A366740 -id:A366740 - cp's OEIS frontend

A367097 Least positive integer whose multiset of prime indices has exactly n distinct semi-sums.

1, 4, 12, 30, 60, 210, 330, 660, 2730, 3570, 6270, 12540, 53130, 79170, 110670, 221340, 514140, 1799490, 2284590, 4196010, 6750870, 13501740, 37532220, 97350330, 131362770, 189620970, 379241940, 735844830, 1471689660

Offset: 0

Gus Wiseman, Nov 09 2023

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
We define a semi-sum of a multiset to be any sum of a 2-element submultiset. This is different from sums of pairs of elements. For example, 2 is the sum of a pair of elements of {1}, but there are no semi-sums.
From David A. Corneth, Nov 15 2023: (Start)
Terms are cubefree.
bigomega(a(n)) = A001222(a(n)) >= A002024(n) + 1 = floor(sqrt(2n) + 1/2) + 1 for n > 0. (End)

			The prime indices of 60 are {1,1,2,3}, with four semi-sums {2,3,4,5}, and 60 is the first number whose prime indices have four semi-sums, so a(4) = 60.
The terms together with their prime indices begin:
       1: {}
       4: {1,1}
      12: {1,1,2}
      30: {1,2,3}
      60: {1,1,2,3}
     210: {1,2,3,4}
     330: {1,2,3,5}
     660: {1,1,2,3,5}
    2730: {1,2,3,4,6}
    3570: {1,2,3,4,7}
    6270: {1,2,3,5,8}
   12540: {1,1,2,3,5,8}
   53130: {1,2,3,4,5,9}
   79170: {1,2,3,4,6,10}
  110670: {1,2,3,4,7,11}
  221340: {1,1,2,3,4,7,11}
  514140: {1,1,2,3,5,8,13}

The non-binary version is A259941, firsts of A299701.
These are the positions of first appearances in A366739.
A001222 counts prime factors (or prime indices), distinct A001221.
A001358 lists semiprimes, squarefree A006881, complement A100959.
A056239 adds up prime indices, row sums of A112798.
A299702 ranks knapsack partitions, counted by A108917.
A366738 counts semi-sums of partitions, strict A366741.
Semiprime divisors are listed by A367096 and have:
- square count: A056170
- sum: A076290
- squarefree count: A079275
- count: A086971
- firsts: A220264
Cf. A000720, A001248, A002024, A004709, A304793, A365920, A366740, A366753, A367093, A367095.

nn=1000;
w=Table[Length[Union[Total/@Subsets[prix[n],{2}]]],{n,nn}];
spnm[y_]:=Max@@NestWhile[Most,y,Union[#]!=Range[0,Max@@#]&];
v=Table[Position[w,k][[1,1]],{k,0,spnm[w]}]

from itertools import count
from sympy import factorint, primepi
from sympy.utilities.iterables import multiset_combinations
def A367097(n): return next(k for k in count(1) if len({sum(s) for s in multiset_combinations({primepi(i):j for i,j in factorint(k).items()},2)}) == n) # Chai Wah Wu, Nov 13 2023

2 | a(n) for n > 0. - David A. Corneth, Nov 13 2023

a(17)-a(22) from Chai Wah Wu, Nov 13 2023

a(23)-a(28) from David A. Corneth, Nov 13 2023

A367098 Number of divisors of n with exactly two distinct prime factors.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 1, 1, 0, 0, 2, 0, 2, 1, 1, 0, 3, 0, 1, 0, 2, 0, 3, 0, 0, 1, 1, 1, 4, 0, 1, 1, 3, 0, 3, 0, 2, 2, 1, 0, 4, 0, 2, 1, 2, 0, 3, 1, 3, 1, 1, 0, 5, 0, 1, 2, 0, 1, 3, 0, 2, 1, 3, 0, 6, 0, 1, 2, 2, 1, 3, 0, 4, 0, 1, 0, 5, 1, 1, 1

Offset: 1

Gus Wiseman, Nov 09 2023

			The a(n) divisors for n = 1, 6, 12, 24, 36, 60, 72, 120, 144, 216, 288, 360:
  .  6  6   6   6   6   6   6   6    6    6    6
        12  12  12  10  12  10  12   12   12   10
            24  18  12  18  12  18   18   18   12
                36  15  24  15  24   24   24   15
                    20  36  20  36   36   36   18
                        72  24  48   54   48   20
                            40  72   72   72   24
                                144  108  96   36
                                     216  144  40
                                          288  45
                                               72

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

For just one distinct prime factor we have A001222 (prime-power divisors).
This sequence counts divisors belonging to A007774.
Counting all prime factors gives A086971, firsts A220264.
Column k = 2 of A146289.
- Positions of zeros are A000961 (powers of primes), complement A024619.
- Positions of ones are A006881 (squarefree semiprimes).
- Positions of twos are A054753.
- Positions of first appearances are A367099.
A001221 counts distinct prime factors.
A001358 lists semiprimes, complement A100959.
A367096 lists semiprime divisors, sum A076290.
Cf. A000005, A001248, A056170, A079275, A090885, A366740.

Table[Length[Select[Divisors[n], PrimeNu[#]==2&]],{n,100}]
a[1] = 0; a[n_] := (Total[(e = FactorInteger[n][[;; , 2]])]^2 - Total[e^2])/2; Array[a, 100] (* Amiram Eldar, Jan 08 2024 *)

a(n) = {my(e = factor(n)[, 2]); (vecsum(e)^2 - e~*e)/2;} \\ Amiram Eldar, Jan 08 2024

a(n) = (A001222(n)^2 - A090885(n))/2. - Amiram Eldar, Jan 08 2024

A367099 Least positive integer such that the number of divisors having two distinct prime factors is n.

