A108227 -id:A108227 - 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

A063124 a(n) = # { primes p | prime(n) <= p < 2*prime(n) } where prime(n) is the n-th prime.

Original entry on oeis.org

2, 2, 2, 3, 4, 4, 5, 5, 6, 7, 8, 10, 10, 10, 10, 12, 14, 13, 14, 15, 14, 16, 16, 17, 20, 21, 20, 20, 19, 19, 24, 24, 26, 26, 28, 27, 29, 29, 29, 29, 31, 31, 33, 33, 33, 33, 36, 39, 39, 39, 40, 40, 40, 42, 43, 44, 43, 43, 43, 43, 43, 45, 50, 51, 50, 50, 55, 55, 57, 56, 56, 56, 58

Offset: 1

Reinhard Zumkeller, Aug 08 2001

nonn

a(n) is the number of primes between prime(n) and 2*prime(n) inclusive. - Sean A. Irvine, Apr 18 2023

Also for x = Product_{i=n..n+k} A000040(i), the least k such that A003961(x) > 2*x. - Antti Karttunen, Dec 08 2024

			a(10) = 7 as there are 7 primes between prime(10) = 29 and 58 = 29*2: 29, 31, 37, 41, 43, 47, 53.

N. J. A. Sloane, Table of n, a(n) for n = 1..20000 [First 2000 terms from Harry J. Smith]

Related sequences:
Primes (p) and composites (c): A000040, A002808, A000720, A065855.
Primes between p(n) and 2*p(n): A063124, A070046; between c(n) and 2*c(n): A376761; between n and 2*n: A035250, A060715, A077463, A108954.
Composites between p(n) and 2*p(n): A246514; between c(n) and 2*c(n): A376760; between n and 2*n: A075084, A307912, A307989, A376759.
Cf. A003961, A108227, A331677, A334051, A378746.

A062134 := proc(n) numtheory:-pi(2*ithprime(n))-n+1; end; # N. J. A. Sloane, Oct 19 2024
[seq(A062134(n),n=1..100)];

Table[PrimePi[2*Prime[n]] - n + 1, {n, 100}] (* Paolo Xausa, Oct 22 2024 *)

a(n)={1 + primepi(2*prime(n)) - n} \\ Harry J. Smith, Aug 19 2009

a(n) = A035250(prime(n)).
a(n) = A070046(n) + 1. - Sean A. Irvine, Apr 18 2023
From Antti Karttunen, Dec 08 2024: (Start)
a(n) = n-A331677(n) = 1+n-A334051(n).
a(n) = 1+A000720(2*A000040(n))-n. [After Harry J. Smith's PARI-program]
a(n) < A108227(n). [Assuming M. F. Hasler's interpretation in May 08 2017 comment in the latter]
a(n) = A001222(A378746(n)).
(End)

Definition clarified by N. J. A. Sloane, Oct 04 2024

A007741 a(n) = prime(n)...prime(m), the least product of consecutive primes which is abundant.

Original entry on oeis.org

30, 15015, 33426748355, 1357656019974967471687377449, 7105630242567996762185122555313528897845637444413640621

Offset: 1

Walter Nissen

nonn

Essentially, i.e., except for a(1), identical to A007702. All terms are primitive abundant numbers (A091191) and thus, except for the first term, odd primitive abundant (A006038). The next term is too large to be displayed here, see A007707 (and formula) for many more terms, using a more compact encoding. - M. F. Hasler, Apr 30 2017

Amiram Eldar, Table of n, a(n) for n = 1..14
G. Thimm, Emails to N. J. A. Sloane, Sep. 1994

Cf. A005101, A006038, A007702, A007707, A007708, A091191.

