A007684 -id:A007684 - cp's OEIS frontend

A112642 Primorial number quotients arising in A007684: a(n) = A002110(A007684(n))/A002110(n-1).

6, 15015, 33426748355, 1357656019974967471687377449, 7105630242567996762185122555313528897845637444413640621, 1924344668948998025181489521338230544342953524990122861050411878226909135705454891961917517

Offset: 1

Labos Elemer, Sep 19 2005

nonn

These numbers are (perhaps the smallest) squarefree solutions to Puzzle 329 of Rivera; a(n) is abundant, not divisible by the first n-1 prime numbers, i.e., the least prime divisor of a(n) is the n-th prime number.

Duplicate of A007702.

			The corresponding sigma(a(n))/a(n) abundance ratios are as follows: 2, 2.14825, 2.00097, 2.01433, 2.00587, 2.00101, ...;
the terms have 2,3,5,7,11,... as least prime divisors.

Carlos Rivera, Puzzle 329. Odd abundant numbers not divided by 2 or 3, The Prime Puzzles and Problems Connection.

Cf. A007702, A002110, A007684, A064001, A110585, A112640.

a(n) = A002110(A007684(n))/A002110(n-1).

Term a(2) and name corrected by Andrey Zabolotskiy, Jul 16 2022

A108227 a(n) is the least number of prime factors for any abundant number with p_n (the n-th prime) as its least factor.

Original entry on oeis.org

3, 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

Offset: 1

Hugo van der Sanden, Jun 17 2005

nonn

If we replace "abundant" in the definition with "non-deficient", we get the same sequence with an initial 2 instead of 3, barring an astronomically unlikely coincidence with some as-yet-undiscovered odd perfect number. [This is sequence A107705. - M. F. Hasler, Jun 14 2017]
It appears that all terms >= 5 correspond to the odd primitive abundant numbers (A006038) which are products of consecutive primes (cf. A285993), i.e., of the form N = Product_{0<=iM. F. Hasler, May 08 2017
From Jianing Song, Apr 21 2021: (Start)
Let x_1 < x_2 < ... < x_k < ... be the numbers of the form p of p^2 + p, where p is a prime >= prime(n). Then a(n) is the smallest N such that Product_{i=1..N} (1 + 1/x_i) > 2. See my link below for a proof.
For example, for n = 3, we have {x_1, x_2, ..., x_k, ...} = {5, 7, 11, 13, 17, 19, 23, 29, 5^2 + 5, ...}, we have Product_{i=1..8} (1 + 1/x_i) < 2 and Product_{i=1..9} (1 + 1/x_i) > 2, so a(3) = 9. (End)

			a(2) = 5 since 945 = 3^3*5*7 is an abundant number with p_2 = 3 as its smallest prime factor, and no such number exists with fewer than 5 prime factors.

Amiram Eldar, Table of n, a(n) for n = 1..500
Jianing Song, Notes for A108227

Cf. A000040, A005101, A006038, A001222.
Cf. A107705.
Cf. A007707 (~ A007684), A007708, A007741.
Cf. A001276 (least number of prime factors for a (p_n)-rough abundant number, counted without multiplicity).

A108227(n, s=1+1/prime(n))=for(a=1, 9e9, if(2M. F. Hasler, Jun 15 2017

isform(k,q) = my(p=prime(k)); if(isprime(q) && (q>=p), 1, if(issquare(4*q+1), my(r=(sqrtint(4*q+1)-1)/2); isprime(r) && (r>=p), 0))
a(n) = my(Prod=1, Sum=0); for(i=prime(n), oo, if(isform(n,i), Prod *= (1+1/i); Sum++); if(Prod>2, return(Sum))) \\ Jianing Song, Apr 21 2021

a(n) = A007684(n)-n+1, for n>1. A007741(n) = Product_{0<=iM. F. Hasler, Jun 15 2017

Data corrected by Amiram Eldar, Aug 08 2019

A007707 Prime(n)...prime(a(n)) is the least product of consecutive primes which is abundant.

