A174090 -id:A174090 - cp's OEIS frontend

A065770 Number of prime cascades to reach 1, where a prime cascade (A065769) is multiplicative with a(p(m)^k) = p(m-1) * p(m)^(k-1).

0, 1, 2, 2, 3, 2, 4, 3, 3, 3, 5, 3, 6, 4, 3, 4, 7, 3, 8, 3, 4, 5, 9, 4, 4, 6, 4, 4, 10, 3, 11, 5, 5, 7, 4, 4, 12, 8, 6, 4, 13, 4, 14, 5, 4, 9, 15, 5, 5, 4, 7, 6, 16, 4, 5, 4, 8, 10, 17, 4, 18, 11, 4, 6, 6, 5, 19, 7, 9, 4, 20, 5, 21, 12, 4, 8, 5, 6, 22, 5, 5, 13, 23, 4, 7, 14, 10, 5, 24, 4, 6, 9, 11

Offset: 1

Henry Bottomley, Nov 19 2001

nonn

It seems that a(n) <= A297113(n) for all n. Of the first 10000 positive natural numbers, 6454 are such that a(n) = A297113(n). - Antti Karttunen, Dec 31 2017

Also one plus the maximum number of unit steps East or South in the Young diagram of the integer partition with Heinz number n > 1, starting from the upper-left square and ending in a boundary square in the lower-right quadrant. The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). - Gus Wiseman, Apr 06 2019

			a(50) = 4 since the cascade goes from 50 = 2^1 * 5^2 to 15 = 3^1 * 5^1 to 6 = 2^1 * 3^1 to 2 = 2^1 to 1.
From _Gus Wiseman_, Apr 06 2019: (Start)
The partition with Heinz number 7865 is (6,5,5,3), with diagram
  o o o o o o
  o o o o o
  o o o o o
  o o o
which has longest path from (1,1) to (5,3) of length 6, so a(7865) = 7.
(End)

Antti Karttunen, Table of n, a(n) for n = 1..16384
FindStat, St000384: The maximal part of the shifted composition of an integer partition

Cf. A065769.
Differs from A297113 for the first time at n=20, where a(20) = 3, while A297113(20) = 4.
Cf. A001221, A001222, A052126, A056239, A064989, A112798, A174090, A257990, A325166, A325167, A325169.

Table[If[n==1,0,Max@@Total/@Position[PadRight[ConstantArray[1,#]&/@Sort[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]],Greater]],1]-1],{n,100}] (* Gus Wiseman, Apr 06 2019 *)

(definec (A065770 n) (if (= 1 n) 0 (+ 1 (A065770 (A065769 n))))) ;; Antti Karttunen, Dec 31 2017

Inverse of primes, powers of 2 and primorials in sense that a(A000040(n))=n; a(A000079(n))=n; a(A002110(n))=n. If n>0: a(3^n)=n+1; a(2^n*3^k)=n+k; a(p(k)^n)=n+k-1; a(n!)=A022559(n).

a(1) = 0; and for n > 1, a(n) = 1 + A065769(n). - Antti Karttunen, Dec 31 2017

A325169 Origin-to-boundary graph-distance of the Young diagram of the integer partition with Heinz number n.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 3, 1, 1, 2, 2, 2, 2, 1, 2, 2, 2, 1, 3, 1, 2, 3, 2, 1, 2, 2, 3, 2, 2, 1, 2, 2, 2, 2, 2, 1, 3, 1, 2, 3, 1, 2, 3, 1, 2, 2, 3, 1, 2, 1, 2, 3, 2, 2, 3, 1, 2, 2, 2, 1, 3, 2, 2, 2

Offset: 1

Gus Wiseman, Apr 05 2019

nonn

The origin-to-boundary graph-distance of a Young diagram is the minimum number of unit steps left or down from the upper-left square to a nonsquare in the lower-right quadrant. It is also the side-length of the minimum triangular partition contained inside the diagram.

