A128301 -id:A128301 - cp's OEIS frontend

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

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).

A358105 Unreduced denominator of the n-th divisible pair, where pairs are ordered by Heinz number. Lesser prime index of A318990(n).

Original entry on oeis.org

1, 1, 2, 1, 1, 2, 1, 3, 1, 1, 1, 2, 1, 4, 2, 1, 1, 3, 1, 1, 1, 2, 1, 1, 2, 3, 1, 5, 1, 2, 4, 1, 1, 1, 1, 2, 1, 6, 1, 2, 3, 1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 4, 1, 2, 1, 1, 7, 1, 1, 2, 3, 1, 5, 2, 1, 1, 2, 1, 1, 8, 1, 3, 4, 1, 1, 2, 1, 1, 2, 1, 3, 1, 2, 1, 1, 1, 1

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) = 2.

The divisible pairs are ranked by A318990, proper A339005.
For all semiprimes we have A338912, greater A338913.
The quotient of the pair is A358103.
The reduced version for all semiprimes is A358193, numerator 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.
A318991 ranks divisor-chains.
Cf. A000720, A027751, 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]:>x,{0}],{n,1000}]

A358103(n) = A358104(n)/a(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&]

A128302 The indices of cubes (of primes) in the 3-almost primes.

Original entry on oeis.org

1, 5, 30, 82, 328, 551, 1243, 1763, 3112, 6276, 7619, 12972, 17615, 20322, 26514, 37977, 52220, 57703, 76200, 90701, 98470, 124541, 144229, 177395, 229275, 258410, 273908, 306750, 324149, 360724, 510225, 559384, 638657, 666743, 819645, 852588

Offset: 1

Rick L. Shepherd, Feb 25 2007

nonn

the primepi function might be used to find terms. But it is expensive for larger numbers so then one might use A335331 to ease finding primepi(m) for larger m. - David A. Corneth, Apr 13 2025

			a(4) = 82 as 343 = 7^3 = prime(4)^3, the fourth cube in the 3-almost primes, is the eighty-second 3-almost prime.

Amiram Eldar, Table of n, a(n) for n = 1..590
David A. Corneth, PARI program

Cf. A000040, A014612, A030078, A128301, A128303, A335331.

Position[Select[Range[10^6], PrimeOmega[#] == 3 &], ?(PrimeNu[#] == 1 &)] // Flatten (* _Amiram Eldar, Apr 13 2025 *)

list(lim) = {my(f, c); for(k = 1, lim, f = factor(k); if(bigomega(f) == 3, c++; if(omega(f) == 1, print1(c, ", "))));} \\ Amiram Eldar, Apr 13 2025

PARI
```
\\ See Corneth link
```

A014612(a(n)) = A030078(n) = A000040(n)^3.

A131188 Indices of products of twin primes in the semiprimes.

Original entry on oeis.org

6, 13, 48, 103, 270, 508, 1001, 1413, 2724, 3052, 4859, 5668, 8029, 9062, 9608, 12558, 13828, 17319, 18823, 22781, 28077, 40162, 42359, 48113, 60703, 71793, 79161, 83792, 90129, 94954, 140436, 144445, 146452, 156704, 165199, 218110, 223095

Offset: 1

Zak Seidov, Sep 25 2007

nonn

			Ignoring (2, 3), the first twin prime pair is (3, 5). We have 3 * 5 = 15, which is the sixth semiprime (the previous five semiprimes being 4, 6, 9, 10, 14). Hence 6 is the first term of this sequence.
The second twin prime pair is (5, 7). Then 5 * 7 = 35, which is the thirteenth semiprime (following 21, 22, 25, 26, 33, 34). Hence 13 is the second term of this sequence.

Zak Seidov, Table of n, a(n) for n = 1..300

Cf. A128301.

N:= 10^7: # to use semiprimes <= N
P:= select(isprime, [2,seq(i,i=3..N/2,2)]):
count:= 0:
for i from 1 to numtheory:-pi(floor(sqrt(N))) do
  for j from i to nops(P) while P[i]*P[j] <= N do
    count:= count+1;
    S[count]:= [P[i]*P[j],evalb(P[j]-P[i]=2)]
od od:
S:= sort(convert(S,list),(a,b) -> a[1] S[t][2],[$1..nops(S)]); # Robert Israel, Dec 30 2015

s = Select[Range[10^6], PrimeOmega@ # == 2 &]; Map[Position[s, #] &, # (# + 2) &@ Select[Prime@ Range@ 160, NextPrime@ # - # == 2 &]] // Flatten (* Michael De Vlieger, Dec 31 2015 *)
Module[{upto=2*10^6,sp,tp},sp=Select[Range[upto],PrimeOmega[#]==2&]; tp= Times@@@Select[Partition[Prime[Range[upto/2]],2,1],#[[2]]-#[[1]] == 2&]; Table[Position[sp,n],{n,tp}]]//Flatten (* Harvey P. Dale, Nov 03 2016 *)

{i: A001358(i) = A001359(k) * A006512(k), for some k > 0}. - R. J. Mathar, Oct 26 2007

More terms from R. J. Mathar, Oct 26 2007

A283235 Triangle read by rows: n-th row gives the numbers of primes p such that p*prime(k) <= prime(n)^2, k=1..n.

