A243056 -id:A243056 - cp's OEIS frontend

A243055 Difference between the indices of the smallest and the largest prime dividing n: If n = p_i * ... * p_k, where p_i <= ... <= p_k, where p_h = A000040(h), then a(n) = (k-i), a(1) = 0 by convention.

0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 1, 0, 3, 1, 0, 0, 1, 0, 2, 2, 4, 0, 1, 0, 5, 0, 3, 0, 2, 0, 0, 3, 6, 1, 1, 0, 7, 4, 2, 0, 3, 0, 4, 1, 8, 0, 1, 0, 2, 5, 5, 0, 1, 2, 3, 6, 9, 0, 2, 0, 10, 2, 0, 3, 4, 0, 6, 7, 3, 0, 1, 0, 11, 1, 7, 1, 5, 0, 2, 0, 12, 0, 3, 4, 13, 8, 4, 0, 2

Offset: 1

Antti Karttunen, May 31 2014

nonn

For n>=1, A100484(n+1) gives the position where n occurs for the first time (setting also the records for the sequence).

a(n) = the difference between the largest and the smallest parts of the partition having Heinz number n. We define the Heinz number of a partition p = [p_1, p_2, ..., p_r] as Product(p_j-th prime, j=1...r) (concept used by Alois P. Heinz in A215366 as an "encoding" of a partition). For example, for the partition [1, 1, 2, 4, 10] we get 2*2*3*7*29 = 2436. Example: a(57) = 6; indeed, the partition having Heinz number 57 = 3*19 is [2, 8]. - Emeric Deutsch, Jun 04 2015

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences computed from indices in prime factorization

Differs from A242411 for the first time at n=30.
A000961 gives the positions of zeros.
Cf. A243056, A241917, A241919, A049084, A027748, A055396, A061395, A100484, A215366.

with(numtheory):
a:= n-> `if`(n=1, 0, (f-> pi(max(f[]))-pi(min(f[])))(factorset(n))):
seq(a(n), n=1..100);  # Alois P. Heinz, Jun 04 2015

a[1]=0; a[n_] := Function[{f}, PrimePi[Max[f]] - PrimePi[Min[f]]][FactorInteger[n][[All, 1]]]; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Jul 29 2015, after Alois P. Heinz *)

from sympy import primepi, primefactors
def A243055(n): return primepi(max(p:=primefactors(n),default=0))-primepi(min(p,default=0)) # Chai Wah Wu, Oct 10 2023

(define (A243055 n) (- (A061395 n) (A055396 n)))

If n = p_i * ... * p_k, where p_i <= ... <= p_k are not necessarily distinct primes (sorted into nondescending order) in the prime factorization of n, where p_i = A000040(i), then a(n) = (k-i).

a(n) = A061395(n) - A055396(n).

A355536 Irregular triangle read by rows where row n lists the differences between adjacent prime indices of n; if n is prime, row n is empty.

Original entry on oeis.org

0, 1, 0, 0, 0, 2, 0, 1, 3, 1, 0, 0, 0, 1, 0, 0, 2, 2, 4, 0, 0, 1, 0, 5, 0, 0, 0, 3, 1, 1, 0, 0, 0, 0, 3, 6, 1, 0, 1, 0, 7, 4, 0, 0, 2, 1, 2, 0, 4, 0, 1, 8, 0, 0, 0, 1, 0, 2, 0, 5, 0, 5, 1, 0, 0, 2, 0, 0, 3, 6, 9, 0, 1, 1, 10, 0, 2, 0, 0, 0, 0, 0, 3, 1, 3, 0, 6

Offset: 2

Gus Wiseman, Jul 12 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 version where zero is prepended to the prime indices is A287352.
One could argue that row n = 1 is empty, but adding it changes only the offset, not the data.

