of A112798.
Cf. A000720, A076610, A296150.

a003961 1 = 1
a003961 n = product $ map (a000040 . (+ 1) . a049084) $ a027746_row n
-- Reinhard Zumkeller, Apr 09 2012, Oct 09 2011
(MIT/GNU Scheme, with Aubrey Jaffer's SLIB Scheme library)
(require 'factor)
(define (A003961 n) (apply * (map A000040 (map 1+ (map A049084 (factor n))))))
;; Antti Karttunen, May 20 2014

a:= n-> mul(nextprime(i[1])^i[2], i=ifactors(n)[2]):
seq(a(n), n=1..80);  # Alois P. Heinz, Sep 13 2017

a[p_?PrimeQ] := a[p] = Prime[ PrimePi[p] + 1]; a[1] = 1; a[n_] := a[n] = Times @@ (a[#1]^#2& @@@ FactorInteger[n]); Table[a[n], {n, 1, 65}] (* Jean-François Alcover, Dec 01 2011, updated Sep 20 2019 *)
Table[Times @@ Map[#1^#2 & @@ # &, FactorInteger[n] /. {p_, e_} /; e > 0 :> {Prime[PrimePi@ p + 1], e}] - Boole[n == 1], {n, 65}] (* Michael De Vlieger, Mar 24 2017 *)

a(n)=local(f); if(n<1,0,f=factor(n); prod(k=1,matsize(f)[1],nextprime(1+f[k,1])^f[k,2]))

a(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ Michel Marcus, May 17 2014

use ntheory ":all";  sub a003961 { vecprod(map { next_prime($) } factor(shift)); }  # _Dana Jacobsen, Mar 06 2016

from sympy import factorint, prime, primepi, prod
def a(n):
    f=factorint(n)
    return 1 if n==1 else prod(prime(primepi(i) + 1)**f[i] for i in f)
[a(n) for n in range(1, 11)] # Indranil Ghosh, May 13 2017

If n = Product p(k)^e(k) then a(n) = Product p(k+1)^e(k).
Multiplicative with a(p^e) = A000040(A000720(p)+1)^e. - David W. Wilson, Aug 01 2001
a(n) = Product_{k=1..A001221(n)} A000040(A049084(A027748(n,k))+1)^A124010(n,k). - Reinhard Zumkeller, Oct 09 2011 [Corrected by Peter Munn, Nov 11 2019]
A064989(a(n)) = n and a(A064989(n)) = A000265(n). - Antti Karttunen, May 20 2014 & Nov 01 2019
A001221(a(n)) = A001221(n) and A001222(a(n)) = A001222(n). - Michel Marcus, Jun 13 2014
From Peter Munn, Oct 31 2019: (Start)
a(n) = A225546((A225546(n))^2).
a(A225546(n)) = A225546(n^2).
(End)
Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * Product_{p prime} ((p^2-p)/(p^2-nextprime(p))) = 2.06399637... . - Amiram Eldar, Nov 18 2022

A048673 Permutation of natural numbers: a(n) = (A003961(n)+1) / 2 [where A003961(n) shifts the prime factorization of n one step towards larger primes].

Original entry on oeis.org

1, 2, 3, 5, 4, 8, 6, 14, 13, 11, 7, 23, 9, 17, 18, 41, 10, 38, 12, 32, 28, 20, 15, 68, 25, 26, 63, 50, 16, 53, 19, 122, 33, 29, 39, 113, 21, 35, 43, 95, 22, 83, 24, 59, 88, 44, 27, 203, 61, 74, 48, 77, 30, 188, 46, 149, 58, 47, 31, 158, 34, 56, 138, 365, 60, 98, 36, 86, 73