Original entry on oeis.org

1, 6, 12, 24, 36, 60, 72, 120, 144, 216, 288, 360, 432, 960, 720, 864, 1296, 1440, 1728, 2160, 2592, 3456, 7560, 4320, 5184, 7776, 10800, 8640, 10368, 12960, 15552, 17280, 20736, 40320, 25920, 31104, 41472, 60480, 64800, 51840, 62208, 77760, 93312

Offset: 0

Gus Wiseman, Nov 09 2023

nonn

Does this contain every power of six, namely 1, 6, 36, 216, 1296, 7776, ...?

Yes, every power of six is a term, since 6^k = 2^k * 3^k is the least positive integer having n = tau(6^k) - (2k+1) divisors with two distinct prime factors. - Ivan N. Ianakiev, Nov 11 2023

			The divisors of 60 having two distinct prime factors are: 6, 10, 12, 15, 20. Since 60 is the first number having five such divisors, we have a(5) = 60.
The terms together with their prime indices begin:
     1: {}
     6: {1,2}
    12: {1,1,2}
    24: {1,1,1,2}
    36: {1,1,2,2}
    60: {1,1,2,3}
    72: {1,1,1,2,2}
   120: {1,1,1,2,3}
   144: {1,1,1,1,2,2}
   216: {1,1,1,2,2,2}
   288: {1,1,1,1,1,2,2}
   360: {1,1,1,2,2,3}
   432: {1,1,1,1,2,2,2}
   960: {1,1,1,1,1,1,2,3}
   720: {1,1,1,1,2,2,3}
   864: {1,1,1,1,1,2,2,2}

Amiram Eldar, Table of n, a(n) for n = 0..4469

The version for all divisors is A005179 (firsts of A000005).
For all prime factors (A001222) we have A220264, firsts of A086971.
Positions of first appearances in A367098 (counts divisors in A007774).
A000961 lists prime powers, complement A024619.
A001221 counts distinct prime factors.
A001358 lists semiprimes, squarefree A006881, complement A100959.
A367096 lists semiprime divisors, sum A076290.
Cf. A001248, A054753, A056170, A079275, A146289, A366740, A367093.

nn=1000;
w=Table[Length[Select[Divisors[n],PrimeNu[#]==2&]],{n,nn}];
spnm[y_]:=Max@@NestWhile[Most,y,Union[#]!=Range[0,Max@@#]&];
Table[Position[w,k][[1,1]],{k,0,spnm[w]}]

a(n) = my(k=1); while (sumdiv(k, d, omega(d)==2) != n, k++); k; \\ Michel Marcus, Nov 11 2023

A367105 Least positive integer with n more divisors than distinct subset-sums of prime indices.

Original entry on oeis.org

1, 12, 24, 48, 60, 192, 144, 120, 180, 336, 240, 630, 420, 360, 900, 1344, 960, 1008, 720, 840, 2340, 1980, 1260, 1440, 3120, 2640, 1680, 4032, 2880, 6840, 3600, 4620, 3780, 2520, 6480, 11700, 8820, 6300, 7200, 10560, 6720, 12240, 9360, 7920, 5040, 10920, 9240

Offset: 1

Gus Wiseman, Nov 09 2023

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.An integer n is a subset-sum (A299701, A304792) of a multiset y if there exists a submultiset of y with sum n.

			The divisors of 60 are {1,2,3,4,5,6,10,12,15,20,30,60}, and the distinct subset-sums of its prime indices {1,1,2,3} are {0,1,2,3,4,5,6,7}, so the difference is 12 - 8 = 4. Since 60 is the first number with this difference, we have a(4) = 60.
The terms together with their prime indices begin:
     1: {}
    12: {1,1,2}
    24: {1,1,1,2}
    48: {1,1,1,1,2}
    60: {1,1,2,3}
   120: {1,1,1,2,3}
   144: {1,1,1,1,2,2}
   180: {1,1,2,2,3}
   192: {1,1,1,1,1,1,2}
   240: {1,1,1,1,2,3}
   336: {1,1,1,1,2,4}
   360: {1,1,1,2,2,3}
   420: {1,1,2,3,4}
   630: {1,2,2,3,4}
   720: {1,1,1,1,2,2,3}
   840: {1,1,1,2,3,4}
   900: {1,1,2,2,3,3}
   960: {1,1,1,1,1,1,2,3}

The first part (divisors) is A000005.
The second part (subset-sums of prime indices) is A299701, positive A304793.
These are the positions of first appearances in the difference A325801.
The binary version is A367093, firsts of  A086971 - A366739.
A001222 counts prime factors (or prime indices), distinct A001221.
A056239 adds up prime indices, row sums of A112798.
Cf. A000720, A220264, A304792, A365920, A366740, A367097.

nn=1000;
w=Table[DivisorSigma[0,n]-Length[Union[Total/@Subsets[prix[n]]]],{n,nn}];
spnm[y_]:=Max@@Select[Union[y],Function[i,Union[Select[y,#<=i&]]==Range[0,i]]];
Table[Position[w,k][[1,1]],{k,0,spnm[w]}]

A000005(a(n)) - A299701(a(n)) = n.

cp's OEIS Frontend

A367097 Least positive integer whose multiset of prime indices has exactly n distinct semi-sums.

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A367098 Number of divisors of n with exactly two distinct prime factors.

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A367099 Least positive integer such that the number of divisors having two distinct prime factors is n.

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Examples

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Crossrefs

Programs

Mathematica

PARI

A367105 Least positive integer with n more divisors than distinct subset-sums of prime indices.

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