A358105 -id:A358105 - cp's OEIS frontend

A318990 Numbers of the form prime(x) * prime(y) where x divides y.

4, 6, 9, 10, 14, 21, 22, 25, 26, 34, 38, 39, 46, 49, 57, 58, 62, 65, 74, 82, 86, 87, 94, 106, 111, 115, 118, 121, 122, 129, 133, 134, 142, 146, 158, 159, 166, 169, 178, 183, 185, 194, 202, 206, 213, 214, 218, 226, 235, 237, 254, 259, 262, 267, 274, 278, 289

Offset: 1

Gus Wiseman, Sep 06 2018

nonn

			The sequence of all dividing pairs (columns) begins:
  1  1  2  1  1  2  1  3  1  1  1  2  1  4  2  1  1  3  1  1  1  2  1  1
  1  2  2  3  4  4  5  3  6  7  8  6  9  4  8 10 11  6 12 13 14 10 15 16

Andrew Howroyd, Table of n, a(n) for n = 1..10000

A subset of A001358 (semiprimes), squarefree A006881.
The squarefree version is A339005.
The quotient is A358103 = A358104 / A358105.
A000040 lists the primes.
A001222 counts prime indices, distinct A001221.
A003963 multiplies together prime indices.
A056239 adds up prime indices.
A358192/A358193 gives quotients of semiprime indices.
Cf. A000837, A300912, A318991, A318992, A318993.
Cf. A000720, A027751, A032741, A215366, A296150.
Cf. A289508, A289509, A358106.

Select[Range[100],And[PrimeOmega[#]==2,Or[PrimePowerQ[#],Divisible@@Reverse[PrimePi/@FactorInteger[#][[All,1]]]]]&]

ok(n)={my(f=factor(n)); bigomega(f)==2 && (#f~==1 || primepi(f[2,1]) % primepi(f[1,1]) == 0)} \\ Andrew Howroyd, Oct 26 2018

A358103 Quotient of the n-th divisible pair, where pairs are ordered by Heinz number. Quotient of prime indices of A318990(n).

Original entry on oeis.org

1, 2, 1, 3, 4, 2, 5, 1, 6, 7, 8, 3, 9, 1, 4, 10, 11, 2, 12, 13, 14, 5, 15, 16, 6, 3, 17, 1, 18, 7, 2, 19, 20, 21, 22, 8, 23, 1, 24, 9, 4, 25, 26, 27, 10, 28, 29, 30, 5, 11, 31, 3, 32, 12, 33, 34, 1, 35, 36, 13, 6, 37, 2, 14, 38, 39, 15, 40, 41, 1, 42, 7, 4, 43

Offset: 1

Gus Wiseman, Nov 02 2022

nonn

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

			The 12th divisible pair is (2,6) so a(12) = 3.

The divisible pairs are ranked by A318990, proper A339005.
Quotient of A358104 and A358105.
A different ordering is A358106.
A000040 lists the primes.
A001222 counts prime indices, distinct A001221.
A001358 lists the semiprimes, squarefree A006881.
A003963 multiplies together prime indices.
A056239 adds up prime indices.
A358192/A358193 gives quotients of semiprime indices.
Cf. A000720, A027751, A032741, A215366, A289508, A289509, A296150, A300912, A318991, A318992.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Join@@Table[Cases[primeMS[n],{x_,y_}/;Divisible[y,x]:>y/x,{0}],{n,100}]

a(n) = A358104(n)/A358105(n).

A358104 Unreduced numerator of the n-th divisible pair, where pairs are ordered by Heinz number. Greater prime index of A318990(n).

Original entry on oeis.org

1, 2, 2, 3, 4, 4, 5, 3, 6, 7, 8, 6, 9, 4, 8, 10, 11, 6, 12, 13, 14, 10, 15, 16, 12, 9, 17, 5, 18, 14, 8, 19, 20, 21, 22, 16, 23, 6, 24, 18, 12, 25, 26, 27, 20, 28, 29, 30, 15, 22, 31, 12, 32, 24, 33, 34, 7, 35, 36, 26, 18, 37, 10, 28, 38, 39, 30, 40, 41, 8, 42

Offset: 1

Gus Wiseman, Nov 02 2022

nonn

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

			The 12th divisible pair is (2,6) so a(12) = 6.

The divisible pairs are ranked by A318990, proper A339005.
For all semiprimes we have A338913.
The quotient of the pair is A358103.
The denominator is A358105.
The reduced version for all semiprimes is A358192, denominator A358193.
A000040 lists the primes.
A001222 counts prime indices, distinct A001221.
A001358 lists the semiprimes, squarefree A006881.
A003963 multiplies together prime indices.
A056239 adds up prime indices.
A318991 ranks divisor-chains.
Cf. A000720, A032741, A128301, A215366, A289508, A289509, A296150, A300912, A318992, A358106.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Join@@Table[Cases[primeMS[n],{x_,y_}/;Divisible[y,x]:>y,{0}],{n,1000}]

A358103(n) = a(n)/A358105(n).

