A339005 -id:A339005 - cp's OEIS frontend

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

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&]

A347047 Smallest squarefree semiprime whose prime indices sum to n.

Original entry on oeis.org

6, 10, 14, 21, 26, 34, 38, 46, 58, 62, 74, 82, 86, 94, 106, 118, 122, 134, 142, 146, 158, 166, 178, 194, 202, 206, 214, 218, 226, 254, 262, 274, 278, 298, 302, 314, 326, 334, 346, 358, 362, 382, 386, 394, 398, 422, 446, 454, 458, 466, 478, 482, 502, 514, 526

Offset: 3

Gus Wiseman, Aug 22 2021

nonn

Compared to A001747, we have 21 instead of 22 and lack 2 and 4.
Compared to A100484 (shifted) we have 21 instead of 22 and lack 4.
Compared to A161344, we have 21 instead of 22 and lack 4 and 8.
Compared to A339114, we have 11 instead of 9 and lack 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.
A squarefree semiprime (A006881) is a product of any two distinct prime numbers.

			The initial terms and their prime indices:
    6: {1,2}
   10: {1,3}
   14: {1,4}
   21: {2,4}
   26: {1,6}
   34: {1,7}
   38: {1,8}
   46: {1,9}

The opposite version (greatest instead of smallest) is A332765.
These are the minima of rows of A338905.
The nonsquarefree version is A339114 (opposite: A339115).
A001358 lists semiprimes (squarefree: A006881).
A024697 adds up semiprimes by weight (squarefree: A025129).
A056239 adds up prime indices, row sums of A112798.
A246868 gives the greatest squarefree number whose prime indices sum to n.
A320655 counts factorizations into semiprimes (squarefree: A320656).
A338898, A338912, A338913 give the prime indices of semiprimes.
A338899, A270650, A270652 give the prime indices of squarefree semiprimes.
A339116 groups squarefree semiprimes by greater factor, sums A339194.
A339362 adds up prime indices of squarefree semiprimes.
Cf. A001221, A087112, A089994, A098350, A176504, A338900, A338901, A338904, A338907/A338908, A339005, A339191.

Table[Min@@Select[Table[Times@@Prime/@y,{y,IntegerPartitions[n,{2}]}],SquareFreeQ],{n,3,50}]

from sympy import prime, sieve
def a(n):
    p = [0] + list(sieve.primerange(1, prime(n)+1))
    return min(p[i]*p[n-i] for i in range(1, (n+1)//2))
print([a(n) for n in range(3, 58)]) # Michael S. Branicky, Sep 05 2021

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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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Mathematica

A347047 Smallest squarefree semiprime whose prime indices sum to n.

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