A076290 -id:A076290 - cp's OEIS frontend

A227476 Numbers whose sum of semiprime divisors (A076290) is a positive square.

4, 8, 9, 16, 25, 27, 32, 49, 64, 81, 121, 125, 128, 138, 169, 225, 243, 256, 289, 306, 343, 361, 426, 512, 516, 529, 625, 644, 675, 729, 841, 918, 961, 975, 1002, 1024, 1032, 1125, 1140, 1146, 1150, 1220, 1230, 1288, 1305, 1316, 1331, 1369, 1681, 1849, 2025

Offset: 1

Michel Lagneau, Jul 13 2013

nonn

Except for the number 1, the sequence A195942 (Zeroless prime powers (excluding primes)) is a subsequence of this sequence because the set of divisors of the numbers of the form p^m with p prime and m >= 2 contains only one semiprime divisor, p^2.

The subset of the nonprime powers is {138, 225, 306, 426, 516, 644, 675, 918, ...}.

			138 is in the sequence because the divisors of 138 are {1, 2, 3, 6, 23, 46, 69, 138} and the sum of the semiprime divisors is 2*3 + 2*23 + 3*23 = 11^2.

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

Cf. A001358, A025475, A076290, A164722.

semipSigma[n_] := DivisorSum[n, # &, PrimeOmega[#] == 2 &]; Select[Range[2000], (s = semipSigma[#]) > 0 && IntegerQ @ Sqrt[s] &] (* Amiram Eldar, May 10 2020 *)

isok(n) = issquare(s = sumdiv(n, d, d*(bigomega(d)==2))) && (s>0); \\ Michel Marcus, Sep 16 2017

Definition corrected by Michel Marcus, Sep 16 2017

A366740 Positive integers whose semiprime divisors do not all have different Heinz weights (sum of prime indices, A056239).

Original entry on oeis.org

90, 180, 210, 270, 360, 420, 450, 462, 525, 540, 550, 630, 720, 810, 840, 858, 900, 910, 924, 990, 1050, 1080, 1100, 1155, 1170, 1260, 1326, 1350, 1386, 1440, 1470, 1530, 1575, 1620, 1650, 1666, 1680, 1710, 1716, 1800, 1820, 1848, 1870, 1890, 1911, 1938, 1980

Offset: 1

Gus Wiseman, Nov 05 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.
From Robert Israel, Nov 06 2023: (Start)
Positive integers divisible by the product of four primes, prime(i)*prime(j)*prime(k)*prime(l), i < j <= k < l, with i + l = j + k.
All positive multiples of terms are terms. (End)

			The semiprime divisors of 90 are (6,9,10,15), with prime indices ({1,2},{2,2},{1,3},{2,3}) with sums (3,4,4,5), which are not all different, so 90 is in the sequence.
The terms together with their prime indices begin:
    90: {1,2,2,3}
   180: {1,1,2,2,3}
   210: {1,2,3,4}
   270: {1,2,2,2,3}
   360: {1,1,1,2,2,3}
   420: {1,1,2,3,4}
   450: {1,2,2,3,3}
   462: {1,2,4,5}
   525: {2,3,3,4}
   540: {1,1,2,2,2,3}
   550: {1,3,3,5}
   630: {1,2,2,3,4}
   720: {1,1,1,1,2,2,3}

The complement is too dense.
For all divisors instead of just semiprimes we have A299729, strict A316402.
Distinct semi-sums of prime indices are counted by A366739.
Partitions of this type are counted by A366753, non-binary A366754.
A001222 counts prime factors (or prime indices), distinct A001221.
A001358 lists semiprimes, squarefree A006881, conjugate A065119.
A056239 adds up prime indices, row sums of A112798.
A299701 counts distinct subset-sums of prime indices, positive A304793.
A299702 ranks knapsack partitions, counted by A108917, strict A275972.
Semiprime divisors are listed by A367096 and have:
- square count: A056170
- sum: A076290
- squarefree count: A079275
- count: A086971
- firsts: A220264
Cf. A000720, A001248, A008967, A365541, A365920, A366737, A366738, A366741, A367093, A367095, A367097.

N:= 10^4: # for terms <= N
P:= select(isprime, [$1..N]): nP:= nops(P):
R:= {}:
for i from 1 while P[i]*P[i+1]^2*P[i+2] < N do
  for j from i+1 while P[i]*P[j]^2 * P[j+1] < N do
    for k from j do
      l:= j+k-i;
      if l <= k or l > nP then break fi;
      v:= P[i]*P[j]*P[k]*P[l];
      if v <= N then
        R:= R union {seq(t,t=v..N,v)};
      fi
od od od:
sort(convert(R,list)); # Robert Israel, Nov 06 2023

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1000],!UnsameQ@@Total/@Union[Subsets[prix[#],{2}]]&]

These are numbers k such that A086971(k) > A366739(k).