			Triangle begins (showing n, prime indices, differences*):
   2:    (1)       .
   3:    (2)       .
   4:   (1,1)      0
   5:    (3)       .
   6:   (1,2)      1
   7:    (4)       .
   8:  (1,1,1)    0 0
   9:   (2,2)      0
  10:   (1,3)      2
  11:    (5)       .
  12:  (1,1,2)    0 1
  13:    (6)       .
  14:   (1,4)      3
  15:   (2,3)      1
  16: (1,1,1,1)  0 0 0

Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Row-lengths are A001222 minus one.
The prime indices are A112798, sum A056239.
Row-sums are A243055.
Constant rows have indices A325328.
The Heinz numbers of the rows plus one are A325352.
Strict rows have indices A325368.
Row minima are A355524.
Row maxima are A286470, also A355526.
An adjusted version is A358169, reverse A355534.
Cf. A066312, A124010, A129654, A243056, A287352, A325394, A355523, A355528, A355531.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Differences[primeMS[n]],{n,2,100}]

A355534 Irregular triangle read by rows where row n lists the augmented differences of the reversed prime indices of n.

Original entry on oeis.org

1, 2, 1, 1, 3, 2, 1, 4, 1, 1, 1, 1, 2, 3, 1, 5, 2, 1, 1, 6, 4, 1, 2, 2, 1, 1, 1, 1, 7, 1, 2, 1, 8, 3, 1, 1, 3, 2, 5, 1, 9, 2, 1, 1, 1, 1, 3, 6, 1, 1, 1, 2, 4, 1, 1, 10, 2, 2, 1, 11, 1, 1, 1, 1, 1, 4, 2, 7, 1, 2, 3, 1, 2, 1, 1, 12, 8, 1, 5, 2, 3, 1, 1, 1

Offset: 2

Gus Wiseman, Jul 12 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 augmented differences aug(q) of a (usually weakly decreasing) sequence q of length k are given by aug(q)i = q_i - q{i+1} + 1 if i < k and aug(q)_k = q_k. For example, we have aug(6,5,5,3,3,3) = (2,1,3,1,1,3).
One could argue that row n = 1 is empty, but adding it changes only the offset, not the data.

			Triangle begins:
   2: 1
   3: 2
   4: 1 1
   5: 3
   6: 2 1
   7: 4
   8: 1 1 1
   9: 1 2
  10: 3 1
  11: 5
  12: 2 1 1
  13: 6
  14: 4 1
  15: 2 2
  16: 1 1 1 1
For example, the reversed prime indices of 825 are (5,3,3,2), which have augmented differences (3,1,2,2).

Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Crossrefs found in the link are not repeated here.
Row-lengths are A001222.
Row-sums are A252464
Other similar triangles are A287352, A091602.
Constant rows have indices A307824.
The Heinz numbers of the rows are A325351.
Strict rows have indices A325366.
Row minima are A355531, non-augmented A355524, also A355525.
Row maxima are A355535, non-augmented A286470, also A355526.
The non-augmented version is A355536, also A355533.
A112798 lists prime indices, sum A056239.
Cf. A066312, A124010, A129654, A243055, A243056, A325352, A325394, A355523, A355528, A355531.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
aug[y_]:=Table[If[i

A251719 a(n) = the least k such that A250474(k) > n.

Original entry on oeis.org

1, 1, 1, 2, 3, 3, 3, 3, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10

Offset: 1

Antti Karttunen, Dec 15 2014

nonn

Provided that A250474 is strictly increasing (implied for example if either Legendre's or Brocard's conjecture is true) then all natural numbers occur in this sequence, in order, and after three 1's, each n+1 appears for the first time at A250474(n). Thus from n=2 onward, each n occurs A251723(n-1) times.

With the same provision, we have for n>1: a(n) = smallest positive integer k such that A083221(k, n) is a semiprime and A083221(k+1, n) = A003961(A083221(k, n)), where A003961 shifts the prime factorization one step towards larger primes, thus the latter value is also a semiprime.

Antti Karttunen, Table of n, a(n) for n = 1..10351

Cf. A001222, A003961, A055396, A083221, A250474, A243056, A251717, A251718, A251723, A251724.

Equally: a(1) = a(2) = a(3) = 1; and for n>=4: a(n) = the largest k such that A250474(k-1) <= n.
Other identities. For all n >= 1:
a(n) >= A251718(n) >= A251717(n).
a(n) = A055396(A251724(n)), or equally, A251724(n) = A083221(a(n), n). [This sequence gives the row-index of the first "settled semiprime" in column n of the sieve of Eratosthenes.]

