A002100 -id:A002100 - cp's OEIS frontend

A324381 Number of nonzero digits when the n-th highly composite number is written in primorial base: a(n) = A267263(A002182(n)).

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

Offset: 1

Antti Karttunen, Feb 26 2019

nonn

			For n=12, A002182(12) = 240, which is written as "11000" in primorial base (A049345) because 240 = 1*A002110(4) + 1*A002110(3) = 210+30, thus a(12) = 2, as there are two nonzero digits.
For n=18, A002182(18) = 2520 = "110000" in primorial base because 2520 = 1*A002110(5) + 1*A002110(4) = 2310+210, thus a(18) = 2.
For n=26, A002182(26) = 45360 = "1670000" in primorial base because 45360 = 1*A002110(6) + 6*A002110(5) + 7*A002110(4), thus a(26) = 3, as there are three nonzero digits.

Cf. A002100, A002182, A049345, A267263, A324382.

Cf. also A324341.

A267263(n) = { my(s=0); forprime(p=2, , if(n%p, s++, if(n==0, return(s))); n\=p); }; \\ From A267263
A324381(n) = A267263(A002182(n));

a(n) = A267263(A002182(n)).

a(n) <= A324382(n).

A371453 Numbers whose binary indices are all squarefree semiprimes.

Original entry on oeis.org

32, 512, 544, 8192, 8224, 8704, 8736, 16384, 16416, 16896, 16928, 24576, 24608, 25088, 25120, 1048576, 1048608, 1049088, 1049120, 1056768, 1056800, 1057280, 1057312, 1064960, 1064992, 1065472, 1065504, 1073152, 1073184, 1073664, 1073696, 2097152, 2097184

Offset: 1

Gus Wiseman, Apr 02 2024

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.

			The terms together with their binary expansions and binary indices begin:
       32:                 100000 ~ {6}
      512:             1000000000 ~ {10}
      544:             1000100000 ~ {6,10}
     8192:         10000000000000 ~ {14}
     8224:         10000000100000 ~ {6,14}
     8704:         10001000000000 ~ {10,14}
     8736:         10001000100000 ~ {6,10,14}
    16384:        100000000000000 ~ {15}
    16416:        100000000100000 ~ {6,15}
    16896:        100001000000000 ~ {10,15}
    16928:        100001000100000 ~ {6,10,15}
    24576:        110000000000000 ~ {14,15}
    24608:        110000000100000 ~ {6,14,15}
    25088:        110001000000000 ~ {10,14,15}
    25120:        110001000100000 ~ {6,10,14,15}
  1048576:  100000000000000000000 ~ {21}

Partitions of this type are counted by A002100, squarefree case of A101048.
For primes instead of squarefree semiprimes we get A326782.
For prime indices instead of binary indices we have A339113, A339112.
Allowing any squarefree numbers gives A368533.
This is the squarefree case of A371454.
A001358 lists squarefree semiprimes, squarefree A006881.
A005117 lists squarefree numbers.
A048793 lists binary indices, reverse A272020, length A000120, sum A029931.
A070939 gives length of binary expansion.
A096111 gives product of binary indices.
Cf. A087086, A112798, A296119, A302478, A326031, A367905, A368109, A371450.

M:= 26: # for terms < 2^M
P:= select(isprime, [$2..(M+1)/2]): nP:= nops(P):
S:= select(`<`,{seq(seq(P[i]*P[j],i=1..j-1),j=1..nP)},M+1):
R:= map(proc(s) local i; add(2^(i-1),i=s) end proc, combinat:-powerset(S) minus {{}}):
sort(convert(R,list)); # Robert Israel, Apr 04 2024

bix[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
sqfsemi[n_]:=SquareFreeQ[n]&&PrimeOmega[n]==2;
Select[Range[10000],And@@sqfsemi/@bix[#]&]

def A371453(n): return sum(1<<A006881(i)-1 for i, j in enumerate(bin(n)[:1:-1],1) if j=='1')