Cf. A108227, A285993.

a[n_] := Module[{p = Prime[n]}, r = 1; prod = 1; While[r <= 2, r *= 1 + 1/p; prod *= p; p = NextPrime[p]]; prod]; Array[a, 5] (* Amiram Eldar, Jun 29 2019 *)

a(n) = {p = prime(n); sig = p+1; prd = p; while (sig <= 2*prd, p = nextprime(p+1); sig *= p+1; prd *= p;); return (prd);} \\ Michel Marcus, Mar 10 2013

a(n) = Product_{k=n..A007707(n)} prime(k) = Product_{0 <= i < A108227(n)} prime(n+i). - M. F. Hasler, Apr 30 2017 and Jun 15 2017

More terms from Don Reble, Nov 10 2005

A001276 Smallest k such that the product of q/(q-1) over the primes from prime(n) to prime(n+k-1) is greater than 2.

Original entry on oeis.org

2, 3, 7, 15, 27, 41, 62, 85, 115, 150, 186, 229, 274, 323, 380, 443, 509, 577, 653, 733, 818, 912, 1010, 1114, 1222, 1331, 1448, 1572, 1704, 1845, 1994, 2138, 2289, 2445, 2609, 2774, 2948, 3127, 3311, 3502, 3699, 3900, 4112, 4324, 4546, 4775, 5016, 5255, 5493

Offset: 1

N. J. A. Sloane

nonn

A perfect (or abundant) number with prime(n) as its lowest prime factor must be divisible by at least a(n) distinct primes.

In fact, a(n) is the least possible number of distinct prime factors for a (prime(n))-rough abundant number: (prime(n))^(e_n) * ... * (prime(n+a(n)-1))^(e_(n+a(n)-1)) is abundant for sufficiently large e_n, ..., e_(n+a(n)-1). - Jianing Song, Apr 13 2021

			Every odd abundant number has at least 3 distinct prime factors, and 945 = 3^3 * 5 * 7 has exactly 3, so a(2) = 3. - _Jianing Song_, Apr 13 2021

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Amiram Eldar, Table of n, a(n) for n = 1..650
Karl K. Norton, Remarks on the number of factors of an odd perfect number, Acta Arith., 6 (1961), 365-374.

Cf. A001275, A005101.

Cf. A108227 (least number of prime factors for a (prime(n))-rough abundant number, counted with multiplicity).

a[n_] := Module[{p = Prime[n], r = 1, k = 0}, While[r <= 2, r *= p/(p - 1); p = NextPrime[p]; k++]; k]; Array[a, 50] (* Amiram Eldar, Jul 12 2019 *)

a(n)=my(pr=1.,k=0);forprime(p=prime(n),default(primelimit),pr*=p/(p-1);k++;if(pr>2,return(k))) \\ Charles R Greathouse IV, May 09 2011

a(n) = li(prime(n)^2) + O(n^2/exp((log n)^(4/7 - e))) for any e > 0.

a(n) = pi(A001275(n)) - n + 1. - Amiram Eldar, Jul 12 2019

Comment, formula, program, and new definition from Charles R Greathouse IV, May 10 2011

A007702 a(n) = prime(n)...prime(m), the least product of consecutive primes which is non-deficient.

Original entry on oeis.org

6, 15015, 33426748355, 1357656019974967471687377449, 7105630242567996762185122555313528897845637444413640621

Offset: 1

Walter Nissen

nonn

Amiram Eldar, Table of n, a(n) for n = 1..14

Cf. A005100, A007741, A007684, A007686.

Cf. A107705, A108227, A285993.

a[n_] := Module[{p = Prime[n]}, r = 1; prod = 1; While[r < 2, r *= 1 + 1/p; prod *= p; p = NextPrime[p]]; prod]; Array[a, 5] (* Amiram Eldar, Jun 29 2019 *)

A007702(n, p=prime(n), s=1+1/p, P=p)={until(2<=s*=1+1/p,P*=p=nextprime(p+1));P} \\ M. F. Hasler, Jun 15 2017

a(n) = Product_{k = n..A007684(n)} prime(k) = Product_{0 <= i < A107705(n)} prime(n+i). - M. F. Hasler, Jun 15 2017

More terms from Don Reble, Nov 10 2005

A107705 a(n) is the least number of prime factors in any non-deficient number that has the n-th prime as its least prime factor.