Offset: 1

Antti Karttunen, Jul 14 1999

nonn

Inverse of sequence A064216 considered as a permutation of the positive integers. - Howard A. Landman, Sep 25 2001
From Antti Karttunen, Dec 20 2014: (Start)
Permutation of natural numbers obtained by replacing each prime divisor of n with the next prime and mapping the generated odd numbers back to all natural numbers by adding one and then halving.
Note: there is a 7-cycle almost right in the beginning: (6 8 14 17 10 11 7). (See also comments at A249821. This 7-cycle is endlessly copied in permutations like A250249/A250250.)
The only 3-cycle in range 1 .. 402653184 is (2821 3460 5639).
For 1- and 2-cycles, see A245449.
(End)
The first 5-cycle is (1410, 2783, 2451, 2703, 2803). - Robert Israel, Jan 15 2015
From Michel Marcus, Aug 09 2020: (Start)
(5194, 5356, 6149, 8186, 10709), (46048, 51339, 87915, 102673, 137205) and (175811, 200924, 226175, 246397, 267838) are other 5-cycles.
(10242, 20479, 21413, 29245, 30275, 40354, 48241) is another 7-cycle. (End)
From Antti Karttunen, Feb 10 2021: (Start)
Somewhat artificially, also this permutation can be represented as a binary tree. Each child to the left is obtained by multiplying the parent by 3 and subtracting one, while each child to the right is obtained by applying A253888 to the parent:
                                       1
                                       |
                    ................../ \..................
                   2                                       3
         5......../ \........4                   8......../ \........6
        / \                 / \                 / \                 / \
       /   \               /   \               /   \               /   \
      /     \             /     \             /     \             /     \
    14       13         11       7          23       9          17       18
  41 10    38  12     32  28   20 15      68  25   26 63      50  16   53  19
etc.
Each node's (> 1) parent can be obtained with A253889. Sequences A292243, A292244, A292245 and A292246 are constructed from the residues (mod 3) of the vertices encountered on the path from n to the root (1).
(End)

			For n = 6, as 6 = 2 * 3 = prime(1) * prime(2), we have a(6) = ((prime(1+1) * prime(2+1))+1) / 2 = ((3 * 5)+1)/2 = 8.
For n = 12, as 12 = 2^2 * 3, we have a(12) = ((3^2 * 5) + 1)/2 = 23.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for sequences that are permutations of the natural numbers.

Inverse: A064216.
Row 1 of A251722, Row 2 of A249822.
One more than A108228, half the terms of A243501.
Fixed points: A048674.
Positions of records: A029744, their values: A246360 (= A007051 interleaved with A057198).
Positions of subrecords: A247283, their values: A247284.
Cf. A246351 (Numbers n such that a(n) < n.)
Cf. A246352 (Numbers n such that a(n) >= n.)
Cf. A246281 (Numbers n such that a(n) <= n.)
Cf. A246282 (Numbers n such that a(n) > n.), A252742 (their char. function)
Cf. A246261 (Numbers n for which a(n) is odd.)
Cf. A246263 (Numbers n for which a(n) is even.)
Cf. A246260 (a(n) reduced modulo 2), A341345 (modulo 3), A341346, A292251 (3-adic valuation), A292252.
Cf. A246342 (Iterates starting from n=12.)
Cf. A246344 (Iterates starting from n=16.)
Cf. also A003602, A003961 (A045965), A108951, A151800, A245449, A249735, A249821, A250471, A250249, A250250.
Cf. A245447 (This permutation "squared", a(a(n)).)
Other permutations whose formulas refer to this sequence: A122111, A243062, A243066, A243500, A243506, A244154, A244319, A245605, A245608, A245610, A245612, A245708, A246265, A246267, A246268, A246363, A249745, A249824, A249826, and also A183209, A254103 that are somewhat similar.
Cf. also prime-shift based binary trees A005940, A163511, A245612 and A244154.
Cf. A253888, A253889, A292243, A292244, A292245 and A292246 for other derived sequences.
Cf. A323893 (Dirichlet inverse), A323894 (sum with it), A336840 (inverse Möbius transform).

a048673 = (`div` 2) . (+ 1) . a045965
-- Reinhard Zumkeller, Jul 12 2012

f:= proc(n)
local F,q,t;
  F:= ifactors(n)[2];
  (1 + mul(nextprime(t[1])^t[2], t = F))/2
end proc:
seq(f(n),n=1..1000); # Robert Israel, Jan 15 2015

Table[(Times @@ Power[If[# == 1, 1, NextPrime@ #] & /@ First@ #, Last@ #] + 1)/2 &@ Transpose@ FactorInteger@ n, {n, 69}] (* Michael De Vlieger, Dec 18 2014, revised Mar 17 2016 *)

A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
A048673(n) = (A003961(n)+1)/2; \\ Antti Karttunen, Dec 20 2014