Original entry on oeis.org

1, 2, 2, 5, 4, 3, 9, 6, 4, 4, 17, 12, 9, 7, 5, 23, 16, 11, 9, 6, 6, 34, 24, 16, 13, 9, 8, 7, 41, 30, 20, 15, 11, 9, 8, 8, 56, 40, 27, 21, 15, 12, 11, 9, 9, 81, 59, 39, 30, 21, 18, 15, 14, 11, 10

Offset: 1

Zak Seidov, Mar 03 2017

Sequence is related to A128301 = indices of squares (of primes) in the semiprimes.

			Triangle begins:
   1;
   2,  2;
   5,  4,  3;
   9,  6,  4,  4;
  17, 12,  9,  7,  5;
  23, 16, 11,  9,  6,  6;
  34, 24, 16, 13,  9,  8,  7;
  41, 30, 20, 15, 11,  9,  8,  8;
  56, 40, 27, 21, 15, 12, 11,  9,  9;
  81, 59, 39, 30, 21, 18, 15, 14, 11, 10;
  ...

Cf. A128301, A348836 (1st column).

Table[PrimePi[Prime[n]^2/Prime[k]],{n,10},{k,n}]//Flatten

row(n) = my(p=prime(n)); vector(n, k, primepi(p^2/prime(k))); \\ Michel Marcus, Nov 01 2021

A289053 Array T(i,k) read by antidiagonals: position of prime(i)*prime(k) in A001358.

Original entry on oeis.org

1, 3, 2, 9, 6, 4, 17, 13, 7, 5, 40, 26, 19, 11, 8, 56, 48, 31, 23, 15, 10, 90, 75, 61, 39, 28, 18, 12, 114, 103, 79, 68, 44, 34, 20, 14, 164, 135, 122, 94, 81, 54, 37, 24, 16, 253, 199, 172, 152, 118, 101, 65, 49, 30, 21

Offset: 1

Zak Seidov, Jun 23 2017

			Array begins:
    1      2      4      5      8     10    12    14    16    21
    3      6      7     11     15     18    20    24    30
    9     13     19     23     28     34    37    49
   17     26     31     39     44     54    65
   40     48     61     68     81    101
   56     75     79     94    118
   90    103    122    152
  114    135    172
  164    199
  253

Cf. A000040, A001358, A115392 (1st row), A128301 (1st column).

With[{nn = 11}, Function[s, Table[Position[s, Prime[i] Prime[k]][[1, 1]], {i, nn}, {k, i, 1, -1}] // Flatten]@ Select[Range[Prime[nn]^2], PrimeOmega@ # == 2 &]] (* Michael De Vlieger, Jun 23 2017 *)

A376479 Array read by antidiagonals: T(n,k) is the index of prime(k)^n in the numbers with n prime factors, counted with multiplicity.

Original entry on oeis.org

1, 2, 1, 3, 3, 1, 4, 9, 5, 1, 5, 17, 30, 8, 1, 6, 40, 82, 90, 14, 1, 7, 56, 328, 385, 269, 23, 1, 8, 90, 551, 2556, 1688, 788, 39, 1, 9, 114, 1243, 5138, 18452, 7089, 2249, 64, 1, 10, 164, 1763, 15590, 44329, 126096, 28893, 6340, 103, 1, 11, 253, 3112, 24646, 179313, 361249, 827901, 115180, 17526

Offset: 1

Robert Israel, Sep 24 2024

T(n,k) is the number of numbers j with n prime factors, counted with multiplicity, such that j <= prime(k)^n.

			T(2,3) = 9 because the third prime is 5 and 5^2 = 25 is the 9th semiprime.

Cf. A001222, A078843 (second column), A078844 (third column), A078845 (fourth column), A078846 (fifth column), A128301 (second row), A128302 (third row), A128304 (fourth row).

T:= Matrix(12,12):
with(priqueue);
for m from 1 to 12 do
  initialize(pq);
  insert([-2^m, [2$m]],pq);
  k:= 0:
  for count from 1 do
    t:= extract(pq);
    w:= t[2];
    if nops(convert(w,set))=1 then
      k:= k+1;
      T[m,k]:= count;
      if m+k = 13 then break fi;
    fi;
    p:= nextprime(w[-1]);
    for i from m to 1 by -1 while w[i] = w[m] do
      insert([t[1]*(p/w[-1])^(m+1-i),[op(w[1..i-1]),p$(m+1-i)]],pq);
od od od:
seq(seq(T[i,s-i],i=1..s-1),s=2..13)

cp's OEIS Frontend

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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A358105 Unreduced denominator of the n-th divisible pair, where pairs are ordered by Heinz number. Lesser prime index of A318990(n).

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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

A128302 The indices of cubes (of primes) in the 3-almost primes.

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Mathematica

PARI

PARI

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A131188 Indices of products of twin primes in the semiprimes.

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Maple

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A283235 Triangle read by rows: n-th row gives the numbers of primes p such that p*prime(k) <= prime(n)^2, k=1..n.

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Mathematica

PARI

A289053 Array T(i,k) read by antidiagonals: position of prime(i)*prime(k) in A001358.

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