A366739 Number of distinct semi-sums of the multiset of prime indices of n. Number of distinct sums of prime indices of semiprime divisors of n (counted by A086971).

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 2, 0, 1, 1, 1, 0, 2, 0, 2, 1, 1, 0, 2, 1, 1, 1, 2, 0, 3, 0, 1, 1, 1, 1, 3, 0, 1, 1, 2, 0, 3, 0, 2, 2, 1, 0, 2, 1, 2, 1, 2, 0, 2, 1, 2, 1, 1, 0, 4, 0, 1, 2, 1, 1, 3, 0, 2, 1, 3, 0, 3, 0, 1, 2, 2, 1, 3, 0, 2, 1, 1, 0, 4, 1, 1, 1, 2, 0, 3, 1, 2, 1, 1, 1, 2, 0, 2, 2, 3, 0, 3, 0, 2, 3

Offset: 1

Gus Wiseman, Nov 04 2023

nonn

First differs from A086971 at a(90) = 3, A086971(90) = 4.
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.

			The prime indices of 90 are {1,2,2,3}, with semi-sums
  3 = 1+2
  4 = 1+3 (or 2+2)
  5 = 2+3
so a(90) = 3.
Alternatively, the semiprime divisors of 90 are (6,9,10,15), with prime indices ({1,2},{2,2},{1,3},{2,3}) with sums (3,4,4,5) so a(90) = 3.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences related to prime indices in the factorization of n.

The non-binary version is A299701.
Summing over partitions gives A366738, strict A366741.
For all sums of pairs of elements we have A367095.
Positions of first appearances are A367097.
A001222 counts prime factors (or prime indices), distinct A001221.
A001358 lists semiprimes, squarefree A006881, conjugate A065119.
A056239 adds up prime indices, row sums of A112798.
A299702 ranks knapsack partitions, counted by A108917.
Semiprime divisors are listed by A367096 and have:
- square count: A056170
- sum: A076290
- squarefree count: A079275
- count: A086971
- firsts: A220264
Cf. A000005, A000720, A001248, A008967, A117855, A304792, A304793, A365541, A365920, A366740, A366753, A367093.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[Union[Total/@Subsets[prix[n],{2}]]],{n,100}]

A366739(n) = #Set(apply(d->((f)->sum(i=1,#f~,f[i,2]*primepi(f[i,1])))(factor(d)), select(d->2==bigomega(d), divisors(n)))); \\ Antti Karttunen, Jan 20 2025

a(n) <= A086971(n). - Antti Karttunen, Jan 20 2025

Data section extended to a(105) by Antti Karttunen, Jan 20 2025

A367093 Least positive integer with n more semiprime divisors than semi-sums of prime indices.

Original entry on oeis.org

1, 90, 630, 2310, 6930, 34650, 30030, 90090, 450450, 570570, 510510, 1531530, 7657650, 14804790, 11741730, 9699690, 29099070, 145495350

Offset: 0

Gus Wiseman, Nov 05 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.
Are all primorials after 210 included?

			The terms together with their prime indices begin:
       1: {}
      90: {1,2,2,3}
     630: {1,2,2,3,4}
    2310: {1,2,3,4,5}
    6930: {1,2,2,3,4,5}
   34650: {1,2,2,3,3,4,5}
   30030: {1,2,3,4,5,6}
   90090: {1,2,2,3,4,5,6}
  450450: {1,2,2,3,3,4,5,6}
  570570: {1,2,3,4,5,6,8}
  510510: {1,2,3,4,5,6,7}

The first part (semiprime divisors) is A086971, firsts A220264.
The second part (semi-sums of prime indices) is A366739, firsts A367097.
All sums of pairs of prime indices are counted by A367095.
The non-binary version is A367105.
A001222 counts prime factors (or prime indices), distinct A001221.
A001358 lists semiprimes, squarefree A006881, conjugate A065119.
A056239 adds up prime indices, row sums of A112798.
A299701 counts subset-sums of prime indices, positive A304793.
Semiprime divisors are listed by A367096 and have:
- square count: A056170
- sum: A076290
- squarefree count: A079275
- count: A086971
- firsts: A220264
Cf. A000720, A001248, A008967, A117855, A304792, A365541, A365920, A366738, A366740, A366753.

nn=10000;
w=Table[Length[Union[Subsets[prix[n],{2}]]]-Length[Union[Total/@Subsets[prix[n],{2}]]],{n,nn}];
spnm[y_]:=Max@@NestWhile[Most,y,Union[#]!=Range[0,Max@@#]&];
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 A367093(n):
    for k in count(1):
        c, a = 0, set()
        for s in (sum(p) for p in multiset_combinations({primepi(i):j for i,j in factorint(k).items()},2)):
            if s not in a:
                a.add(s)
            else:
                c += 1
            if c > n:
                break
        if c == n:
            return k # Chai Wah Wu, Nov 13 2023

a(n) is the least positive integer such that A086971(a(n)) - A366739(a(n)) = n.

