A318993 -id:A318993 - cp's OEIS frontend

A318991 Numbers whose consecutive prime indices are divisible. Heinz numbers of integer partitions in which each part is divisible by the next.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 31, 32, 34, 36, 37, 38, 39, 40, 41, 42, 43, 44, 46, 47, 48, 49, 50, 52, 53, 54, 56, 57, 58, 59, 61, 62, 63, 64, 65, 67, 68, 71, 72, 73, 74, 76, 78, 79, 80

Offset: 1

Gus Wiseman, Sep 06 2018

nonn

A prime index of n is a number m such that prime(m) divides n.

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

			The sequence of all dividing partitions (columns) begins:
   1  2  1  3  2  4  1  2  3  5  2  6  4  1  7  2  8  3  4  5  9  2  3  6  2  4
         1     1     1  2  1     1     1  1     2     1  2  1     1  3  1  2  1
                     1           1        1     1     1           1        2  1
                                          1                       1

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

Cf. A000040, A001221, A001222, A003238, A008480, A300912, A318990, A318992, A318993.

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

ok(n)={my(v=apply(primepi, factor(n)[,1])); for(i=2, #v, if(v[i]%v[i-1], return(0))); 1} \\ Andrew Howroyd, Oct 26 2018

A318992 Numbers whose consecutive prime indices are not all divisible.

Original entry on oeis.org

15, 30, 33, 35, 45, 51, 55, 60, 66, 69, 70, 75, 77, 85, 90, 91, 93, 95, 99, 102, 105, 110, 119, 120, 123, 132, 135, 138, 140, 141, 143, 145, 150, 153, 154, 155, 161, 165, 170, 175, 177, 180, 182, 186, 187, 190, 195, 198, 201, 203, 204, 205, 207, 209, 210, 215

Offset: 1

Gus Wiseman, Sep 06 2018

nonn

			The sequence of partitions whose Heinz numbers belong to the sequence begins: (3,2), (3,2,1), (5,2), (4,3), (3,2,2), (7,2), (5,3), (3,2,1,1), (5,2,1), (9,2), (4,3,1), (3,3,2), (5,4), (7,3), (3,2,2,1), (6,4), (11,2), (8,3), (5,2,2).

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

Cf. A000040, A001221, A001222, A008480, A289509, A300912, A318990, A318991, A318993.

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

ok(n)={my(v=apply(primepi, factor(n)[,1])); for(i=2, #v, if(v[i]%v[i-1], return(1))); 0} \\ Andrew Howroyd, Oct 26 2018

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

Original entry on oeis.org

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

cp's OEIS Frontend

A318991 Numbers whose consecutive prime indices are divisible. Heinz numbers of integer partitions in which each part is divisible by the next.

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A318992 Numbers whose consecutive prime indices are not all divisible.

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

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