A355533 Irregular triangle read by rows where row n lists the differences between adjacent prime indices of n; if n is prime(k), then row n is just (k).

Original entry on oeis.org

1, 2, 0, 3, 1, 4, 0, 0, 0, 2, 5, 0, 1, 6, 3, 1, 0, 0, 0, 7, 1, 0, 8, 0, 2, 2, 4, 9, 0, 0, 1, 0, 5, 0, 0, 0, 3, 10, 1, 1, 11, 0, 0, 0, 0, 3, 6, 1, 0, 1, 0, 12, 7, 4, 0, 0, 2, 13, 1, 2, 14, 0, 4, 0, 1, 8, 15, 0, 0, 0, 1, 0, 2, 0

Offset: 2

Gus Wiseman, Jul 12 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 version where zero is prepended to the prime indices before taking differences is A287352.
One could argue that row n = 1 is empty, but adding it changes only the offset, with no effect on the data.

			Triangle begins (showing n, prime indices, differences*):
   2:    (1)       1
   3:    (2)       2
   4:   (1,1)      0
   5:    (3)       3
   6:   (1,2)      1
   7:    (4)       4
   8:  (1,1,1)    0 0
   9:   (2,2)      0
  10:   (1,3)      2
  11:    (5)       5
  12:  (1,1,2)    0 1
  13:    (6)       6
  14:   (1,4)      3
  15:   (2,3)      1
  16: (1,1,1,1)  0 0 0
For example, the prime indices of 24 are (1,1,1,2), with differences (0,0,1).

Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Crossrefs found in the link are not repeated here.
Row sums are A243056.
The version for prime indices prepended by 0 is A287352.
Constant rows have indices A325328.
Strict rows have indices A325368.
Number of distinct terms in each row are 1 if prime, otherwise A355523.
Row minima are A355525, augmented A355531.
Row maxima are A355526, augmented A355535.
The augmented version is A355534, Heinz number A325351.
The version with prime-indexed rows empty is A355536, Heinz number A325352.
A112798 lists prime indices, sum A056239.
Cf. A001222, A066312, A124010, A286469, A286470, A325160, A325390, A355524.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[If[PrimeQ[n],{PrimePi[n]},Differences[primeMS[n]]],{n,2,30}]

Row lengths are 1 or A001222(n) - 1 depending on whether n is prime.

A355531 Minimal augmented difference between adjacent reversed prime indices of n; a(1) = 0.

Original entry on oeis.org

0, 1, 2, 1, 3, 1, 4, 1, 1, 1, 5, 1, 6, 1, 2, 1, 7, 1, 8, 1, 2, 1, 9, 1, 1, 1, 1, 1, 10, 1, 11, 1, 2, 1, 2, 1, 12, 1, 2, 1, 13, 1, 14, 1, 1, 1, 15, 1, 1, 1, 2, 1, 16, 1, 3, 1, 2, 1, 17, 1, 18, 1, 1, 1, 3, 1, 19, 1, 2, 1, 20, 1, 21, 1, 1, 1, 2, 1, 22, 1, 1, 1

Offset: 1

Gus Wiseman, Jul 14 2022

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 augmented differences aug(q) of a (usually weakly decreasing) sequence q of length k are given by aug(q)i = q_i - q{i+1} + 1 if i < k and aug(q)_k = q_k. For example, we have aug(6,5,5,3,3,3) = (2,1,3,1,1,3).

			The reversed prime indices of 825 are (5,3,3,2), with augmented differences (3,1,2,2), so a(825) = 1.

Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Crossrefs found in the link are not repeated here.
Positions of first appearances are A008578.
Positions of 1's are 2 followed by A013929.
The non-augmented maximal version is A286470, also A355526.
The non-augmented version is A355524, also A355525.
Row minima of A355534, which has Heinz number A325351.
The maximal version is A355535.
A001222 counts prime indices.
A112798 lists prime indices, sum A056239.
Cf. A124010, A129654, A243055, A243056, A307824, A325366, A325394, A355528, A355533, A355536.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
aug[y_]:=Table[If[i

A243057 If n = p_a * p_b * ... * p_h * p_i * p_j * p_k, with p_a <= p_b <= ... <= p_k, then a(n) = p_{k-j} * p_{k-i} * p_{k-h} * ... * p_{k-a} * p_k, where p_{0} = 1 and for k>=1, p_{k} = A000040(k).