a(12)-a(16) from Chai Wah Wu, Nov 13 2023

a(17) from Chai Wah Wu, Nov 18 2023

A366753 Number of integer partitions of n without all different sums of two-element submultisets.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 3, 4, 9, 11, 22, 27, 48, 61, 98, 123, 188, 237, 345, 435, 611, 765, 1046, 1305, 1741, 2165, 2840, 3502, 4527, 5562, 7083, 8650, 10908, 13255, 16545, 20016, 24763, 29834, 36587, 43911, 53514, 63964, 77445, 92239, 111015, 131753

Offset: 0

Gus Wiseman, Nov 07 2023

nonn

			The two-element submultisets of y = {1,1,1,2,2,3} are {1,1}, {1,2}, {1,3}, {2,2}, {2,3}, with sums 2, 3, 4, 4, 5, which are not all different, so y is counted under a(10).
The a(8) = 1 through a(13) = 11 partitions:
  (3221)  (32211)  (4321)    (33221)    (4332)      (43321)
                   (32221)   (43211)    (5331)      (53221)
                   (322111)  (322211)   (5421)      (53311)
                             (3221111)  (43221)     (54211)
                                        (322221)    (332221)
                                        (332211)    (432211)
                                        (432111)    (3222211)
                                        (3222111)   (3322111)
                                        (32211111)  (4321111)
                                                    (32221111)
                                                    (322111111)

Semiprime divisors are counted by A086971, distinct sums A366739.
The non-binary complement is A108917, strict A275972, ranks A299702.
These partitions have ranks A366740.
The non-binary version is A366754, strict A316402, ranks A299729.
A276024 counts positive subset-sums of partitions, strict A284640.
A304792 counts subset-sum of partitions, strict A365925.
A365543 counts partitions with a subset-sum k, complement A046663.
A365661 counts strict partitions with a subset-sum k, complement A365663.
A366738 counts semi-sums of partitions, strict A366741.
A367096 lists semiprime divisors, row sums A076290.
Cf. A002033, A001358, A006827, A008967, A122768, A126796, A365923, A365924, A367093, A367095.

Table[Length[Select[IntegerPartitions[n],!UnsameQ@@Total/@Union[Subsets[#,{2}]]&]],{n,0,30}]

A367096 Irregular triangle read by rows where row n lists the semiprime divisors of n. Alternatively, row n lists the semiprime divisors of A002808(n).

Original entry on oeis.org

4, 6, 4, 9, 10, 4, 6, 14, 15, 4, 6, 9, 4, 10, 21, 22, 4, 6, 25, 26, 9, 4, 14, 6, 10, 15, 4, 33, 34, 35, 4, 6, 9, 38, 39, 4, 10, 6, 14, 21, 4, 22, 9, 15, 46, 4, 6, 49, 10, 25, 51, 4, 26, 6, 9, 55, 4, 14, 57, 58, 4, 6, 10, 15, 62, 9, 21, 4, 65, 6, 22, 33, 4, 34

Offset: 1

Gus Wiseman, Nov 08 2023

On the first interpretation, the first three rows are empty. On the second, the first row is (4).

			The semiprime divisors of 30 are {6,10,15}, so row 30 is (6,10,15). Without empty rows, this is row 19.
Triangle begins (empty rows indicated by dots):
   1: .
   2: .
   3: .
   4: 4
   5: .
   6: 6
   7: .
   8: 4
   9: 9
  10: 10
  11: .
  12: 4,6
Without empty rows:
   1: 4
   2: 6
   3: 4
   4: 9
   5: 10
   6: 4,6
   7: 14
   8: 15
   9: 4
  10: 6,9
  11: 4,10
  12: 21

For all divisors we have A027750.
Square terms are counted by A056170.
Row sums are A076290.
Squarefree terms are counted by A079275.
Row lengths are A086971, firsts A220264.
A000005 counts divisors.
A001222 counts prime factors (or prime indices), distinct A001221.
A001358 lists semiprimes, squarefree A006881, complement A100959.
Cf. A008967, A365829, A366738, A366739, A366740, A367093, A367098.

Table[Select[Divisors[n],PrimeOmega[#]==2&],{n,100}]

row(n) = select(x -> bigomega(x) == 2, divisors(n)); \\ Amiram Eldar, May 02 2025

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

Original entry on oeis.org

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

A163407 Sum of semiprime divisors of n with repetition.

