A367097 -id:A367097 - cp's OEIS frontend

A366741 Number of semi-sums of strict integer partitions of n.

0, 0, 0, 1, 1, 2, 5, 6, 9, 13, 21, 26, 37, 48, 63, 86, 108, 139, 175, 223, 274, 350, 422, 527, 638, 783, 939, 1146, 1371, 1648, 1957, 2341, 2770, 3285, 3867, 4552, 5353, 6262, 7314, 8529, 9924, 11511, 13354, 15423, 17825, 20529, 23628, 27116, 31139, 35615

Offset: 0

Gus Wiseman, Nov 05 2023

nonn

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 strict partitions of 9 and their a(9) = 13 semi-sums:
    (9) ->
   (81) -> 9
   (72) -> 9
   (63) -> 9
  (621) -> 3,7,8
   (54) -> 9
  (531) -> 4,6,8
  (432) -> 5,6,7

The non-strict non-binary version is A304792.
The non-binary version is A365925.
The non-strict version is A366738.
A000041 counts integer partitions, strict A000009.
A001358 lists semiprimes, squarefree A006881, conjugate A065119.
A126796 counts complete partitions, ranks A325781, strict A188431.
A276024 counts positive subset-sums of partitions, strict A284640.
A365543 counts partitions with a subset summing to k, complement A046663.
A365661 counts strict partitions w/ subset summing to k, complement A365663.
A365924 counts incomplete partitions, ranks A365830, strict A365831.
A366739 counts semi-sums of prime indices, firsts A367097.
Cf. A008967, A122768, A299701, A364272, A364350, A365541, A365832, A366753.

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

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

A367095 Number of distinct sums of pairs (repeats allowed) of prime indices of n.

Original entry on oeis.org

0, 1, 1, 1, 1, 3, 1, 1, 1, 3, 1, 3, 1, 3, 3, 1, 1, 3, 1, 3, 3, 3, 1, 3, 1, 3, 1, 3, 1, 5, 1, 1, 3, 3, 3, 3, 1, 3, 3, 3, 1, 6, 1, 3, 3, 3, 1, 3, 1, 3, 3, 3, 1, 3, 3, 3, 3, 3, 1, 5, 1, 3, 3, 1, 3, 6, 1, 3, 3, 6, 1, 3, 1, 3, 3, 3, 3, 6, 1, 3, 1, 3, 1, 6, 3, 3, 3, 3, 1, 5, 3, 3, 3, 3, 3, 3, 1, 3, 3, 3, 1, 6, 1, 3, 5

Offset: 1

Gus Wiseman, Nov 06 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.

Is the image missing only 2 and 4?

			The prime indices of 15 are {2,3}, with sums of pairs:
  2+2 = 4
  2+3 = 5
  3+3 = 6
so a(15) = 3.
The prime indices of 180 are {1,1,2,2,3}, with sums of pairs:
  1+1 = 2
  1+2 = 3
  1+3 = 4
  2+2 = 4
  2+3 = 5
  3+3 = 6
so a(180) = 5.

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

Depends only on squarefree kernel A007947. (Even more exactly, on A322591 - Antti Karttunen, Jan 20 2025)
Positions of first appearances appear to be a subset of A325986.
For 2-element submultisets we have A366739, for all submultisets A299701.
A001222 counts prime factors (also indices), distinct A001221.
A001358 lists semiprimes, squarefree A006881, conjugate A065119.
A056239 adds up prime indices, row sums of A112798.
A304793 counts positive subset-sums of prime indices.
A367096 lists semiprime divisors, count A086971.
Cf. A001248, A004526, A008967, A366737, A366738, A366740, A367093, A367097.

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

A367095(n) = if(1==n, 0, my(pis=apply(primepi,factor(n)[,1]), pairsums = vector(binomial(1+#pis,2)), k=0); for(i=1,#pis,for(j=i,#pis,k++; pairsums[k] = pis[i]+pis[j])); #Set(pairsums)); \\ 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

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

A366741 Number of semi-sums of strict integer partitions of n.

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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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A367095 Number of distinct sums of pairs (repeats allowed) of prime indices of n.

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

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

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