A320911 -id:A320911 - cp's OEIS frontend

A339361 Product of prime indices of the n-th squarefree semiprime.

2, 3, 4, 6, 8, 5, 6, 10, 7, 12, 8, 12, 9, 14, 15, 16, 10, 11, 18, 18, 12, 20, 13, 21, 14, 20, 24, 22, 15, 24, 16, 24, 27, 17, 28, 18, 26, 28, 32, 19, 30, 20, 30, 30, 21, 33, 22, 32, 36, 23, 34, 24, 36, 36, 35, 25, 38, 26, 40, 39, 27, 40, 40, 28, 42, 44, 29, 42

Offset: 1

Gus Wiseman, Dec 06 2020

nonn

A squarefree semiprime (A006881) is a product of any two distinct prime numbers. A prime index of n is a number m such that the m-th prime number divides n. The multiset of prime indices of n is row n of A112798.

			The sequence of all squarefree semiprimes together with the products of their prime indices begins:
   6: 1 * 2 = 2
  10: 1 * 3 = 3
  14: 1 * 4 = 4
  15: 2 * 3 = 6
  21: 2 * 4 = 8
  22: 1 * 5 = 5
  26: 1 * 6 = 6
  33: 2 * 5 = 10
  34: 1 * 7 = 7
  35: 3 * 4 = 12

A001358 lists semiprimes.
A003963 gives the product of prime indices of n.
A005117 lists squarefree numbers.
A006881 lists squarefree semiprimes.
A025129 is the sum of squarefree semiprimes of weight n.
A332765/A339114 give the greatest/least squarefree semiprime of weight n.
A338898/A338912/A338913 give the prime indices of semiprimes, with product/sum/difference A087794/A176504/A176506.
A338899/A270650/A270652 give the prime indices of squarefree semiprimes, with product/sum/difference A339361/A339362/A338900.
A338905 groups squarefree semiprimes by weight.
A338907/A338908 list squarefree semiprimes of odd/even weight.
A339116 groups squarefree semiprimes by greater prime factor.
Cf. A001221, A046388, A056239 (weight), A112798, A166237, A320656, A320911, A338901, A339002, A339003, A339004.

Table[Times@@PrimePi/@First/@FactorInteger[n],{n,Select[Range[100],SquareFreeQ[#]&&PrimeOmega[#]==2&]}]

a(n) = A003963(A006881(n)).

a(n) = A270650(n) * A270652(n).

A340104 Products of distinct primes of nonprime index (A007821).

Original entry on oeis.org

1, 2, 7, 13, 14, 19, 23, 26, 29, 37, 38, 43, 46, 47, 53, 58, 61, 71, 73, 74, 79, 86, 89, 91, 94, 97, 101, 103, 106, 107, 113, 122, 131, 133, 137, 139, 142, 146, 149, 151, 158, 161, 163, 167, 173, 178, 181, 182, 193, 194, 197, 199, 202, 203, 206, 214, 223, 226

Offset: 1

Gus Wiseman, Mar 12 2021

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.

			The sequence of terms together with the corresponding prime indices of prime indices begins:
     1: {}              58: {{},{1,3}}        113: {{1,2,3}}
     2: {{}}            61: {{1,2,2}}         122: {{},{1,2,2}}
     7: {{1,1}}         71: {{1,1,3}}         131: {{1,1,1,1,1}}
    13: {{1,2}}         73: {{2,4}}           133: {{1,1},{1,1,1}}
    14: {{},{1,1}}      74: {{},{1,1,2}}      137: {{2,5}}
    19: {{1,1,1}}       79: {{1,5}}           139: {{1,7}}
    23: {{2,2}}         86: {{},{1,4}}        142: {{},{1,1,3}}
    26: {{},{1,2}}      89: {{1,1,1,2}}       146: {{},{2,4}}
    29: {{1,3}}         91: {{1,1},{1,2}}     149: {{3,4}}
    37: {{1,1,2}}       94: {{},{2,3}}        151: {{1,1,2,2}}
    38: {{},{1,1,1}}    97: {{3,3}}           158: {{},{1,5}}
    43: {{1,4}}        101: {{1,6}}           161: {{1,1},{2,2}}
    46: {{},{2,2}}     103: {{2,2,2}}         163: {{1,8}}
    47: {{2,3}}        106: {{},{1,1,1,1}}    167: {{2,6}}
    53: {{1,1,1,1}}    107: {{1,1,4}}         173: {{1,1,1,3}}