Original entry on oeis.org

2, 5, 9, 18, 31, 46, 67, 91, 122, 158, 194, 238, 284, 334, 392, 456, 522, 591, 668, 749, 835, 929, 1028, 1133, 1242, 1352, 1469, 1594, 1727, 1869, 2019, 2163, 2315, 2471, 2636, 2802, 2977, 3157, 3342, 3534, 3731, 3933, 4145, 4358, 4581, 4811, 5053, 5293

Offset: 1

Hugo van der Sanden, Jun 10 2005

nonn

Barring unforeseen odd perfect numbers (which it has been proved must have at least 29 prime factors if they exist at all), if we replace "non-deficient" in the description with "abundant", the value of a(1) becomes 3 and all other values stay the same.

The above mentioned sequence is A108227, see there for a comment on the relation of this sequence to that of primitive abundant numbers (A006038) which are products of consecutive primes, i.e., of the form N = Product_{0<=iA007702. - M. F. Hasler, Jun 15 2017

			a(2) is 5 since 1) there are abundant numbers with a(2)=5 prime factors of which p_2=3 is the least prime factor (such as 945 = 3^3.5.7); 2) there are no non-deficient numbers with fewer than 5 prime factors, of which 3 is the least prime factor.

Amiram Eldar, Table of n, a(n) for n = 1..500

Cf. A000040, A023196, A005101, A001222.

Cf. A007684 (~ A007707), A007708, A007741.

A107705(n,s=1+1/prime(n))=for(a=1,9e9,2>(s*=1+1/prime(n+a))||return(a+1)) \\ M. F. Hasler, Jun 15 2017

a(n) = A007684(n)-n+1. A007702(n) = Product_{0<=iM. F. Hasler, Jun 15 2017

Data corrected by Amiram Eldar, Aug 08 2019

A285993 Largest odd abundant number (A005231) equal to the product of n consecutive primes.

Original entry on oeis.org

15015, 255255, 4849845, 111546435, 33426748355, 1236789689135, 50708377254535, 2180460221945005, 102481630431415235, 5431526412865007455, 320460058359035439845, 19548063559901161830545, 1309720258513377842646515, 1357656019974967471687377449, 107254825578022430263302818471

Offset: 5

M. F. Hasler, Apr 30 2017

nonn

The smallest term is a(5) = 3*5*7*11*13, there is no odd abundant number (A005231) equal to the product of less than 5 consecutive primes.
The smallest odd abundant number (A005231) equal to the product of n consecutive primes is equal (when it exists, i.e., for n >= 5) to the least odd number with n (distinct) prime divisors, equal to the product of the first n odd primes = A070826(n+1) = A002110(n+1)/2.
See A188342 = (945, 3465, 15015, 692835, 22309287, ...) for the least odd primitive abundant number (A006038) with n distinct prime factors, and A275449 for the least odd primitive abundant number with n prime factors counted with multiplicity.
The terms are in general not primitive abundant numbers (A091191), in particular this cannot be the case when a(n) is a multiple of a(n-1), as is the case for most of the terms, for which a(n) = a(n-1)*A117366(a(n-1)). In the other event, spf(a(n)) = nextprime(spf(a(n-1))), and a(n) is in A007741(2,3,4...). These are exactly the primitive terms in this sequence.

			For n < 5, there is no odd abundant number equal to the product of n distinct primes.
For 5 <= n <= 8, the largest odd abundant number equal to the product of n consecutive primes is 3*...*prime(n+1).
For 9 <= n <= 17, the largest odd abundant number equal to the product of n consecutive primes is 5*...*prime(n+2).
For 18 <= n <= 30, the largest odd abundant number equal to the product of n consecutive primes is 7*...*prime(n+3).
For 31 <= n <= 45, the largest odd abundant number equal to the product of n consecutive primes is 11*...*prime(n+4).
For 46 <= n <= 66, the largest odd abundant number equal to the product of n consecutive primes is 13*...*prime(n+5).

