A262228 -id:A262228 - cp's OEIS frontend

A336253 Exponential barely deficient numbers: exponential deficient numbers whose exponential abundancy is closer to 2 than that of any smaller exponential deficient number.

1, 4, 72, 100, 144, 3528, 12100, 15876, 24336, 441000, 1334025, 2205000, 5664400, 24206400, 71267364, 151880976, 3252372552, 9346201200, 13319078472, 26828235000, 347372082000, 1851803856100, 2260121356900, 3198696480100, 5202286387272, 10330374528100, 16316106062400

Offset: 1

Amiram Eldar, Jul 14 2020

nonn

The exponential abundancy of a number k is esigma(k)/k, where esigma is the sum of exponential divisors of k (A051377).
Exponential deficient numbers are numbers k with esigma(k)/k < 2. These are numbers that are neither e-perfect (A054979) nor exponential abundant (A129575).
The corresponding values of the exponential abundancy are 1, 1.5, 1.666..., 1.8..., 1.833..., ...
All the terms are powerful numbers (A001694) because esigma(k)/k depends only on the powerful part of k (A057521). - Amiram Eldar, May 06 2025

			4 is a term since it is exponential deficient, and esigma(4)/4 = 3/2 is higher than esigma(k)/k for all the exponential deficient numbers k < 4.

Amiram Eldar, Table of n, a(n) for n = 1..36

Cf. A051377, A054979, A057521, A129575.
Subsequence of A001694.
Similar sequences: A302572, A228450, A262228, A307122, A336252, A336254.

fun[p_, e_] := DivisorSum[e, p^# &]; esigma[1] = 1; esigma[n_] := Times @@ fun @@@ FactorInteger[n]; rm = 0; s={}; Do[r = esigma[n]/n; If[r >= 2, Continue[]]; If[r > rm, rm = r; AppendTo[s, n]], {n, 1, 10^6}]; s

a(21)-a(27) from Amiram Eldar, May 06 2025

A352739 Primitive nondeficient numbers satisfying a stronger condition that compares abundancy with related numbers as detailed in the comments.

Original entry on oeis.org

6, 28, 104, 496, 836, 1952, 2002, 3230, 4030, 5830, 7912, 8128, 8415, 8925, 11096, 17816, 32445, 35650, 45356, 45885, 46035, 47804, 51850, 55796, 61904, 63388, 66928, 76516, 77572, 77744, 83265, 83312, 91388, 93148, 101475, 107198, 111644, 114256, 130304, 131054, 133042

Offset: 1

Peter Munn, Mar 31 2022

nonn

In this entry we list a primitive nondeficient number, m, if and only if there is no nondeficient number k = (m/p)*q whose abundancy is less than the abundancy of m, where p and q are prime numbers.

The primitive nondeficient number list (A006039) abbreviates the nondeficient number list (A023196) by omitting multiples of numbers that have already appeared. Here we abbreviate further, although typically by omission due to the lesser abundancy of a larger number. For instance, 315 * p is a primitive nondeficient number for prime p, 3 <= p <= 103; of these only 32445 = 315 * 103 is listed. See the example section for further detail.