A048673(n) = if(1==n,n,if(n%2,A253888(A048673((n-1)/2)),(3*A048673(n/2))-1)); \\ (Not practical, but demonstrates the construction as a binary tree). - Antti Karttunen, Feb 10 2021

from sympy import factorint, nextprime, prod
def a(n):
    f = factorint(n)
    return 1 if n==1 else (1 + prod(nextprime(i)**f[i] for i in f))//2 # Indranil Ghosh, May 09 2017

(define (A048673 n) (/ (+ 1 (A003961 n)) 2)) ;; Antti Karttunen, Dec 20 2014

From Antti Karttunen, Dec 20 2014: (Start)
a(1) = 1; for n>1: If n = product_{k>=1} (p_k)^(c_k), then a(n) = (1/2) * (1 + product_{k>=1} (p_{k+1})^(c_k)).
a(n) = (A003961(n)+1) / 2.
a(n) = floor((A045965(n)+1)/2).
Other identities. For all n >= 1:
a(n) = A108228(n)+1.
a(n) = A243501(n)/2.
A108951(n) = A181812(a(n)).
a(A246263(A246268(n))) = 2*n.
As a composition of other permutations involving prime-shift operations:
a(n) = A243506(A122111(n)).
a(n) = A243066(A241909(n)).
a(n) = A241909(A243062(n)).
a(n) = A244154(A156552(n)).
a(n) = A245610(A244319(n)).
a(n) = A227413(A246363(n)).
a(n) = A245612(A243071(n)).
a(n) = A245608(A245605(n)).
a(n) = A245610(A244319(n)).
a(n) = A249745(A249824(n)).
For n >= 2, a(n) = A245708(1+A245605(n-1)).
(End)
From Antti Karttunen, Jan 17 2015: (Start)
We also have the following identities:
a(2n) = 3*a(n) - 1. [Thus a(2n+1) = 0 or 1 when reduced modulo 3. See A341346]
a(3n) = 5*a(n) - 2.
a(4n) = 9*a(n) - 4.
a(5n) = 7*a(n) - 3.
a(6n) = 15*a(n) - 7.
a(7n) = 11*a(n) - 5.
a(8n) = 27*a(n) - 13.
a(9n) = 25*a(n) - 12.
and in general:
a(x*y) = (A003961(x) * a(y)) - a(x) + 1, for all x, y >= 1.
(End)
From Antti Karttunen, Feb 10 2021: (Start)
For n > 1, a(2n) = A016789(a(n)-1), a(2n+1) = A253888(a(n)).
a(2^n) = A007051(n) for all n >= 0. [A property shared with A183209 and A254103].
(End)
a(n) = A003602(A003961(n)). - Antti Karttunen, Apr 20 2022
Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/4) * Product_{p prime} ((p^2-p)/(p^2-nextprime(p))) = 1.0319981... , where nextprime is A151800. - Amiram Eldar, Jan 18 2023

New name and crossrefs to derived sequences added by Antti Karttunen, Dec 20 2014

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).

Original entry on oeis.org

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

A048674 Fixed points of A048673 and A064216: Numbers n such that if n = product_{k >= 1} (p_k)^(c_k), then Product_{k >= 1} (p_{k+1})^(c_k) = (2*n)-1, where p_k indicates the k-th prime, A000040(k).

Original entry on oeis.org

1, 2, 3, 25, 26, 33, 93, 1034, 970225, 8550146, 325422273, 414690595, 1864797542, 2438037206

Offset: 1

Antti Karttunen, Jul 14 1999

Equally: after 1, numbers n such that, if the prime factorization of 2n-1 = Product_{k >= 1} (p_k)^(c_k) then Product_{k >= 1} (p_{k-1})^(c_k) = n.
Factorization of the initial terms: 1, 2, 3, 5^2, 2*13, 3*11, 3*31, 2*11*47, 5^2*197^2, 2*11*47*8269, 3*11*797*12373, 5*11^2*433*1583, 2*23*59*101*6803, 2*11*53*1201*1741.
The only 3-cycle of permutation A048673 in range 1 .. 402653184 is (2821 3460 5639).
For 2-cycles, take setwise difference of A245449 and this sequence.
Numbers k for which A336853(k) = k-1. - Antti Karttunen, Nov 26 2021

			25 is present, as 2*25 - 1 = 49 = p_4^2, and p_3^2 = 5*5 = 25.
26 is present, as 2*26 - 1 = 51 = 3*17 = p_2 * p_8, and p_1 * p_7 = 2*13 = 26.
Alternatively, as 26 = 2*13 = p_1 * p_7, and ((p_2 * p_8)+1)/2 = ((3*17)+1)/2 = 26 also, thus 26 is present.