Original entry on oeis.org

3, 6, 11, 21, 35, 51, 73, 98, 130, 167, 204, 249, 296, 347, 406, 471, 538, 608, 686, 768, 855, 950, 1050, 1156, 1266, 1377, 1495, 1621, 1755, 1898, 2049, 2194, 2347, 2504, 2670, 2837, 3013, 3194, 3380, 3573, 3771, 3974, 4187, 4401, 4625, 4856

Offset: 1

Walter Nissen

nonn

Essentially (except the first term) the same as A007684, where the product is only required to be non-deficient, i.e., possibly a perfect number. This happens for the first term, but can't happen later any more. - M. F. Hasler, Jul 30 2016

Amiram Eldar, Table of n, a(n) for n = 1..1000

Cf. A005101, A007684 (essentially the same), A007708, A007741.

a[n_] := Module[{p = Prime[n]}, k = n; r = 1 + 1/p; While[r <= 2,  p = NextPrime[p]; r *= 1 + 1/p; k++]; k]; Array[a, 46] (* Amiram Eldar, Jun 29 2019 *)

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

a=1;i=0;for(n=1,99,until(2M. F. Hasler, Jul 30 2016

More terms from Don Reble, Nov 10 2005

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

A007686 Prime(n)...a(n) is the least product of consecutive primes which is non-deficient.

Original entry on oeis.org

3, 13, 31, 73, 149, 233, 367, 521, 733, 991, 1249, 1579, 1949, 2341, 2791, 3343, 3881, 4481, 5147, 5849, 6619, 7499, 8387, 9341, 10321, 11411, 12517, 13709, 15013, 16363, 17881, 19381, 20873, 22369, 24007, 25763, 27611, 29399, 31357

Offset: 1

Walter Nissen

nonn

Differs from A007708 only for n=1. - Michel Marcus, Mar 10 2013

a(n) is approximately n^2 log^2 n. - Charles R Greathouse IV, Feb 26 2025

Amiram Eldar, Table of n, a(n) for n = 1..1000
H. W. Gould, A class of binomial sums and a series transform, Utilitas Math., 45 (1994), 71-83. (Annotated scanned copy) [An annotation on p. 81 references this A-number, but the sequence with that annotation is actually A003046 (the last term is erroneous), unrelated to this entry. - _Andrey Zabolotskiy_, Feb 26 2025]

Cf. A005100, A007684, A007702, A007708.

a[n_] := Module[{p = Prime[n]}, r = 1 + 1/p; While[r < 2,  p = NextPrime[p]; r *= 1 + 1/p]; p]; Array[a, 39] (* 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 (p);} \\ Michel Marcus, Mar 10 2013

More terms from Don Reble, Nov 10 2005

A126105 Prime(n)^2prime(n+1)...prime(a(n)) is the least product of consecutive primes which is abundant. Note that only the first term is squared.

Original entry on oeis.org

2, 5, 10, 20, 34, 50, 72, 97, 129, 165, 203, 248, 295, 346, 405, 469, 537, 607, 685, 766, 853, 949, 1049, 1155, 1264, 1376, 1494, 1620, 1754, 1897, 2048, 2193, 2346, 2503, 2669, 2836, 3012, 3193, 3378, 3572, 3770, 3973, 4186, 4400, 4624, 4855, 5098, 5339, 5578

Offset: 1

Walter Kehowski, Mar 04 2007

			a(3)=10 since x=5^2*7*11*13*17*19*23*29=5391411025 is abundant with sigma(x)=10799308800 and sigma(x)-2*x=16486750.

Amiram Eldar, Table of n, a(n) for n = 1..200

Cf. A005101, A007684 (a very similar sequence), A007708, A007741.

a[n_] := Module[{p = Prime[n]}, c = 1; pr = 1 + 1/p + 1/p^2; While[pr < 2, p = NextPrime[p]; pr *= (1 + 1/p); c++]; c + n - 1]; Array[a, 50] (* Amiram Eldar, Aug 14 2019 *)

More terms from Stefan Steinerberger, May 11 2007

a(21) corrected and more terms added by Amiram Eldar, Aug 14 2019

cp's OEIS Frontend

A112642 Primorial number quotients arising in A007684: a(n) = A002110(A007684(n))/A002110(n-1).

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A108227 a(n) is the least number of prime factors for any abundant number with p_n (the n-th prime) as its least factor.

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A007707 Prime(n)*...*prime(a(n)) is the least product of consecutive primes which is abundant.

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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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A007686 Prime(n)*...*a(n) is the least product of consecutive primes which is non-deficient.

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A126105 Prime(n)^2*prime(n+1)...*prime(a(n)) is the least product of consecutive primes which is abundant. Note that only the first term is squared.

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Mathematica

Extensions

A007707 Prime(n)...prime(a(n)) is 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.

A007686 Prime(n)...a(n) is the least product of consecutive primes which is non-deficient.

A126105 Prime(n)^2prime(n+1)...prime(a(n)) is the least product of consecutive primes which is abundant. Note that only the first term is squared.