			6 is a perfect number, and a perfect number is a primitive nondeficient number with the least abundancy (1.0) for classification as nondeficient, from which we see there is no nondeficient k with lesser abundancy. So 6 meets the stronger condition to be listed here.
12 is nondeficient, but is not primitive (as a multiple of nondeficient 6). So 12 is not listed.
104 = 2^3*13 is nondeficient, but 52 & 8 (and so other proper divisors) are deficient. So 104 is primitive nondeficient. To evaluate the stronger condition, we replace a factor 2 by prime p_1 <> 2, giving k = 2^2 * 13 * p_1 (maybe k = 2^2 * 13^2), or the factor 13 by prime p_2 <> 13, giving k = 2^3 * p_2 (maybe k = 2^4). If p_1 > 11, p_2 = 2, or p_2 > 13, k is deficient. Otherwise (p_1 <= 11 or 2 < p_2 < 13, and) calculation shows abundancy of k is greater than the abundancy of 104. So there is no nondeficient k with lesser abundancy. So 104 is listed.
(The remaining examples explore a little about calculating relative abundancies.)
88 = 2^3*11 is nondeficient, but is not listed because 104 = 2^3*13 is nondeficient and 13 contributes a smaller factor to abundancy (14/13) than 11 does (12/11). Likewise, 272, 304, 368, 464 (each 16 times a prime) are omitted due to the lesser abundancy of 496 = 16*31 (which is listed).
70 is nondeficient, but is not listed due to the lesser abundancy of nondeficient 28. 70 = 2*7*5, whereas 28 = 2*7*2, and a second factor of 2 contributes a smaller factor to abundancy (7/6) than the first factor of 5 does (6/5).
The following table of terms shows prime factors listed in order of decreasing factor, (m+1)/m, that they contribute to the number's abundancy (e.g., a 2nd factor of 3 increases abundancy by 13/12, so appears after 11 in the factorization of 8415). The relevant values of m are shown on the right.
      6  2 * 3                          2, 3
     28  2 * 2 * 7                      2, 6,  7
    104  2 * 2 *13 * 2                  2, 6, 13, 14
    496  2 * 2 * 2 * 2 * 31             2, 6, 14, 30,  31
    836  2 * 2 *11 *19                  2, 6, 11, 19
   1952  2 * 2 * 2 * 2 * 61 * 2         2, 6, 14, 30,  61,  62
   2002  2 * 7 *11 *13                  2, 7, 11, 13
   3230  2 * 5 *17 *19                  2, 5, 17, 19
   4030  2 * 5 *13 *31                  2, 5, 13, 31
   5830  2 * 5 *11 *53                  2, 5, 11, 53
   7912  2 * 2 * 2 *23 * 43             2, 6, 14, 23,  43
   8128  2 * 2 * 2 * 2 *  2 * 2 *127    2, 6, 14, 30,  62, 126, 127
   8415  3 * 5 *11 * 3 * 17             3, 5, 11, 12,  17
   8925  3 * 5 * 7 *17 *  5             3, 5,  7, 17,  30
  11096  2 * 2 * 2 *19 * 73             2, 6, 14, 19,  73
  17816  2 * 2 * 2 *17 *113             2, 6, 14, 17, 113
  32445  3 * 5 * 7 * 3 *103             3, 5,  7, 12, 103

Peter Munn, Table of n, a(n) for n = 1..5000
Peter Munn, PARI program
Index entries for sequences where any odd perfect numbers must occur
Index entries for sequences related to sigma(n)

Cf. A262228, A335557.
Subsequence of A006039, A023196.
A000396 is a subsequence.

PARI
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A295001 a(n) = nextprime(1/(2/sigma[-1](P(n)) - 1)) where P(n) = Product_{0 <= k < n} a(k), sigma[-1](x) = sigma(x)/x, a(0) = 4.

Original entry on oeis.org

4, 11, 23, 257, 13007, 44512049, 46880563785749, 125637016478802067649031191, 652182699863469019760217209096329987925268834143233, 1800254420479597976179975458181139131985404009703136640765845238082635790500153934999846722641241849

Offset: 0

M. F. Hasler, Nov 23 2017

Here, nextprime(x) = min { p > x; p prime }, prevprime(x) = max { p < x; p prime }.
The next term, a(10) ~ 3.1*10^196, is too large to be displayed above.
From a(3) on, a(n+1) has roughly twice the number of digits of a(n).
For n >= 1, a(n) is the least prime such that Product_{k=0..n} a(k) is deficient. This implies that (Product_{k=0..n-1} a(k))*prevprime(a(n)) is perfect for n = 1, and a primitive weird number (A002975) for some but not all larger n.

			Let Q(x) = 1/(2/sigma[-1](x) - 1), P(n) = Product(a(k), k=0..n-1), and start with a(0) = 4 = P(1). Then:
Q(P(1)) = 7, a(1) = 11. (4*7 is perfect, P(2) = 4*11 is deficient.)
Q(P(2)) = 21, a(2) = 23. (4*11*19 is weird, P(3) = 4*11*23 is deficient.)
Q(P(3)) = 252, a(3) = 257. (P(3)*251 is weird, P(4) = 4*11*23*257 is deficient.)
Q(P(4)) = 13003.2, a(4) = 13007. (P(4)*13003 is weird, P(5) = 4*11*23*257*13007 is deficient.)
Q(P(5)) = 44512006.7..., a(5) = 44512049. (P(5)*44511949 is weird ; P(6) = 4*11*257*44512049 is deficient.)
P(6)*prevprime(a(6)) is semiperfect, i.e., no more weird.

