A133812 -id:A133812 - cp's OEIS frontend

A047802 Smallest abundant number (sigma(x) > 2x) which is not divisible by any of the first n primes.

12, 945, 5391411025, 20169691981106018776756331, 49061132957714428902152118459264865645885092682687973, 7970466327524571538225709545434506255970026969710012787303278390616918473506860039424701

Offset: 0

Ulrich Schimke (ulrschimke(AT)aol.com)

nonn

a(n) exists for every n, since the sum of the inverses of the primes is infinite.
From Fred Schneider, Sep 20 2006; edited by Danny Rorabaugh, Nov 26 2018: (Start)
Heuristic: Add the squares of several successive primes and then add successive primes until the number is abundant.
a(2) = 5^2 * 7 * 11 * 13 * 17 * 19 * 23 * 29;
a(3) = 7^2 * 11^2 * 13 * 17 * ... * 61 * 67;
a(4) = 11^2 * 13^2 * 17 * 19 * ... * 131 * 137;
a(5) = 13^2 * 17^2 * 19 * 23 * ... * 223 * 227. (End)
a(6) = 17^2 * 19^2 * 23^2 * 29 * 31 * ... * 347 * 349;
a(7) = 19^2 * 23^2 * 29^2 * 31 * 37 * ... * 491 * 499 (both coming from the D. Iannucci paper). - Michel Marcus, May 01 2013
The known terms of this sequence provide Egyptian decompositions of unity in which all the denominators lack the first n primes, as follows: Every term listed in this sequence is a semiperfect number, which means that a subset of its divisors add up to the number itself. The decomposition 1 = 1/a + 1/b + ... + 1/m, where the denominators are a(n) divided by those divisors, is the desired decomposition. - Javier Múgica, Nov 15 2017
a(n) is the product of consecutive primes starting from prime(n+1) raised to nonincreasing powers. - Jianing Song, Apr 10 2021
From Jianing Song, Apr 14 2021: (Start)
By definition, Omega(a(n)) >= A108227(n+1) for all n, where Omega = A001222. For 0 <= n <= 12 we have Omega(a(n)) = A108227(n+1), but this is not true for n = 13, where Omega(a(13)) = 335 > A108227(14) = 334.
We also have omega(a(n)) >= A001276(n+1) for all n, where omega = A001221. The differences for known terms are 0, 0, 1, 1, 2, 3, 2, 3, 4, 4, 5, 6, 6, 6 respectively.
Conjecture: other than a(1) = 945, all terms are cubefree. (End)

			a(0) = 12, the first abundant number; a(1) = 945, the first odd abundant number; a(5) is the first abundant number not divisible by 2,3,5,7 or 11.

M. T. Whalen and C. L. Miller, Odd abundant numbers: some interesting observations, Journal of Recreational Mathematics 22 (1990), pp. 257-261.

Jeppe Stig Nielsen, Table of n, a(n) for n = 0..13
Thomas Fink, Recursively abundant and recursively perfect numbers, arXiv:2008.10398 [math.NT], 2020. Mentions this sequence.
Douglas Iannucci, On the smallest abundant number not divisible by the first k primes, Bulletin of the Belgian Mathematical Society 12:1 (2005), pp. 39-44.

Subsequence of A005101 and A133812; cf. A005231.

Iannucci shows that log a(n) = (n log n)^(2 + o(1)). - Charles R Greathouse IV, Feb 16 2011

2 more terms from Fred Schneider, Sep 20 2006

A133813 Numbers that are primally tight and have strictly descending powers.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 16, 17, 19, 23, 24, 25, 27, 29, 31, 32, 37, 41, 43, 45, 47, 48, 49, 53, 59, 61, 64, 67, 71, 72, 73, 79, 81, 83, 89, 96, 97, 101, 103, 107, 109, 113, 121, 125, 127, 128, 131, 135, 137, 139, 144, 149, 151, 157, 163, 167, 169, 173, 175

Offset: 1

Olivier Gérard, Sep 23 2007

nonn

All numbers of the form p_1^k1*p_2^k2*...*p_n^k_n, where k1 > k2 > ... > k_n and the p_i are n successive primes.
Subset of A073491, A133812.
Differs from A085233 starting n=22.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A025487, A087980, A073491, A133808-A133812.

Cf. A027748, A124010, A049084, A020639, A000040, A087980.

import Data.List (isPrefixOf)
a133813 n = a133813_list !! (n-1)
a133813_list = 1 : filter f [2..] where
   f x = isPrefixOf ps (dropWhile (< a020639 x) a000040_list) &&
           all (< 0) (zipWith (-) (tail es) es)
         where ps = a027748_row x; es = a124010_row x
-- Reinhard Zumkeller, Apr 14 2015

A049084(A027748(a(n),k+1)) = A049084(A027748(a(n),k)) + 1 and A124010(a(n),k+1) < A124010(a(n),k), 1 <= k < A001221(a(n)). - Reinhard Zumkeller, Apr 14 2015

A145108 Multiples of 4 that are primally tight and have strictly ascending powers.

Original entry on oeis.org

4, 8, 16, 32, 64, 108, 128, 256, 324, 512, 648, 972, 1024, 1944, 2048, 2916, 3888, 4096, 5832, 8192, 8748, 11664, 16384, 17496, 23328, 26244, 32768, 34992, 52488, 65536, 67500, 69984, 78732, 104976, 131072, 139968, 157464, 209952, 236196, 262144

Offset: 1

Reikku Kulon, Oct 02 2008

nonn

All numbers of the form 2^k0*p_1^k1*p_2^k2*...*p_n^k_n, where 2 <= k0 < k1 < k2 < ... < k_n and the p_i are n successive primes.

Subset of A073491, A133808, A133809, A133810 and A133811.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A025487, A087980, A073491, A133808, A133809, A133810, A133811, A133812, A133813.

Cf. A008586.

a145108 n = a145108_list !! (n-1)
a145108_list = filter ((== 0) . (`mod` 4)) a133809_list
-- Reinhard Zumkeller, Apr 14 2015

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A047802 Smallest abundant number (sigma(x) > 2x) which is not divisible by any of the first n primes.

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A133813 Numbers that are primally tight and have strictly descending powers.

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A145108 Multiples of 4 that are primally tight and have strictly ascending powers.

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Haskell