Original entry on oeis.org

1, 2, 3, 2, 5, 6, 7, 2, 3, 15, 11, 12, 13, 35, 10, 2, 17, 6, 19, 45, 21, 77, 23, 24, 5, 143, 3, 175, 29, 30, 31, 2, 55, 221, 14, 12, 37, 323, 91, 135, 41, 105, 43, 539, 20, 437, 47, 48, 7, 15, 187, 1573, 53, 6, 33, 875, 247, 667, 59, 90, 61, 899, 63, 2, 65, 385

Offset: 1

Antti Karttunen, May 31 2014

nonn

A243058 gives all n such that a(n) = n (the fixed points of this sequence, which include primes).
A102750 gives such n that a(a(n)) = n. A243286 is the self-inverse permutation induced when the domain is restricted to A102750. Cf. also A242420.
A070003 gives all n such that a(a(n)) <> n. Another variant, A243059, is defined to be zero when n is one of the terms of A070003.

			For n = 9 = 3*3 = p_2 * p_2, we have a(n) = p_{3-3} * p_3 = 1*3 = 3. [Like all terms in A070003 this is an example of "degenerate case", where some p's in the product get index 0, and thus are set to 1 by convention used here.]
For n = 10 = 2*5 = p_1 * p_3, we have a(n) = p_{3-1} * p_3 = 3*5 = 15.
For n = 12 = 2*2*3 = p_1 * p_1 * p_2, we have a(n) = p_{2-1} * p{2-1} * p_2 = p_1^2 * p_2 = 12.
For n = 15 = 3*5 = p_2 * p_3, we have a(n) = p_{3-2} * p_3 = 2*5 = 10.
For n = 2200 = 2*2*2*5*5*11 = p_1 * p_1 * p_1 * p_3 * p_3 * p_5, we have a(n) = p_{5-3} * p_{5-3} * p_{5-1} * p_{5-1} * p_{5-1} * p_5 = 3*3*7*7*7*11 = 33957.
For n = 33957 = 3*3*7*7*7*11 = p_2 * p_2 * p_4 * p_4 * p_4 * p_5, we have a(n) = p_{5-4} * p_{5-4} * p_{5-4} * p_{5-2} * p_{5-2} * p_5 = 2*2*2*5*5*11 = 2200.

Antti Karttunen, Table of n, a(n) for n = 1..10001

Fixed points: A243058 (includes primes).

Cf. A243059, A000040, A032742, A243056, A242419, A242420, A243286, A070003, A102750, A243074.

If n = p_a * p_b * ... * p_h * p_i * p_j * p_k, where p_a <= p_b <= ... <= p_k are (not necessarily distinct) primes (sorted into nondescending order) in the prime factorization of n, then a(n) = p_{k-j} * p_{k-i} * p_{k-h} * ... * p_{k-a} * p_k, where p_{0} = 1 and for k>=1, p_{k} = A000040(k).
a(1)=1, and for n>1, a(n) = p_{A243056(n)} * a(A032742(n)). Here p_{k} stands for 1 when k=0, and otherwise for the k-th prime, A000040(k).
For all n, a(n) = a(A243074(n)).
For all k in A102750, a(k) = A242420(k).

A243059 If n = p_a * p_b * ... * p_h * p_i * p_j * p_k, with p_a <= p_b <= ... <= p_k, then a(n) = p_{k-j} * p_{k-i} * p_{k-h} * ... * p_{k-a} * p_k, where p_{0} = 0 and for k>=1, p_{k} = A000040(k). a(1)=1 by convention.

Original entry on oeis.org

1, 2, 3, 0, 5, 6, 7, 0, 0, 15, 11, 12, 13, 35, 10, 0, 17, 0, 19, 45, 21, 77, 23, 24, 0, 143, 0, 175, 29, 30, 31, 0, 55, 221, 14, 0, 37, 323, 91, 135, 41, 105, 43, 539, 20, 437, 47, 48, 0, 0, 187, 1573, 53, 0, 33, 875, 247, 667, 59, 90, 61, 899, 63, 0, 65, 385, 67, 2873, 391, 70, 71, 0

Offset: 1

Antti Karttunen, May 31 2014

nonn

A243058 gives all n such that a(n) = n (the fixed points of this sequence, which include primes).