M. F. Hasler, Table of n, a(n) for n = 0..13

Cf. A002975 (primitive weird numbers), A000203 (sigma).
The nextprime and prevprime functions are here used for possibly non-integral arguments, but rounding these down or up allows the use of the nextprime and prevprime functions for integer arguments, A151800 and A151799.
See A262228 for the variant starting with a(0) = 1.

A295001=List(m=4);for(n=1,13,listput(A295001,p=nextprime(1\(2/sigma(m,-1)-1)+1));p>default(primelimit)&&addprimes(p);m*=p)

A339344 Lexicographically earliest sequence of odd primes such that the asymptotic density of the numbers which are divisible by at least one of these primes is 1/2.

Original entry on oeis.org

3, 5, 17, 257, 65537, 4294967311, 1229782942255939601, 88962710886098567818446141338419231, 255302062200114858892457591448999891874349780170241684791167583265041

Offset: 1

Amiram Eldar, Nov 30 2020

nonn

Given a set of prime numbers P, finite or infinite, the set of numbers which are divisible by at least one of the primes in P has an asymptotic density Product_{p in P} (1 - 1/p). If P is finite, then this density is equal to 1/2 only when P = {2}. Otherwise, the density is 1/2 for infinitely many sets P. This sequence is the lexicographically earliest infinite sequence of such primes.
The first 5 terms are the Fermat primes (A019434).
a(10) = 7.455916... * 10^135 is too large to be included in the data section.

Amiram Eldar, Table of n, a(n) for n = 1..12

Cf. A000215, A019434, A080307, A125045, A262228, A339345.

s = {}; r = 1; p = 3; Do[AppendTo[s, p]; r *= 1 - 1/p; p = NextPrime[r/(r - 1/2)], {9}]; s

a(1) = 3, a(n) = nextprime(r(n-1)/(r(n-1) - 1/2)), where r(n) = Product_{k=1..n-1} 1 - 1/a(n).

Product_{n=>1} (1 - 1/a(n)) = 1/2.

A336252 Infinitary barely deficient numbers: infinitary deficient numbers whose infinitary abundancy is closer to 2 than that of any smaller infinitary deficient number.

Original entry on oeis.org

1, 2, 8, 84, 110, 128, 1155, 3680, 6490, 8200, 8648, 12008, 18632, 32768, 724000, 1495688, 2095208, 3214090, 3477608, 3660008, 5076008, 12026888, 16102808, 26347688, 29322008, 33653888, 73995392, 615206030, 815634435, 2147483648, 42783299288, 80999455688

Offset: 1

Amiram Eldar, Jul 14 2020

nonn

The infinitary abundancy of a number k is isigma(k)/k, where isigma is the sum of infinitary divisors of k (A049417).

The corresponding values of the infinitary abundancy are 1, 1.5, 1.875, 1.904..., 1.963..., ...

			8 is a term since it is infinitary deficient (A129657), and isigma(8)/8 = 15/8 is higher than isigma(k)/k for all the infinitary deficient numbers k < 8.

Cf. A049417, A129657, A335054.

Similar sequences: A228450, A262228, A302572, A307122, A336253.

fun[p_, e_] := Module[{b = IntegerDigits[e, 2]}, m = Length[b]; Product[If[b[[j]] > 0, 1 + p^(2^(m - j)), 1], {j, 1, m}]]; isigma[1] = 1; isigma[n_] := Times @@ fun @@@ FactorInteger[n]; seq = {}; r = 0; Do[s = isigma[n]/n; If[s < 2 && s > r, AppendTo[seq, n]; r = s], {n, 1, 10^6}]; seq

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A336253 Exponential barely deficient numbers: exponential deficient numbers whose exponential abundancy is closer to 2 than that of any smaller exponential deficient number.

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A352739 Primitive nondeficient numbers satisfying a stronger condition that compares abundancy with related numbers as detailed in the comments.

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A295001 a(n) = nextprime(1/(2/sigma[-1](P(n)) - 1)) where P(n) = Product_{0 <= k < n} a(k), sigma[-1](x) = sigma(x)/x, a(0) = 4.

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A339344 Lexicographically earliest sequence of odd primes such that the asymptotic density of the numbers which are divisible by at least one of these primes is 1/2.

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A336252 Infinitary barely deficient numbers: infinitary deficient numbers whose infinitary abundancy is closer to 2 than that of any smaller infinitary deficient number.

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