A007702 -id:A007702 - cp's OEIS frontend

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

30, 15015, 33426748355, 1357656019974967471687377449, 7105630242567996762185122555313528897845637444413640621

Offset: 1

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

A007684 Prime(n)...prime(a(n)) is the least product of consecutive primes that is non-deficient.

Original entry on oeis.org

2, 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

Subscript of the smallest primorial number that when divided by the (n-1)-th primorial number gives an abundant number.
Products of consecutive primes started with prime(a) up to prime(b) result in abundant squarefree numbers if b is large enough and provides perhaps the least squarefree solutions to Rivera Puzzle 329 and its generalization.
Adding a new prime p to the product increases the relative abundancy sigma(N)/N by a factor 1+1/p. This leads to a simple and fast algorithm, see the PARI code. - M. F. Hasler, Jul 30 2016

			n=1: a(1)=2 means that primorial(2)=6 divided by primorial(1-1)=1 gives the quotient 6/1=6 which is just non-deficient (being a perfect number);
n=3: a(n)=11 because prime(3)=5, primorial(11) = 2*3*5*...*29*31, primorial(3-1) = 2*3 = 6.
p#(11)/p#(2) = 3*5*7*11*13*17*19*23*29*31 = 33426748355 = q and sigma(q)/q = 2.00097 > 2 so q is an abundant number. Also p#(10)/p#(3-1) is not yet abundant.

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) [Annotation on p. 82 references this A-number, but the triangle with that annotation is apparently A046900, unrelated to this entry. - _Andrey Zabolotskiy_, Jul 16 2022]
Carlos Rivera, Puzzle 329. Odd abundant numbers not divided by 2 or 3, The Prime Puzzles and Problems Connection.

Cf. A005100, A007686, A007702, A007707 (an essentially identical sequence).

Cf. A002110, A064001, A112640, A005101, A005231.

spr[x_, y_] :=Apply[Times, Table[(Prime[w]+1)/(Prime[w]), {w, x, y}]];
Table[Min[Flatten[Position[Table[Floor[spr[n, w]], {w, 1, 1000}], 2]]], {n, 1, 20}] (* Labos Elemer, Sep 19 2005 *)

a=1;i=0;for(n=1,99,while(2>a*=1+1/prime(i++),);print1(i",");a/=1+1/prime(n)) \\ M. F. Hasler, Jul 30 2016

a(n) is the minimal x such that floor(sigma(p#(x)/p#(n-1)) / (p#(x)/p#(n-1))) = 2, where p#(w) is the w-th primorial number, the product of first w prime numbers. For a>b, the p#(a)/p#(b)=A002110(a)/A002110(b) quotients are prime(b+1)*prime(b+2)*...*prime(a).

Additional comments from Labos Elemer, Sep 19 2005
More terms from Don Reble, Nov 10 2005
Edited by N. J. A. Sloane, Dec 22 2006

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

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

Original entry on oeis.org

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

cp's OEIS Frontend

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

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A007684 Prime(n)*...*prime(a(n)) is the least product of consecutive primes that 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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Programs

Mathematica

PARI

Extensions

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

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

A007684 Prime(n)...prime(a(n)) is the least product of consecutive primes that is non-deficient.

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