Amiram Eldar, Table of n, a(n) for n = 5..341

Cf. A005231, A006038, A007707, A007708, A007741.

A subsequence of A112643 (odd squarefree abundant numbers); see also A108227 (~ A107705) which give indices of primitive terms = those with smallest prime factor larger than that of earlier terms.

a(r,f=vector(r,i,prime(i+1)),o)={ while(sigma(factorback(f),-1)>2, o=f; f=concat(f[^1],nextprime(f[r]+1)));factorback(o)} \\ Intentionally throws an error when n < 5.

a(n) >= a(n-1)*p where p = A117366(a(n-1)) = A151800(A006530(a(n-1))) = nextprime(gpf(a(n-1))), an odd abundant number equal to the product of n consecutive primes. We have strict inequality for n = 9, 18, 31, 46, 67, ..., in which case a(n) = a(n-1)*p*p'/q, where p' = nextprime(p), q = least prime factor of a(n-1). This is the case if a(n) is in A007741.

A337476 Position of the first occurrence of n in A337474.

Original entry on oeis.org

1, 3, 11, 23, 61, 127, 199, 331, 467, 673, 929, 1181, 1493, 1861, 2243, 2693, 3221, 3739, 4327, 4993, 5689, 6421, 7283, 8191, 9137, 10111, 11161, 12281, 13451, 14747, 16067, 17569, 19037, 20509, 22051, 23687, 25411, 27179, 29023, 31019, 32971, 34963, 37097, 39371, 41651, 44021, 46559, 49169, 51719

Offset: 0

Antti Karttunen, Aug 28 2020

nonn

Cf. A000040, A107705, A108227, A337474, A337478.

Cf. also A337477.

For all n >= 0, A337474(a(n)) = n.
For all n >= 1, a(n) = A000040(A107705(n)). [Conjectured, see comment in A108227]
For all n >= 0, a(n) <= A337478(n).

A343357 7-rough abundant numbers.

Original entry on oeis.org

20169691981106018776756331, 21373852696395930345517903, 21975933054040886129898689, 23476198863254546445077041, 23782174126975753483041047, 23836908704943476736166573, 24137500239684251978741183, 24272002214551310731350839, 24955720586792192723783257, 24986334842265665051802619

Offset: 1

David A. Corneth, Apr 12 2021

nonn

Each term has at least A001276(4) = 15 distinct prime factors and A108227(4) = 18 prime factors counted with multiplicity. - Jianing Song, Apr 13 2021

The smallest term with exactly 15 distinct prime factors is a(830) = 465709156638373299218537971 = 7^3 * 11^2 * 13^2 * 17^2 * 19 * 23 * ... * 61. - Jianing Song, Apr 14 2021

			k = 20169691981106018776756331 is in the sequence as its smallest prime factor is at least 7 and it is abundant as sigma(k) > 2*k.

David A. Corneth, Table of n, a(n) for n = 1..2187

Cf. A005101, A007775, A047802, A115414.

is(n) = gcd(n, 30) == 1 && sigma(n) > 2*n

cp's OEIS Frontend

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

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A063124 a(n) = # { primes p | prime(n) <= p < 2*prime(n) } where prime(n) is the n-th prime.

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A007741 a(n) = prime(n)*...*prime(m), the least product of consecutive primes which is abundant.

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A001276 Smallest k such that the product of q/(q-1) over the primes from prime(n) to prime(n+k-1) is greater than 2.

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A007702 a(n) = prime(n)*...*prime(m), the least product of consecutive primes which is non-deficient.

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A107705 a(n) is the least number of prime factors in any non-deficient number that has the n-th prime as its least prime factor.

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A285993 Largest odd abundant number (A005231) equal to the product of n consecutive primes.

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A007741 a(n) = prime(n)...prime(m), the least product of consecutive primes which is abundant.

A007702 a(n) = prime(n)...prime(m), the least product of consecutive primes which is non-deficient.