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

FindStat, St000783: The maximal number of occurrences of a colour in a proper colouring of a Ferrers diagram

Cf. A001221, A001222, A056239, A061395, A065770, A112798, A115994, A174090, A257990, A297113, A325166, A325167, A325170.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
otb[ptn_]:=Min@@MapIndexed[#1+#2[[1]]-1&,Append[ptn,0]];
Table[otb[Reverse[primeMS[n]]],{n,100}]

A257990(n) <= a(n) <= 2 * A257990(n).

A297113 a(1) = 0, a(2) = 1, after which, a(n) = a(n/2) if n is of the form 4k+2, and otherwise a(n) = 1+a(A252463(n)).

Original entry on oeis.org

0, 1, 2, 2, 3, 2, 4, 3, 3, 3, 5, 3, 6, 4, 3, 4, 7, 3, 8, 4, 4, 5, 9, 4, 4, 6, 4, 5, 10, 3, 11, 5, 5, 7, 4, 4, 12, 8, 6, 5, 13, 4, 14, 6, 4, 9, 15, 5, 5, 4, 7, 7, 16, 4, 5, 6, 8, 10, 17, 4, 18, 11, 5, 6, 6, 5, 19, 8, 9, 4, 20, 5, 21, 12, 4, 9, 5, 6, 22, 6, 5, 13, 23, 5, 7, 14, 10, 7, 24, 4, 6, 10, 11, 15, 8, 6, 25

Offset: 1

Antti Karttunen, Dec 26 2017

nonn

From Gus Wiseman, Apr 06 2019: (Start)
Also the number of squares in the Young diagram of the integer partition with Heinz number n that are graph-distance 1 from the lower-right boundary, where the Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). For example, the partition (6,5,5,3) with Heinz number 7865 has diagram
  o o o o o o
  o o o o o
  o o o o o
  o o o
with inner rim
            o
          o
        o o
  o o o
of size 7, so a(7867) = 7.
(End)

Antti Karttunen, Table of n, a(n) for n = 1..12721
Index entries for sequences computed from indices in prime factorization

One more than A297167 (after the initial term).
Cf. A001511, A008683, A033265, A064989, A156552, A252463, A252464, A297112, A297155.
Cf. also A297157, A297161, A297162.
Cf. A052126, A065770, A112798, A115994, A174090, A325166, A325167, A325169.

Table[If[n==1,0,PrimePi[FactorInteger[n][[-1,1]]]+PrimeOmega[n]-PrimeNu[n]],{n,100}] (* Gus Wiseman, Apr 06 2019 *)

A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A297113(n) = if(n<=2,n-1,if(n%2,1+A297113(A064989(n)), !(n%4)+A297113(n/2)));

\\ More complex way, after Moebius transform:
A156552(n) = if(1==n, 0, if(!(n%2), 1+(2*A156552(n/2)), 2*A156552(A064989(n))));
A297112(n) = sumdiv(n,d,moebius(n/d)*A156552(d));
A297113(n) = if(1==n,0,1+valuation(A297112(n),2));

;; With memoization-macro definec.
(definec (A297113 n) (cond ((<= n 2) (- n 1)) ((= 2 (modulo n 4)) (A297113 (/ n 2))) (else (+ 1 (A297113 (A252463 n))))))

a(1) = 0, a(2) = 1, after which, a(n) = a(n/2) if n is of the form 4k+2, and otherwise a(n) = 1+a(A252463(n)) .
For n > 1, a(n) = A001511(A297112(n)), where A297112(n) = Sum_{d|n} moebius(n/d)*A156552(d).
a(n) = A252464(n) - A297155(n).
For n > 1, a(n) = 1+A033265(A156552(n)) = 1+A297167(n) = A046660(n) + A061395(n). - Last two sums added by Antti Karttunen, Sep 02 2018
Other identities. For all n >= 1:
a(A000040(n)) = n. [Each n occurs for the first time at the n-th prime.]