A358106 Quotient of the n-th divisible pair, where pairs are ordered first by sum and then by denominator.

Original entry on oeis.org

1, 2, 3, 1, 4, 5, 2, 1, 6, 7, 3, 1, 8, 2, 9, 4, 1, 10, 11, 5, 3, 2, 1, 12, 13, 6, 1, 14, 4, 2, 15, 7, 3, 1, 16, 17, 8, 5, 2, 1, 18, 19, 9, 4, 3, 1, 20, 6, 2, 21, 10, 1, 22, 23, 11, 7, 5, 3, 2, 1, 24, 4, 25, 12, 1, 26, 8, 2, 27, 13, 6, 3, 1, 28, 29, 14, 9, 5, 4, 2, 1

Offset: 2

Gus Wiseman, Nov 03 2022

			Grouping by sum gives:
   2:  1
   3:  2
   4:  3 1
   5:  4
   6:  5 2 1
   7:  6
   8:  7 3 1
   9:  8 2
  10:  9 4 1
  11: 10
  12: 11 5 3 2 1
  13: 12
  14: 13 6 1
  15: 14 4 2
  16: 15 7 3 1
  17: 16
  18: 17 8 5 2 1

Row-lengths are A032741.
This is A208460/A027751.
A ranking of divisible pairs is A318990, proper A339005.
A different ordering is A358103 = A358104 / A358105.
A000041 counts partitions, strict A000009.
A001358 lists semiprimes, squarefree A006881.
A318991 ranks divisor-chains.
A358192/A358193 gives quotients of semiprime indices.
Cf. A000837, A003238, A122934.

Table[Divide@@@Select[IntegerPartitions[n,{2}],Divisible@@#&],{n,2,30}]

a(n) = A208460(n)/A027751(n).

A358192 Numerator of the quotient of the prime indices of the n-th semiprime.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Nov 03 2022

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 31st semiprime has prime indices (4,6), so the quotient is 4/6 = 2/3; hence a(31) = 2.

The divisible pairs are ranked by A318990, proper A339005.
The unreduced pair is (A338912, A338913).
The quotients of divisible pairs are A358103.
The restriction to divisible pairs is A358104, denominator A358105.
The denominator is A358193.
A000040 lists the primes.
A001222 counts prime indices, distinct A001221.
A001358 lists the semiprimes, squarefree A006881.
A003963 multiplies together prime indices.
A056239 adds up prime indices.
Cf. A000720, A027751, A128301, A215366, A289508, A289509, A296150, A300912, A318991, A358106.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Numerator/@Divide@@@primeMS/@Select[Range[100],PrimeOmega[#]==2&]

A358193 Denominator of the quotient of the prime indices of the n-th semiprime.

Original entry on oeis.org

1, 2, 1, 3, 4, 3, 2, 5, 1, 6, 5, 7, 4, 8, 3, 9, 1, 7, 5, 4, 10, 11, 2, 9, 12, 5, 13, 7, 14, 5, 3, 11, 15, 8, 16, 6, 3, 17, 7, 1, 18, 13, 7, 2, 19, 15, 20, 6, 10, 21, 11, 22, 8, 9, 23, 1, 17, 24, 9, 4, 7, 25, 19, 26, 5, 13, 27, 8, 10, 28, 14, 11, 29, 21, 7, 30

Offset: 1

Gus Wiseman, Nov 03 2022

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 31-st semiprime has prime indices (4,6), so the quotient is 4/6 = 2/3; hence a(31) = 3.

The divisible pairs are ranked by A318990, proper A339005.
The unreduced pair is (A338912, A338913).
The quotients of divisible pairs are A358103.
The restriction to divisible pairs is A358105, numerator A358104.
The numerator is A358192.
A000040 lists the primes.
A001222 counts prime indices, distinct A001221.
A001358 lists the semiprimes, squarefree A006881.
A003963 multiplies together prime indices.
A056239 adds up prime indices.
Cf. A000720, A027751, A032741, A128301, A215366, A289508, A289509, A296150, A300912, A318991, A358106.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Denominator/@Divide@@@primeMS/@Select[Range[100],PrimeOmega[#]==2&]

cp's OEIS Frontend

A318990 Numbers of the form prime(x) * prime(y) where x divides y.

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A358103 Quotient of the n-th divisible pair, where pairs are ordered by Heinz number. Quotient of prime indices of A318990(n).

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A358104 Unreduced numerator of the n-th divisible pair, where pairs are ordered by Heinz number. Greater prime index of A318990(n).

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A358106 Quotient of the n-th divisible pair, where pairs are ordered first by sum and then by denominator.

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A358192 Numerator of the quotient of the prime indices of the n-th semiprime.

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A358193 Denominator of the quotient of the prime indices of the n-th semiprime.

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