Differs from A243057 in that the "degenerate cases" A070003 are here zeros, but is otherwise equal to it (at the points given by A102750), i.e. for all n, a(A102750(n)) = A243057(A102750(n)) = A242420(A102750(n)).

			For n = 9 = 3*3 = p_2 * p_2, we have a(n) = p_{3-3} * p_3 = 0*3 = 0. [Like all terms in A070003 this is an example of "degenerate case", where some p's in the product get index 0, and thus are set to 0 by the convention used here.]
For n = 10 = 2*5 = p_1 * p_3, we have a(n) = p_{3-1} * p_3 = 3*5 = 15.
For n = 12 = 2*2*3 = p_1 * p_1 * p_2, we have a(n) = p_{2-1} * p{2-1} * p_2 = p_1^2 * p_2 = 12.
For n = 15 = 3*5 = p_2 * p_3, we have a(n) = p_{3-2} * p_3 = 2*5 = 10.
For n = 2200 = 2*2*2*5*5*11 = p_1 * p_1 * p_1 * p_3 * p_3 * p_5, we have a(n) = p_{5-3} * p_{5-3} * p_{5-1} * p_{5-1} * p_{5-1} * p_5 = 3*3*7*7*7*11 = 33957.
For n = 33957 = 3*3*7*7*7*11 = p_2 * p_2 * p_4 * p_4 * p_4 * p_5, we have a(n) = p_{5-4} * p_{5-4} * p_{5-4} * p_{5-2} * p_{5-2} * p_5 = 2*2*2*5*5*11 = 2200.

Antti Karttunen, Table of n, a(n) for n = 1..10001

Fixed points: A243058 (includes primes).
Positions of zeros: A070003.
Cf. A243057, A000040, A032742, A243056, A242419, A242420, A243286.

a(1)=1, and for n>1, a(n) = q_{A243056(n)} * a(A032742(n)). Here q_{k} stands for 0 when k=0, and otherwise for the k-th prime, A000040(k).

If n = p_a * p_b * ... * p_h * p_i * p_j * p_k, where p_a <= p_b <= ... <= p_k are (not necessarily distinct) primes (sorted into nondescending order) in the prime factorization of n, then a(n) = p_{k-j} * p_{k-i} * p_{k-h} * ... * p_{k-a} * p_k, where p_{0} = 0 and for k>=1, p_{k} = A000040(k).

A355532 Maximal augmented difference between adjacent reversed prime indices of n; a(1) = 0.

Original entry on oeis.org

0, 1, 2, 1, 3, 2, 4, 1, 2, 3, 5, 2, 6, 4, 2, 1, 7, 2, 8, 3, 3, 5, 9, 2, 3, 6, 2, 4, 10, 2, 11, 1, 4, 7, 3, 2, 12, 8, 5, 3, 13, 3, 14, 5, 2, 9, 15, 2, 4, 3, 6, 6, 16, 2, 3, 4, 7, 10, 17, 2, 18, 11, 3, 1, 4, 4, 19, 7, 8, 3, 20, 2, 21, 12, 2, 8, 4, 5, 22, 3, 2

Offset: 1

Gus Wiseman, Jul 14 2022

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 augmented differences aug(q) of a (usually weakly decreasing) sequence q of length k are given by aug(q)i = q_i - q{i+1} + 1 if i < k and aug(q)_k = q_k. For example, we have aug(6,5,5,3,3,3) = (2,1,3,1,1,3).

			The reversed prime indices of 825 are (5,3,3,2), with augmented differences (3,1,2,2), so a(825) = 3.

Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Crossrefs found in the link are not repeated here.
Prepending 1 to the positions of 1's gives A000079.
Positions of first appearances are A008578.
Positions of 2's are A065119.
The non-augmented version is A286470, also A355526.
The non-augmented minimal version is A355524, also A355525.
The minimal version is A355531.
Row maxima of A355534, which has Heinz number A325351.
A001222 counts prime indices, distinct A001221.
A112798 lists prime indices, sum A056239.
Cf. A066312, A124010, A129654, A243055, A243056, A307824, A325366, A325394, A355533, A355536.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
aug[y_]:=Table[If[i

A286471 If n is noncomposite, then a(n) = 0, otherwise 1 + difference between indices of the two smallest (not necessarily distinct) prime factors of n.

