A308710 -id:A308710 - cp's OEIS frontend

A338133 Primitive nondeficient numbers sorted by largest prime factor then by increasing size. Irregular triangle T(n, k), n >= 2, k >= 1, read by rows, row n listing those with largest prime factor = prime(n).

6, 20, 28, 70, 945, 1575, 2205, 88, 550, 3465, 5775, 7425, 8085, 12705, 104, 572, 650, 1430, 2002, 4095, 6435, 6825, 9555, 15015, 78975, 81081, 131625, 189189, 297297, 342225, 351351, 570375, 63126063, 99198099, 117234117, 272, 748, 1870, 2210, 5355, 8415, 8925, 11492

Offset: 2

David A. Corneth and Peter Munn, Oct 11 2020

For definitions and further references/links, see A006039, the main entry for primitive nondeficient numbers.
Rows are finite: row n is a subset of the divisors of any of the products formed by multiplying 2^(A035100(n)-1) by a member of the first n finite sets described in the Dickson reference.
Column 1 includes the even perfect numbers.
The largest number in rows 2..n (therefore the largest that is prime(n)-smooth) is A338427(n). - Peter Munn, Sep 07 2021

			Row 1 is empty as there exists no primitive nondeficient number of the form prime(1)^k = 2^k.
Row 2 is (6) as 6 is the only primitive nondeficient number of the form prime(1)^k * prime(2)^m = 2^k * 3^m that is a multiple of prime(2) = 3.
Irregular triangle T(n, k) begins:
  n   prime(n)  row n
  2      3      6;
  3      5      20;
  4      7      28, 70, 945, 1575, 2205;
  5     11      88, 550, 3465, 5775, 7425, 8085, 12705;
  ...
See also the factorization of initial terms below:
      6 = 2 * 3,
     20 = 2^2 * 5,
     28 = 2^2 * 7,
     70 = 2 * 5 * 7,
    945 = 3^3 * 5 * 7,
   1575 = 3^2 * 5^2 * 7,
   2205 = 3^2 * 5 * 7^2,
     88 = 2^3 * 11,
    550 = 2 * 5^2 * 11,
   3465 = 3^2 * 5 * 7 * 11,
   5775 = 3 * 5^2 * 7 * 11,
   7425 = 3^3 * 5^2 * 11,
   8085 = 3 * 5 * 7^2 * 11,
  12705 = 3 * 5 * 7 * 11^2,
    104 = 2^3 * 13,
    572 = 2^2 * 11 * 13,
    650 = 2 * 5^2 * 13,
   1430 = 2 * 5 * 11 * 13,
   2002 = 2 * 7 * 11 * 13,
   4095 = 3^2 * 5 * 7 * 13,
  ...

L. E. Dickson, Finiteness of the Odd Perfect and Primitive Abundant Numbers with n Distinct Prime Factors, Amer. J. Math., 35 (1913), 413-426.
Index entries for sequences where any odd perfect numbers must occur

A000040, A006530 are used to define this sequence.
Permutation of A006039.
A047802\{12}, A308710 are subsequences.
Cf. A000043, A000079, A000668, A035100, A059305, A061652, A338427.

rownupto(n, u) = { my(res = List(), pr = primes(n), e = vector(n, i, logint(u, pr[i]))); vu = vector(n, i, [0, e[i]]); vu[n][1] = 1; forvec(x = vu, c = prod(i = 1, n, pr[i]^x[i]); if(c <= u && isprimitive(c), listput(res, c) ) ); Set(res) }
isprimitive(n) = { my(f = factor(n), c); if(sigma(f) < 2*n, return(0)); for(i = 1, #f~, c = n / f[i,1]; if(sigma(c) >= c * 2, return(0) ) ); 1 }
for(i = 2, 7, print(rownupto(i, 10^9)))

A006530(T(n, k)) = A000040(n).
T(n, 1) = A308710(n-1) [provided there is no least deficient number that is not a power of 2, as described in A000079].
For m >= 1, T(A059305(m), 1) = A000668(m) * 2^(A000043(m)-1) = A000668(m) * A061652(m).

A339343 Abundant pseudoperfect numbers k such that no subset of the nontrivial divisors {d|k : 1 < d < k} sums to k.