A257541 The rank of the partition with Heinz number n.

Original entry on oeis.org

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

Offset: 2

Emeric Deutsch, May 09 2015

sign

The rank of a partition p is the largest part of p minus the number of parts of p.
The Heinz number of a partition p = [p_1, p_2, ..., p_r] is defined 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 [1,1,1] the Heinz number is 2*2*2 = 8. Its rank is 1 - 3 = -2 = a(8). - Emeric Deutsch, Jun 09 2015
This is the Dyson rank (St000145), which is different from the Frobenius rank (St000183); see the FindStat links. - Gus Wiseman, Apr 13 2019

			a(24) = -2. Indeed, the partition corresponding to the Heinz number 24 = 2*2*2*3 is [1,1,1,2]; consequently, a(24)= 2 - 4 = -2.

G. E. Andrews, K. Eriksson, Integer Partitions, Cambridge Univ. Press, Cambridge, 2004.

Alois P. Heinz, Table of n, a(n) for n = 2..10000
FindStat, St000145: The Dyson rank of a partition
FindStat, St000183: The side length of the Durfee square of an integer partition

Positions of 0's are A106529. Positions of 1's are A325233. Positions of -1's are A325234.
Cf. A000720, A001222, A061395, A215366.
Cf. A046660, A047993, A056239, A093641, A101198, A112798, A174090, A243055, A257990, A263297, A325134, A325178, A325225, A325235.

with(numtheory): a := proc(n) options operator, arrow: pi(max(factorset(n)))-bigomega(n) end proc: seq(a(n), n = 2 .. 120);

Table[PrimePi@ FactorInteger[n][[-1, 1]] - PrimeOmega@ n, {n, 2, 76}] (* Michael De Vlieger, May 09 2015 *)

a(n) = q(largest prime factor of n) - bigomega(n), where q(p) is defined by q-th prime = p while bigomega(n) is the number of prime factors of n, including multiplicities.

A372441 Number of binary indices (binary weight) of n minus number of prime indices (bigomega) of n.

Original entry on oeis.org

1, 0, 1, -1, 1, 0, 2, -2, 0, 0, 2, -1, 2, 1, 2, -3, 1, -1, 2, -1, 1, 1, 3, -2, 1, 1, 1, 0, 3, 1, 4, -4, 0, 0, 1, -2, 2, 1, 2, -2, 2, 0, 3, 0, 1, 2, 4, -3, 1, 0, 2, 0, 3, 0, 3, -1, 2, 2, 4, 0, 4, 3, 3, -5, 0, -1, 2, -1, 1, 0, 3, -3, 2, 1, 1, 0, 2, 1, 4, -3, -1

Offset: 1

Gus Wiseman, May 07 2024

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.

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.

Robert Israel, Table of n, a(n) for n = 1..10000

Positions of zeros are A071814.
For sum instead of length we have A372428, zeros A372427.
For minimum instead of length we have A372437, zeros {}.
For maximum instead of length we have A372442, zeros A372436.
Positions of odd terms are A372590, even A372591.
A003963 gives product of prime indices.
A019565 gives Heinz number of binary indices, adjoint A048675.
A029837 gives greatest binary index, least A001511.
A048793 lists binary indices, length A000120, reverse A272020, sum A029931.
A061395 gives greatest prime index, least A055396.
A070939 gives length of binary expansion.
A112798 lists prime indices, length A001222, reverse A296150, sum A056239.
Cf. A000720, A014499, A038802, A061712, A174090, A243055, A304818, A359495, A372429-A372432.

f:= proc(n) convert(convert(n,base,2),`+`)-numtheory:-bigomega(n) end proc:
map(f, [$1..100]); # Robert Israel, May 22 2024

bix[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[bix[n]]-Length[prix[n]],{n,100}]

a(n) = A000120(n) - A001222(n).