from math import isqrt
from sympy import primepi, primerange
def A371453(n):
    def f(x,n): return int(n+x+(t:=primepi(s:=isqrt(x)))+(t*(t-1)>>1)-sum(primepi(x//k) for k in primerange(1, s+1)))
    def A006881(n):
        m, k = n, f(n,n)
        while m != k:
            m, k = k, f(k,n)
        return m
    return sum(1<<A006881(i)-1 for i, j in enumerate(bin(n)[:1:-1],1) if j=='1') # Chai Wah Wu, Aug 16 2024

A339889 Products of distinct primes or semiprimes.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70

Offset: 1

Gus Wiseman, Dec 23 2020

nonn

Numbers that can be factored into distinct primes or semiprimes.

A semiprime (A001358) is a product of any two prime numbers.

			See A339840 for examples.

Robert Israel, Table of n, a(n) for n = 1..10000
Gus Wiseman, Counting and ranking factorizations, factorability, and vertex-degree partitions for groupings into pairs.

See link for additional cross-references.
Allowing only primes gives A005117.
Not allowing squares of primes gives A339741.
Positions of nonzeros in A339839.
Complement of A339840.
A001055 counts factorizations.
A001358 lists semiprimes, with squarefree case A006881.
A320663 counts non-isomorphic multiset partitions into singletons or pairs.
A320732 counts factorizations into primes or semiprimes.
A339742 counts factorizations into distinct primes or squarefree semiprimes.
A339841 have exactly one factorization into primes or semiprimes.
Cf. A002100, A013929, A293511, A320893, A339740, A339846.

N:= 100: # for terms <= N
B:= select(t -> numtheory:-bigomega(t) <= 2, {$2..N}):
S:= {1}:
for b in B do
  S:= S union map(`*`,select(`<=`,S,N/b),b)
od:
sort(convert(S,list)); # Robert Israel, Dec 28 2020

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Select[Range[100],Select[facs[#],UnsameQ@@#&&SubsetQ[{1,2},PrimeOmega/@#]&]!={}&]

A371454 Numbers whose binary indices are all semiprimes.

Original entry on oeis.org

8, 32, 40, 256, 264, 288, 296, 512, 520, 544, 552, 768, 776, 800, 808, 8192, 8200, 8224, 8232, 8448, 8456, 8480, 8488, 8704, 8712, 8736, 8744, 8960, 8968, 8992, 9000, 16384, 16392, 16416, 16424, 16640, 16648, 16672, 16680, 16896, 16904, 16928, 16936, 17152

Offset: 1

Gus Wiseman, Apr 02 2024

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.

			The terms together with their binary expansions and binary indices begin:
     8:           1000 ~ {4}
    32:         100000 ~ {6}
    40:         101000 ~ {4,6}
   256:      100000000 ~ {9}
   264:      100001000 ~ {4,9}
   288:      100100000 ~ {6,9}
   296:      100101000 ~ {4,6,9}
   512:     1000000000 ~ {10}
   520:     1000001000 ~ {4,10}
   544:     1000100000 ~ {6,10}
   552:     1000101000 ~ {4,6,10}
   768:     1100000000 ~ {9,10}
   776:     1100001000 ~ {4,9,10}
   800:     1100100000 ~ {6,9,10}
   808:     1100101000 ~ {4,6,9,10}

Partitions of this type are counted by A101048, squarefree case A002100.
For primes instead of semiprimes we get A326782.
For prime indices instead of binary indices we have A339112, A339113.
The squarefree case is A371453.
A001358 lists semiprimes, squarefree A006881.
A005117 lists squarefree numbers.
A048793 lists binary indices, reverse A272020, length A000120, sum A029931.
A070939 gives length of binary expansion.
A096111 gives product of binary indices.
Cf. A087086, A296119, A302478, A326031, A367905, A368109, A368533, A371450.