Fixed points of permutation pair A048673/A064216.
Positions of zeros in A349573.
Subsequence of the following sequences: A245449, A269860, A319630, A349622, A378980 (see also A379216).
This sequence is also obtained as a setwise difference of the following pairs of sequences: A246281 \ A246351, A246352 \ A246282, A246361 \ A246371, A246372 \ A246362.
Cf. A000040, A003961 (A045965), A064989, A285701, A336853.
Cf. also A348514 (fixed points of map A108228, similar to A048673).

A048673 := n -> (A003961(n)+1)/2;
A048674list := proc(upto_n) local b,i; b := [ ]; for i from 1 to upto_n do if(A048673(i) = i) then b := [ op(b), i ]; fi; od: RETURN(b); end;

Join[{1}, Reap[For[n = 1, n < 10^7, n++, ff = FactorInteger[n]; If[Times @@ Power @@@ (NextPrime[ff[[All, 1]]]^ff[[All, 2]]) == 2 n - 1, Print[n]; Sow[n]]]][[2, 1]]] (* Jean-François Alcover, Mar 04 2016 *)

A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
isA048674(n) = ((n+n)==(1+A003961(n))); \\ Antti Karttunen, Nov 26 2021

Entry revised and the names in Maple-code cleaned by Antti Karttunen, Aug 25 2014

Terms a(11) - a(14) added by Antti Karttunen, Sep 11-13 2014

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)}.

A246351 Numbers k such that A048673(k) < k.

Original entry on oeis.org

5, 7, 11, 13, 17, 19, 22, 23, 29, 31, 34, 37, 38, 41, 43, 46, 47, 51, 53, 55, 58, 59, 61, 62, 65, 67, 71, 73, 74, 77, 79, 82, 83, 85, 86, 87, 89, 94, 95, 97, 101, 103, 106, 107, 109, 111, 113, 115, 118, 119, 121, 122, 123, 127, 129, 131, 133, 134, 137, 139, 141, 142, 143, 145, 146, 149, 151, 155, 157, 158, 159

Offset: 1

Antti Karttunen, Aug 24 2014

nonn

The growth rate of the sequence seems to converge:
       a(100) = 217
      a(1000) = 2231
     a(10000) = 21535
    a(100000) = 214647
   a(1000000) = 2155903
  a(10000000) = 21553153
Please see comments in A246282.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Complement: A246352.
Setwise difference of A246281 and A048674.
Cf. A048673, A246371, A246282.

default(primelimit, 2^22);
A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From Michel Marcus
A048673(n) = (A003961(n)+1)/2;
isA246351(n) = (A048673(n) < n);
n = 0; i = 0; while(i < 10000, n++; if(isA246351(n), i++; write("b246351.txt", i, " ", n)));

;; With Antti Karttunen's IntSeq-library.
(define A246351 (MATCHING-POS 1 1 (lambda (n) (< (A048673 n) n))))

A246361 Numbers n such that if 2n-1 = product_{k >= 1} (p_k)^(c_k), then n >= product_{k >= 1} (p_{k-1})^(c_k), where p_k indicates the k-th prime, A000040(k).

Original entry on oeis.org

1, 2, 3, 5, 8, 11, 13, 14, 17, 18, 23, 25, 26, 28, 32, 33, 38, 39, 41, 43, 50, 53, 58, 59, 61, 63, 68, 73, 74, 77, 83, 86, 88, 93, 94, 95, 98, 104, 113, 116, 122, 123, 128, 131, 137, 138, 140, 143, 149, 158, 163, 167, 172, 173, 176, 179, 182, 185, 188, 193, 194, 200, 203, 212, 213, 215, 218, 221, 228, 230, 233

Offset: 1

Antti Karttunen, Aug 24 2014

nonn

Numbers n such that A064216(n) <= n.
Numbers n such that A064989(2n-1) <= n.
The sequence grows as:
      a(100) = 332
     a(1000) = 3207
    a(10000) = 34213
   a(100000) = 340703
  a(1000000) = 3388490
suggesting that overall, less than one third of natural numbers appear in this sequence, and more than two thirds in the complement, A246362. See also comments in the latter.