Original entry on oeis.org

20, 88, 104, 272, 304, 350, 368, 464, 572, 650, 1184, 1312, 1376, 1504, 1696, 1888, 1952, 3770, 4288, 4544, 4672, 5056, 5312, 5696, 5704, 5810, 6208, 6464, 6592, 6790, 6808, 6848, 6976, 7144, 7232, 7630, 7910, 8024, 8056, 9590, 9730, 10744, 11096, 11288, 13192

Offset: 1

Amiram Eldar, Nov 30 2020

nonn

Numbers that are the sum of a proper subset of their aliquot divisors but are not the sum of any subset of their nontrivial divisors.
The perfect numbers (A000396) which are a subset of the pseudoperfect numbers (A005835) are excluded from this sequence since otherwise they would all be trivial terms: if k is a perfect number then the sum of the divisors {d|k : 1 < d < k} is k-1, so any subset of them has a sum smaller than k.
The pseudoperfect numbers are thus a disjoint union of the perfect numbers, this sequence, and A136446.
The abundant numbers (A005101) are a disjoint union of the weird numbers (A006037), this sequence, and A136446.
All the terms are primitive pseudoperfect (A006036), since if k*m is a pseudoperfect number with k > 1, and m also pseudoperfect, then it is a sum of a subset of its divisors, all of which are multiples of k and therefore larger than 1.
This sequence is infinite. If p is an odd prime that is not a Mersenne prime (A000668), and k is the least number such that 2^k * p is an abundant number (A005101; i.e., the least k such that 2^(k+1) - 1 > p), then 2^k * p is a term (these are the nonperfect terms of A308710). If 2^k * p was not a term, then since it has only 2 odd divisors (1 and p), it would be equal to a sum of its even divisors (if 1 is not in the sum then p also cannot be in it). This would make 2^(k-1) * p also a pseudoperfect number, but by definition of k, 2^(k-1) * p is a deficient number (A005100).
If k is an even abundant number with abundance (A033880) 2, i.e., sigma(k) = A000203(k) = 2*k + 2, then k is a term.
a(157) = A122036(1) = 351351 is the least (and currently the only known) odd term.

			20 is a term since it is a pseudoperfect number, 20 = 1 + 4 + 5 + 10, and the set of nontrivial divisors of 20, {d|20 : 1 < d < 20} = {2, 4, 5, 10}, has no subset that sums to 20.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000203, A000396, A000668, A005100, A005101, A005835, A006037, A033880, A070824, A088831, A122036, A136446, A308710.
Subsequence of A006036.
A122036 is a subsequence.

psQ[n_] := DivisorSigma[1, n] > 2*n && Module[{d = Most@Divisors[n], x}, SeriesCoefficient[Series[Product[1 + x^d[[i]], {i, Length[d]}], {x, 0, n}], n] > 0 && SeriesCoefficient[Series[Product[1 + x^d[[i]], {i, 2, Length[d]}], {x, 0, n}], n] == 0 ]; Select[Range[2000], psQ]

A337814 a(n) is the smallest primitive nondeficient number that divides n or is a multiple of n.

Original entry on oeis.org

6, 6, 6, 20, 20, 6, 28, 88, 945, 20, 88, 6, 104, 28, 945, 272, 272, 6, 304, 20, 945, 88, 368, 6, 550, 104, 945, 28, 464, 6, 496, 1184, 3465, 272, 70, 6, 1184, 304, 4095, 20, 1312, 6, 1376, 88, 945, 368, 1504, 6, 2205, 550