bix[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
semi[n_]:=PrimeOmega[n]==2;
Select[Range[10000],And@@semi/@bix[#]&]

from math import isqrt
from sympy import primepi, primerange
def A371454(n):
    def f(x,n): return int(n+x+((t:=primepi(s:=isqrt(x)))*(t-1)>>1)-sum(primepi(x//k) for k in primerange(1, s+1)))
    def A001358(n):
        m, k = n, f(n,n)
        while m != k:
            m, k = k, f(k,n)
        return m
    return sum(1<<A001358(i)-1 for i, j in enumerate(bin(n)[:1:-1],1) if j=='1') # Chai Wah Wu, Aug 16 2024

A339191 Partial products of squarefree semiprimes (A006881).

Original entry on oeis.org

6, 60, 840, 12600, 264600, 5821200, 151351200, 4994589600, 169816046400, 5943561624000, 225855341712000, 8808358326768000, 405184483031328000, 20664408634597728000, 1136542474902875040000, 64782921069463877280000, 3757409422028904882240000

Offset: 1

Gus Wiseman, Nov 30 2020

nonn

A squarefree semiprime is a product of any two distinct prime numbers.

Do all terms belong to A242031 (weakly decreasing prime signature)?

			The sequence of terms together with their prime indices begins:
          6: {1,2}
         60: {1,1,2,3}
        840: {1,1,1,2,3,4}
      12600: {1,1,1,2,2,3,3,4}
     264600: {1,1,1,2,2,2,3,3,4,4}
    5821200: {1,1,1,1,2,2,2,3,3,4,4,5}
  151351200: {1,1,1,1,1,2,2,2,3,3,4,4,5,6}
The sequence of terms together with their prime signatures begins:
                   6: (1,1)
                  60: (2,1,1)
                 840: (3,1,1,1)
               12600: (3,2,2,1)
              264600: (3,3,2,2)
             5821200: (4,3,2,2,1)
           151351200: (5,3,2,2,1,1)
          4994589600: (5,4,2,2,2,1)
        169816046400: (6,4,2,2,2,1,1)
       5943561624000: (6,4,3,3,2,1,1)
     225855341712000: (7,4,3,3,2,1,1,1)
    8808358326768000: (7,5,3,3,2,2,1,1)
  405184483031328000: (8,5,3,3,2,2,1,1,1)

A000040 lists the primes, with partial products A002110 (primorials).
A001358 lists semiprimes, with partial products A112141.
A002100 counts partitions into squarefree semiprimes (restricted: A338903)
A000142 lists factorial numbers, with partial products A000178.
A005117 lists squarefree numbers, with partial products A111059.
A006881 lists squarefree semiprimes, with partial sums A168472.
A166237 gives first differences of squarefree semiprimes.
A320655 counts factorizations into semiprimes.
A320656 counts factorizations into squarefree semiprimes.
A338898/A338912/A338913 give prime indices of semiprimes.
A338899/A270650/A270652 give prime indices of squarefree semiprimes.
A338901 gives first appearances in the list of squarefree semiprimes.
A339113 gives products of primes of squarefree semiprime index.
Cf. A001221, A112798, A167171, A320732, A320891, A320892, A320894, A320911, A338900, A338902, A339003, A339004.

FoldList[Times,Select[Range[20],SquareFreeQ[#]&&PrimeOmega[#]==2&]]

A340017 Products of squarefree semiprimes that are not products of distinct squarefree semiprimes.

Original entry on oeis.org

36, 100, 196, 216, 225, 360, 441, 484, 504, 540, 600, 676, 756, 792, 936, 1000, 1089, 1156, 1176, 1188, 1224, 1225, 1296, 1350, 1368, 1400, 1404, 1444, 1500, 1521, 1656, 1836, 1960, 2052, 2088, 2116, 2160, 2200, 2232, 2250, 2484, 2600, 2601, 2646, 2664, 2744

Offset: 1

Gus Wiseman, Dec 30 2020

nonn

Of course, every number is a product of squarefree numbers (A050320).
A squarefree semiprime (A006881) is a product of any two distinct prime numbers.
All terms have even Omega (A001222, A028260).