Original entry on oeis.org

0, 0, 0, 1, 0, 2, 0, 1, 1, 3, 0, 1, 0, 4, 2, 1, 0, 2, 0, 1, 3, 5, 0, 1, 1, 6, 1, 1, 0, 2, 0, 1, 4, 7, 2, 1, 0, 8, 5, 1, 0, 2, 0, 1, 1, 9, 0, 1, 1, 3, 6, 1, 0, 2, 3, 1, 7, 10, 0, 1, 0, 11, 1, 1, 4, 2, 0, 1, 8, 3, 0, 1, 0, 12, 2, 1, 2, 2, 0, 1, 1, 13, 0, 1, 5, 14, 9, 1, 0, 2, 3, 1, 10, 15, 6, 1, 0, 4, 1, 1, 0, 2, 0, 1, 2, 16, 0, 1, 0, 3, 11, 1, 0, 2, 7, 1, 1, 17

Offset: 1

Antti Karttunen, May 11 2017

nonn

			For n = 1, 2 and 3, which are all noncomposite numbers, a(n) = 0.
For n = 4 = 2*2 = prime(1)*prime(1), the difference 1-1 = 0, plus one is 1, thus a(4) = 1.
For n = 6 = 2*3 = prime(1)*prime(2), the difference 2-1 = 1, plus one is 2, thus a(6) = 2.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A001222, A032742, A055396.

Cf. also A241917, A241919, A242411, A243055, A243056, A286472.

Table[If[Length@ # < 2, 0, First@ Differences@ PrimePi@ Take[#, 2] + 1] &@ Flatten[FactorInteger[n] /. {p_, e_} /; p > 0 :> ConstantArray[p, e]], {n, 118}] (* Michael De Vlieger, May 12 2017 *)

from sympy import primepi, isprime, primefactors, divisors
def a049084(n): return primepi(n)*(1*isprime(n))
def a055396(n): return 0 if n==1 else a049084(min(primefactors(n)))
def a(n): return 0 if n==1 or isprime(n) else 1 + a055396(divisors(n)[-2]) - a055396(n) # Indranil Ghosh, May 12 2017

(define (A286471 n) (if (or (= 1 n) (= 1 (A001222 n))) 0 (+ 1 (- (A055396 (A032742 n)) (A055396 n)))))

If n is noncomposite, then a(n) = 0, otherwise a(n) = 1 + A055396(A032742(n)) - A055396(n).

cp's OEIS Frontend

A243055 Difference between the indices of the smallest and the largest prime dividing n: If n = p_i * ... * p_k, where p_i <= ... <= p_k, where p_h = A000040(h), then a(n) = (k-i), a(1) = 0 by convention.

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A355536 Irregular triangle read by rows where row n lists the differences between adjacent prime indices of n; if n is prime, row n is empty.

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A355534 Irregular triangle read by rows where row n lists the augmented differences of the reversed prime indices of n.

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A251719 a(n) = the least k such that A250474(k) > n.

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A355533 Irregular triangle read by rows where row n lists the differences between adjacent prime indices of n; if n is prime(k), then row n is just (k).

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A355531 Minimal augmented difference between adjacent reversed prime indices of n; a(1) = 0.

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A243057 If n = p_a * p_b * ... * p_h * p_i * p_j * p_k, with p_a <= p_b <= ... <= p_k, then a(n) = p_{k-j} * p_{k-i} * p_{k-h} * ... * p_{k-a} * p_k, where p_{0} = 1 and for k>=1, p_{k} = A000040(k).

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A243059 If n = p_a * p_b * ... * p_h * p_i * p_j * p_k, with p_a <= p_b <= ... <= p_k, then a(n) = p_{k-j} * p_{k-i} * p_{k-h} * ... * p_{k-a} * p_k, where p_{0} = 0 and for k>=1, p_{k} = A000040(k). a(1)=1 by convention.

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