			1 is present, as 2*1 - 1 = empty product = 1.
12 is not present, as (2*12)-1 = 23 = p_9, and p_8 = 19, with 12 < 19.
14 is present, as (2*14)-1 = 27 = p_2^3 = 8, and 14 >= 8.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Complement: A246362.
Union of A246371 and A048674.
Subsequence: A246360.
Cf. A000040, A064216, A064989, A246281.

default(primelimit, 2^30);
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A064216(n) = A064989((2*n)-1);
isA246361(n) = (A064216(n) <= n);
n = 0; i = 0; while(i < 10000, n++; if(isA246361(n), i++; write("b246361.txt", i, " ", n)));
(Scheme, with Antti Karttunen's IntSeq-library)
(define A246361 (MATCHING-POS 1 1 (lambda (n) (<= (A064216 n) 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).

A378662 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, 1, 0, 2, 1, 1, 0, 2, 0, 1, 1, 3, 0, 2, 0, 2, 1, 0, 0, 3, 0, 0, 2, 3, 0, 3, 0, 4, 0, 0, 1, 3, 0, 0, 1, 3, 0, 3, 0, 2, 3, 0, 0, 4, 1, 2, 0, 2, 0, 3, 0, 4, 1, 0, 0, 4, 0, 0, 3, 5, 0, 1, 0, 2, 1, 3, 0, 4, 0, 0, 2, 2, 0, 2, 0, 4, 3, 0, 0, 5, 0, 0, 0, 3, 0, 5, 1, 2, 0, 0, 0, 5, 0, 3, 2, 3, 0, 1, 0, 3, 4

Offset: 1

Antti Karttunen, Dec 06 2024

nonn

Number of terms of A341614 that divide n.

Claim: a(n) > 0 if and only if A003961(n) > 2*n [i.e., n is in A246282]. That a(n) must be zero when n is in A246281 is obvious, as is also that a(n) > 0 when n is a term of A341614 [as then A378664(n) = n], but that a(n) > 0 for all abundant numbers (A005101) is slightly less clear. So the claim boils down to this: All abundant numbers have at least one (by necessity a proper) divisor d|n such that it is in A341614, i.e., sigma(d) <= 2*d < A003961(d), i.e., that for abundant numbers n, A337345(n) is always strictly greater than A080224(n). Equivalently, of the all nonabundant divisors d of an abundant number, at least one is primeshift-abundant, i.e., A003961(d) > 2*d. This has been proved Dec 11 2024 by Jianing Song in A337372. The claim given in A378658 also follows from that proof.

Inverse Möbius transform of A341612.
Cf. A000203, A003961, A005101, A080224, A080225, A337345, A341614, A378663, A378664.
Cf. A246281 (positions of 0's), A246282 (of terms > 0).
Cf. also A337372, A378658.

Table[Length@ Select[Divisors[n], DivisorSigma[1, #] <= 2 # < (Times @@ Map[Power @@ # &, FactorInteger[#] /. {p_, e_} /; e > 0 :> {Prime[PrimePi[p] + 1], e}] - Boole[# == 1]) &], {n, 105}] (* 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)));
A378662(n) = sumdiv(n,d,A341612(d));

a(n) = Sum_{d|n} A341612(d).
a(n) = A337345(n) - A080224(n).
a(n) = A080225(n) + A378663(n).

cp's OEIS Frontend

A378743 a(n) is the least m for which bigomega(A246281(m)) = n.

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A003961 Completely multiplicative with a(prime(k)) = prime(k+1).

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A048673 Permutation of natural numbers: a(n) = (A003961(n)+1) / 2 [where A003961(n) shifts the prime factorization of n one step towards larger primes].

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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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A048674 Fixed points of A048673 and A064216: Numbers n such that if n = product_{k >= 1} (p_k)^(c_k), then Product_{k >= 1} (p_{k+1})^(c_k) = (2*n)-1, 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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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).