			The sequence of terms together with their prime indices begins:
      36: {1,1,2,2}        1000: {1,1,1,3,3,3}
     100: {1,1,3,3}        1089: {2,2,5,5}
     196: {1,1,4,4}        1156: {1,1,7,7}
     216: {1,1,1,2,2,2}    1176: {1,1,1,2,4,4}
     225: {2,2,3,3}        1188: {1,1,2,2,2,5}
     360: {1,1,1,2,2,3}    1224: {1,1,1,2,2,7}
     441: {2,2,4,4}        1225: {3,3,4,4}
     484: {1,1,5,5}        1296: {1,1,1,1,2,2,2,2}
     504: {1,1,1,2,2,4}    1350: {1,2,2,2,3,3}
     540: {1,1,2,2,2,3}    1368: {1,1,1,2,2,8}
     600: {1,1,1,2,3,3}    1400: {1,1,1,3,3,4}
     676: {1,1,6,6}        1404: {1,1,2,2,2,6}
     756: {1,1,2,2,2,4}    1444: {1,1,8,8}
     792: {1,1,1,2,2,5}    1500: {1,1,2,3,3,3}
     936: {1,1,1,2,2,6}    1521: {2,2,6,6}
For example, a complete list of all factorizations of 7560 into squarefree semiprimes is:
  7560 = (6*6*6*35) = (6*6*10*21) = (6*6*14*15),
but since none of these is strict, 7560 is in the sequence.

Eric Weisstein's World of Mathematics, Degree Sequence.
Gus Wiseman, Counting and ranking factorizations, factorability, and vertex-degree partitions for groupings into pairs.

See link for additional cross references.
The distinct prime shadows (under A181819) of these terms are A339842.
Factorizations into squarefree semiprimes are counted by A320656.
Products of squarefree semiprimes that are not products of distinct semiprimes are A320893.
Factorizations into distinct squarefree semiprimes are A339661.
For the next four lines, we list numbers with even Omega (A028260).
- A320891 cannot be factored into squarefree semiprimes.
- A320894 cannot be factored into distinct squarefree semiprimes.
- A320911 can be factored into squarefree semiprimes.
- A339561 can be factored into distinct squarefree semiprimes.
A001358 lists semiprimes, with squarefree case A006881.
A002100 counts partitions into squarefree semiprimes.
A030229 lists squarefree numbers with even Omega.
A050320 counts factorizations into squarefree numbers.
A050326 counts factorizations into distinct squarefree numbers.
A181819 is the Heinz number of the prime signature of n (prime shadow).
A320656 counts factorizations into squarefree semiprimes.
A339560 can be partitioned into distinct strict pairs.
Cf. A001055, A001222, A005117, A007717, A096373, A112798, A300061, A322353, A339559, A338899, A339740.

strr[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[strr[n/d],Min@@#>=d&]],{d,Select[Divisors[n],And[SquareFreeQ[#],PrimeOmega[#]==2]&]}]];
Select[Range[1000],Select[strr[#],UnsameQ@@#&]=={}&&strr[#]!={}&]

Equals A320894 /\ A320911.

Numbers n such that A320656(n) > 0 but A339661(n) = 0.

cp's OEIS Frontend

A324381 Number of nonzero digits when the n-th highly composite number is written in primorial base: a(n) = A267263(A002182(n)).

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A371453 Numbers whose binary indices are all squarefree semiprimes.

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A339889 Products of distinct primes or semiprimes.

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A371454 Numbers whose binary indices are all semiprimes.

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A339191 Partial products of squarefree semiprimes (A006881).

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A340017 Products of squarefree semiprimes that are not products of distinct squarefree semiprimes.

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