Original entry on oeis.org

0, 0, 0, 4, 0, 6, 0, 12, 9, 10, 0, 16, 0, 14, 15, 24, 0, 21, 0, 24, 21, 22, 0, 30, 25, 26, 27, 32, 0, 31, 0, 40, 33, 34, 35, 37, 0, 38, 39, 42, 0, 41, 0, 48, 39, 46, 0, 48, 49, 45, 51, 56, 0, 45, 55, 54, 57, 58, 0, 51, 0, 62, 51, 60, 65, 61, 0, 72, 69, 59, 0, 57, 0, 74, 55, 80, 77, 71, 0

Offset: 1

Franklin T. Adams-Watters, Jul 26 2009

nonn

We regard each prime divisor of n as distinct, and count each product of an unordered, distinct pair of them as a semiprime divisor.

			For n = 12, the prime divisors with repetition are 2,2,3. Distinguishing the 2s as 2 and 2', we have semiprime divisors 2*2', 2*3, and 2'*3, totaling 4+6+6 = 16.

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A076290, A001358, A001222, A067666.

a(n)=local(fn,p,e,s,ss);fn=factor(n);for(i=1,matsize(fn)[1],p=fn[i,1];e=fn[i,2];s+=p*e;ss+=p^2*e);(s^2-ss)\2

If s is the sum of the prime divisors of n with repetition, and ss is the sum of their squares, a(n) = (s^2 - ss) / 2.

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

A164865 Sum of the distinct semiprime divisors of the n-th number with two or more distinct semiprime divisors.

Original entry on oeis.org

10, 15, 14, 10, 18, 31, 19, 14, 41, 26, 24, 10, 35, 30, 15, 18, 35, 30, 61, 38, 59, 19, 40, 42, 71, 14, 45, 26, 40, 50, 10, 63, 42, 39, 91, 30, 71, 19, 87, 18, 101, 62, 48, 35, 66, 50, 101, 65, 24, 38, 121, 63, 19, 70, 78, 56, 42, 60, 113, 75, 14, 15, 86, 103, 45, 129, 66, 90

Offset: 1

Jonathan Vos Post, Aug 28 2009

The sum of semiprime divisors of all k such that A086971(k) > 1.

This sum is prime for k = 30, 36, 60, 72, and infinitely more values (every prime power of every primitive element).

			a(1) = 10 because the 1st number with 2 or more distinct semiprime divisors is k=12=A102467(2), as A001358(1) = 4, 4|12, A001358(2) = 6, 6|12, and 4+6 = 10.
a(6) = 31 because the 6th number with multiple distinct semiprime factors is k=30=A102467(7), the semiprimes 6, 10, and 15 divide 30, and 6 + 10 + 15 = 31.

Cf. A001358, A068318, A086971, A102467.

isA001358 := proc(n) RETURN( numtheory[bigomega](n) =2 ) ; end:
A086971 := proc(n) local a,d; a := 0 ; for d in numtheory[divisors](n) do if isA001358(d) then a := a+1; fi; od; a ; end:
A102467 := proc(n) local a; if n = 1 then 1; else for a from procname(n-1)+1 do if A086971(a) >= 2 then RETURN(a) ; fi; od: fi; end:
A076290 := proc(n) local a,d; a := 0 ; for d in numtheory[divisors](n) do if isA001358(d) then a := a+d; fi; od; a ; end:
A164865 := proc(n) A076290( A102467(n+1)) ; end: seq(A164865(n),n=1..120) ; # R. J. Mathar, Aug 31 2009

sdsd[n_]:=Module[{spd=Select[Divisors[n],PrimeOmega[#]==2&]}, If[ Length[ spd]> 1,Total[spd],0]]; DeleteCases[Array[sdsd,200],0] (* Harvey P. Dale, Oct 29 2015 *)

a(n) = Sum_(k|A102467(n) and k in A001358) k.

a(n) = A076290( A102467(n+1)). - R. J. Mathar, Aug 31 2009

Corrected and extended by R. J. Mathar, Aug 31 2009

cp's OEIS Frontend

A227476 Numbers whose sum of semiprime divisors (A076290) is a positive square.

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A366740 Positive integers whose semiprime divisors do not all have different Heinz weights (sum of prime indices, A056239).

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A366739 Number of distinct semi-sums of the multiset of prime indices of n. Number of distinct sums of prime indices of semiprime divisors of n (counted by A086971).

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A367093 Least positive integer with n more semiprime divisors than semi-sums of prime indices.

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A366753 Number of integer partitions of n without all different sums of two-element submultisets.

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A367096 Irregular triangle read by rows where row n lists the semiprime divisors of n. Alternatively, row n lists the semiprime divisors of A002808(n).

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

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A163407 Sum of semiprime divisors of n with repetition.