These primes (of nonprime index) are listed by A007821.
The non-strict version is A320628, with odd case A320629.
The odd case is A340105.
The prime instead of nonprime version:
  primes: A006450
  products: A076610
  strict: A302590
The semiprime instead of nonprime version:
  primes: A106349
  products: A339112
  strict: A340020
The squarefree semiprime instead of nonprime version:
  strict: A309356
  primes: A322551
  products: A339113
A056239 gives the sum of prime indices, which are listed by A112798.
A257994 counts prime prime indices.
A302242 is the weight of the multiset of multisets with MM-number n.
A305079 is the number of connected components for MM-number n.
A320911 lists products of squarefree semiprimes (Heinz numbers of A338914).
A320912 lists products of distinct semiprimes (Heinz numbers of A338916).
A330944 counts nonprime prime indices.
A330945 lists numbers with a nonprime prime index (nonprime case: A330948).
A339561 lists products of distinct squarefree semiprimes (A339560).
MM-numbers: A255397 (normal), A302478 (set multisystems), A320630 (set multipartitions), A302494 (sets of sets), A305078 (connected), A316476 (antichains), A318991 (chains), A320456 (covers), A328514 (connected sets of sets), A329559 (clutters), A340019 (half-loop graphs).
Cf. A000040, A000720, A001055, A001222, A003963, A005117, A007097, A018252, A289509, A320461, A320631.

Select[Range[100],SquareFreeQ[#]&&FreeQ[If[#==1,{},FactorInteger[#]],{p_,k_}/;PrimeQ[PrimePi[p]]]&]

Equals A005117 /\ A320628.

A340105 Odd products of distinct primes of nonprime index (A007821).

Original entry on oeis.org

1, 7, 13, 19, 23, 29, 37, 43, 47, 53, 61, 71, 73, 79, 89, 91, 97, 101, 103, 107, 113, 131, 133, 137, 139, 149, 151, 161, 163, 167, 173, 181, 193, 197, 199, 203, 223, 227, 229, 233, 239, 247, 251, 257, 259, 263, 269, 271, 281, 293, 299, 301, 307, 311, 313, 317

Offset: 1

Gus Wiseman, Mar 12 2021

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.

			The sequence of terms together with the corresponding sets of multisets begins:
     1: {}              91: {{1,1},{1,2}}      173: {{1,1,1,3}}
     7: {{1,1}}         97: {{3,3}}            181: {{1,2,4}}
    13: {{1,2}}        101: {{1,6}}            193: {{1,1,5}}
    19: {{1,1,1}}      103: {{2,2,2}}          197: {{2,2,3}}
    23: {{2,2}}        107: {{1,1,4}}          199: {{1,9}}
    29: {{1,3}}        113: {{1,2,3}}          203: {{1,1},{1,3}}
    37: {{1,1,2}}      131: {{1,1,1,1,1}}      223: {{1,1,1,1,2}}
    43: {{1,4}}        133: {{1,1},{1,1,1}}    227: {{4,4}}
    47: {{2,3}}        137: {{2,5}}            229: {{1,3,3}}
    53: {{1,1,1,1}}    139: {{1,7}}            233: {{2,7}}
    61: {{1,2,2}}      149: {{3,4}}            239: {{1,1,6}}
    71: {{1,1,3}}      151: {{1,1,2,2}}        247: {{1,2},{1,1,1}}
    73: {{2,4}}        161: {{1,1},{2,2}}      251: {{1,2,2,2}}
    79: {{1,5}}        163: {{1,8}}            257: {{3,5}}
    89: {{1,1,1,2}}    167: {{2,6}}            259: {{1,1},{1,1,2}}

These primes (of nonprime index) are listed by A007821.
The non-strict version is A320629, with not necessarily odd version A320628.
The not necessarily odd version is A340104.
The prime instead of odd nonprime version:
  primes: A006450
  products: A076610
  strict: A302590
The squarefree semiprime instead of odd nonprime version:
  strict: A309356
  primes: A322551
  products: A339113
The semiprime instead of odd nonprime version:
  primes: A106349
  products: A339112
  strict: A340020
A001358 lists semiprimes.
A056239 gives the sum of prime indices, which are listed by A112798.
A257994 counts prime prime indices.
A302242 is the weight of the multiset of multisets with MM-number n.
A305079 is the number of connected components for MM-number n.
A330944 counts nonprime prime indices.
A330945 lists numbers with a nonprime prime index (nonprime case: A330948).
A339561 lists products of distinct squarefree semiprimes.
MM-numbers: A255397 (normal), A302478 (set multisystems), A320630 (set multipartitions), A302494 (sets of sets), A305078 (connected), A316476 (antichains), A318991 (chains), A320456 (covers), A328514 (connected sets of sets), A329559 (clutters), A340019 (half-loop graphs).
Cf. A000040, A000720, A001055, A001222, A003963, A005117, A007097, A018252, A289509, A320461, A320631, A320911, A320912.