A111774 Numbers that can be written as a sum of at least three consecutive positive integers.

Original entry on oeis.org

6, 9, 10, 12, 14, 15, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 81, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99, 100, 102

Offset: 1

Jaap Spies, Aug 15 2005

In this sequence there are no (odd) primes and there are no powers of 2.
So we have only three kinds of natural numbers: the odd primes, the powers of 2 and the numbers that can be represented as a sum of at least three consecutive integers.
Odd primes can only be written as a sum of two consecutive integers. Powers of 2 do not have a representation as a sum of k consecutive integers (other than the trivial n=n, for k=1).
Numbers of the form (x*(x+1)-y*(y+1))/2 for nonnegative integers x,y with x-y >= 3. - Bob Selcoe, Feb 21 2014
Numbers of the form (x + 1)*(x + 2*y)/2 for integers x,y with x >= 2 and y >= 1. For y = 1 only triangular numbers (A000217) >= 6 occur. - Ralf Steiner, Jun 27 2019
From Ralf Steiner, Jul 09 2019: (Start)
If k >= 1 sequences are c_k(n) = c_k(n - 1) + n + k - 1, c_k(0) = 0, means c_k(n) = n*(n + 2*k - 1)/2: A000217, A000096, A055998, A055999, A056000, ... then this sequence is the union of c_k(n), n >= 3. (End)
From Wolfdieter Lang, Oct 28 2020: (Start)
This sequence gives all positive integers that have at least one odd prime as proper divisor. The proof follows from the first two comments.
The set {a(n)}_{n>=1} equals the set {k positive integer : floor(k/2) - delta(k) >= 1}, where delta(k) = A055034(k). Proof: floor(k/2) gives the number of positive odd numbers < k, and delta(k), gives the number of positive odd numbers coprime to k. delta(1) = 1 but 1 is not < 1, therefore k = 1 is not a member of this set. Hence a member >= 2 of this set has at least one odd number > 1 and < k missing in the set of odd numbers relative prime to k. Therefore there exists at least one odd prime < k dividing k. (End)
For the multiplicity of a(n) see A338428, obtained from triangle A337940 (the array is given Bob Selcoe as example below, and in the Ralf Steiner comment above). - Wolfdieter Lang, Dec 09 2020

			a(1)=6 because 6 is the first number that can be written as a sum of three consecutive positive integers: 6 = 1+2+3.
From _Bob Selcoe_, Feb 23 2014: (Start)
Let the top row of an array be A000217(n). Let the diagonals (reading down and left) be A000217(n)-A000217(1),  A000217(n)-A000217(2), A000217(n)-A000217(3)..., A000217(n)-A000217(n-3).  This is A049777 read as a square array, starting with the third column. The array begins as follows:
   6 10 15 21 28 36 45 55 66
   9 14 20 27 35 44 54 65
  12 18 25 33 42 52 63
  15 22 30 39 49 60
  18 26 35 45 56
  21 30 40 51
  24 34 45
  27 38
  30
This is (x*(x+1)-y*(y+1))/2 for nonnegative integers x,y with x-y >= 3, because it is equivalent to 1+2+3/+4/+5/...+x/-0/-1/-2/-3/-4/-5/...-(x+3)/ for all possible strings of consecutive integers, which represents every possible way to sum three or more consecutive positive integers. So for example, 4+5+6+7 = 1+2+3+4+5+6+7-1-2-3 = 22, which is (x*(x+1)-y*(y+1))/2 when x=7, y=3. Notice that values can appear more than once in the array because some numbers can be represented as sums of more than one string of three or more consecutive positive integers. For example, 30 = (x*(x+1)-y*(y+1))/2 when (a) x=11, y=8: 9+10+11; (b) x=9, y=5: 6+7+8+9; and (c) x=8, y=3: 4+5+6+7+8. By definition, x-y is the number of integers in the string. (End)

Paul Halmos, "Problems for Mathematicians, Young and Old", Dolciani Mathematical Expositions, 1991, Solution to problem 3G p. 179.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Nieuw Archief voor Wiskunde 5/6 nr. 2 Problems/UWC, Problem C, Jun 2005, p. 181-182.
Nieuw Archief voor Wiskunde 5/6 nr. 2 Problems/UWC, Problem C: Solution of this Problem
J. Spies, Sage program for computing A111774

Cf. A000040, A000079, A066542, A174090 (complement), A000217, A049777, A055034.