Offset: 1

Peter Munn, Sep 23 2020

nonn

The list of primitive nondeficient numbers (A006039) starts 6, 20, 28, 70, 88, 104, 272, ..., with 945 the first odd term.
When n is a primitive nondeficient number, a(n) = n; for other values of n, either the set of divisors of n or the set of multiples of n contains a primitive nondeficient number, but not both.
If n is a nondeficient number, a(n) is a divisor: we know n is a multiple of at least one primitive nondeficient number, as it follows directly from the definition of primitive nondeficient number.
For deficient n, a(n) is a multiple. We can always find a multiple that is a primitive nondeficient number by multiplying n by the product of successive primes starting with A007918(2n-1), the first prime >= 2n-1. The smallest nondeficient number that is generated this way will be primitive, therefore an upper bound for a(n). (Reaching a nondeficient number is guaranteed because the sum of the inverses of the primes is infinite.)
More extensive explanation, due to M. F. Hasler, summarized from SeqFan list posting: (Start)
Any deficient number N has abundant multiples; to reach a primitive nondeficient number it is sufficient to choose additional prime factors in such a way that you just get abundancy >= 2, but < 2 whatever factor you omit.
An additional prime factor p increases abundancy by a factor 1 + 1/sum_{k=1..m(p)} p^k if the new multiplicity of p is m(p) >= 1.
Let x = max { sum_{k=1..m(p)} p^k : p | N } so that 1+1/x is the smallest such contribution of any prime factor in N.
Since the infinite product (over primes p) of 1+1/p diverges, a satisfactory method is multiplying N by distinct prime factors greater than x until abundancy is >= 2.
(End)

			6 is the smallest primitive nondeficient number, and is a multiple of 1, 2 and 3; so a(1) = a(2) = a(3) = 6.
For n = 4: we see 6 is not a multiple of 4, but the second smallest primitive nondeficient number, 20, is a multiple of 4; so a(4) = 20.
For n = 9: as 9 is deficient, we seek a suitable multiple. All even multiples of 9 are nondeficient, as they are multiples of nondeficient 6, but that also means they are not primitive. So we seek a qualifying odd multiple. The first odd nondeficient number is 945, which must therefore be primitive, and is also a multiple of 9. So a(9) = 945.
For n = 40: as a nondeficient number, we know 40 must have a primitive nondeficient divisor; the least such is 20, so a(40) = 20.

M. F. Hasler and Peter Munn, A property of k-almost primes, but also A006039?, SeqFan list, Oct 09 2020.
Wikipedia, Divergence of the sum of the reciprocals of the primes.

Sequences with related definitions: A064162, A254572.
Range of values: A006039.
See A000203, A005100 and A023196 for definitions of deficient and nondeficient numbers.
Cf. A000040, A007918, A308710, A338133.

For m >= 2, a(A000040(m)) = A338133(m, 1).

For m >= 1, a(6m+3) == 1 (mod 2).

A367481 Primitive practical numbers of the form 2 * 3^i * prime(k).

Original entry on oeis.org

30, 42, 66, 78, 306, 342, 414, 522, 558, 666, 2214, 2322, 2538, 2862, 3186, 3294, 3618, 3834, 3942, 4266, 4482, 4806, 5238, 5454, 5562, 5778, 5886, 6102, 20574, 21222, 22194, 22518, 24138, 24462, 25434, 26406, 27054, 28026, 28998, 29322, 30942, 31266, 31914

Offset: 1

Miko Labalan, Nov 19 2023

nonn

This sequence and A308710 are both non-overlapping subsets of A267124.
a(n) is a number of the form 2 * 3^i * prime(k) for i > 0 and b(i) < k <= b(i+1) where b(n) = Sum_{m=2..n+1} A233919(m).
Terms are pseudoperfect numbers, A005835, but are not primitive pseudoperfect numbers, A006036.
Since no term is a square or twice a square, there are no terms k such that sigma(k) is odd. Therefore, according to Proposition 10 by Rao/Peng (see their JNT paper at A083207) all terms are Zumkeller numbers. - Ivan N. Ianakiev, Nov 28 2023

Cf. A000040, A000244, A005153, A005835, A006036, A233919, A267124, A308710.

a[n_]:=2*3^(Floor[Log[2*Prime[n+2]]/Log[3]]-1)*Prime[n+2]; Array[a,43] (* Stefano Spezia, Nov 19 2023 *)

a(n) = 2 * 3^(floor(log_3(2*prime(n+2)))-1) * prime(n+2).

cp's OEIS Frontend

A338133 Primitive nondeficient numbers sorted by largest prime factor then by increasing size. Irregular triangle T(n, k), n >= 2, k >= 1, read by rows, row n listing those with largest prime factor = prime(n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A339343 Abundant pseudoperfect numbers k such that no subset of the nontrivial divisors {d|k : 1 < d < k} sums to k.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A337814 a(n) is the smallest primitive nondeficient number that divides n or is a multiple of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A367481 Primitive practical numbers of the form 2 * 3^i * prime(k).

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Formula