Select[Range[1,100,2],SquareFreeQ[#]&&FreeQ[If[#==1,{},FactorInteger[#]],{p_,k_}/;PrimeQ[PrimePi[p]]]&]

Equals A005117 /\ A005408 /\ A320628 = A005117 /\ A320629.

A368728 Numbers whose prime indices are 1, prime, or semiprime.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 54, 55, 56, 58, 59, 60, 62, 63, 64, 65, 66, 67, 68, 69, 70, 72, 73, 75

Offset: 1

Gus Wiseman, Jan 07 2024

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.

These are products of primes indexed by elements of A037143.
For just primes we have A076610, strict A302590.
For just semiprimes we have A339112, strict A340020.
For squarefree semiprimes we have A339113, strict A309356.
The odd case is A368729, strict A340019.
The complement is A368833.
A000607 counts partitions into primes, A034891 with ones allowed.
A001358 lists semiprimes, squarefree A006881.
A006450, A106349, A322551, A368732 list selected primes.
A056239 adds up prime indices, row sums of A112798.
A101048 counts partitions into semiprimes.
Cf. A000040, A000720, A001222, A003963, A005117, A302242, A320628, A320911, A320912.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Max@@Length/@prix/@prix[#]<=2&]

Closed under multiplication.

A368732 Primes whose index is one, another prime number, or a semiprime.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 23, 29, 31, 41, 43, 47, 59, 67, 73, 79, 83, 97, 101, 109, 127, 137, 139, 149, 157, 163, 167, 179, 191, 199, 211, 227, 233, 241, 257, 269, 271, 277, 283, 293, 313, 331, 347, 353, 367, 373, 389, 401, 421, 431, 439, 443, 449, 461, 467, 487

Offset: 1

Gus Wiseman, Jan 08 2024

nonn

For just primes we have A006450, products A076610, strict A302590.
These indices are A037143.
For just semiprimes we have A106349, products A339112, strict A340020.
Products of these primes are A368728, odd A368729, odd strict A340019.
Products of the complementary primes are A368833.
A000607 counts partitions into primes, with ones allowed A034891.
A001358 lists semiprimes, squarefree A006881.
A056239 adds up prime indices, row sums of A112798.
A101048 counts partitions into semiprimes.
A322551 lists primes of squarefree semiprime index.
Cf. A000040, A000720, A001222, A003963, A005117, A302242, A309356, A320628, A320911, A320912, A339113.

Prime/@Select[Range[100],PrimeOmega[#]<=2&]

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&]]

A320913 Numbers with an even number of prime factors (counted with multiplicity) that cannot be factored into squarefree semiprimes (A320891) but can be factored into distinct semiprimes (A320912).

Original entry on oeis.org

4, 9, 24, 25, 40, 49, 54, 56, 88, 104, 121, 135, 136, 152, 169, 184, 189, 232, 240, 248, 250, 289, 296, 297, 328, 336, 344, 351, 361, 375, 376, 424, 459, 472, 488, 513, 528, 529, 536, 560, 568, 584, 621, 624, 632, 664, 686, 712, 776, 783, 808, 810, 816, 824

Offset: 1

Gus Wiseman, Oct 23 2018

nonn

A semiprime (A001358) is a product of any two not necessarily distinct primes.

If A025487(k) is contained in this sequence then so is every positive integer with its prime signature. - David A. Corneth, Oct 24 2018

Cf. A001055, A001222, A001358, A005117, A006881, A007717, A025487, A028260, A320655, A320656, A320891, A320892, A320893, A320894, A320911, A320912.

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

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

A339361 Product of prime indices of the n-th squarefree semiprime.

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A340104 Products of distinct primes of nonprime index (A007821).

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A340105 Odd products of distinct primes of nonprime index (A007821).

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A368728 Numbers whose prime indices are 1, prime, or semiprime.

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A368732 Primes whose index is one, another prime number, or a semiprime.

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A339191 Partial products of squarefree semiprimes (A006881).

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A320913 Numbers with an even number of prime factors (counted with multiplicity) that cannot be factored into squarefree semiprimes (A320891) but can be factored into distinct semiprimes (A320912).

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

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