Cf. A337940, A338428.

ispoweroftwo := proc(n) local a, t; t := 1; while (n > t) do t := 2*t end do; if (n = t) then a := true else a := false end if; return a; end proc; f:= proc(n) if (not isprime(n)) and (not ispoweroftwo(n)) then return n end if; end proc; seq(f(i),i = 1..150);

max=6!;lst={};Do[z=n+(n+1);Do[z+=(n+x);If[z>max,Break[]];AppendTo[lst,z],{x,2,max}],{n,max}];Union[lst] (* Vladimir Joseph Stephan Orlovsky, Mar 06 2010 *)

isok(n) = !(n == 1) && !isprime(n) && !(isprimepower(n, &p) && (p == 2)); \\ Michel Marcus, Jul 02 2019

from sympy import primepi
def A111774(n):
    def f(x): return int(n+(0 if x<=1 else primepi(x)-1)+x.bit_length())
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m # Chai Wah Wu, Sep 19 2024

A372442 (Greatest binary index of n) minus (greatest prime index of n).

Original entry on oeis.org

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

Offset: 2

Gus Wiseman, May 07 2024

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.

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.

For sum instead of maximum we have A372428, zeros A372427.
Positions of zeros are A372436.
For minimum instead of maximum we have A372437, zeros {}.
For length instead of maximum we have A372441, zeros A071814.
Positions of odd terms are A372588, even A372589.
A019565 gives Heinz number of binary indices, adjoint A048675.
A029837 gives greatest binary index, least A001511.
A048793 lists binary indices, length A000120, reverse A272020, sum A029931.
A061395 gives greatest prime index, least A055396.
A070939 gives length of binary expansion.
A112798 lists prime indices, length A001222, reverse A296150, sum A056239.
Cf. A000720, A003963, A014499, A174090, A243055, A355536, A359495, A372429-A372432, A372589-A372591.

bix[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Max[bix[n]]-Max[prix[n]],{n,2,100}]

a(n) = A070939(n) - A061395(n) = A029837(n) - A061395(n) for n > 1.

A325225 Lesser of the number of prime factors of n counted with multiplicity and the maximum prime index of n.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 3, 2, 2, 1, 2, 2, 2, 2, 3, 1, 3, 1, 1, 2, 2, 2, 2, 1, 2, 2, 3, 1, 3, 1, 3, 3, 2, 1, 2, 2, 3, 2, 3, 1, 2, 2, 4, 2, 2, 1, 3, 1, 2, 3, 1, 2, 3, 1, 3, 2, 3, 1, 2, 1, 2, 3, 3, 2, 3, 1, 3, 2, 2, 1, 4, 2, 2, 2, 4, 1, 3, 2, 3, 2, 2, 2, 2, 1, 3, 3, 3, 1, 3, 1, 4, 3

Offset: 1

Gus Wiseman, Apr 12 2019

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.

			88 has 4 prime indices {1,1,1,5}, the maximum of which is 5, so a(88) = min(4,5) = 4.

Antti Karttunen, Table of n, a(n) for n = 1..65537
FindStat, St000533: The maximal number of non-attacking rooks on a Ferrers shape

Positions of 1's are A174090. Positions of 2's are A325229.

Cf. A001222, A056239, A061395, A065770, A093641, A112798, A252464, A257541, A257990, A263297, A325169, A325224, A325227, A325232.

Table[Min[PrimeOmega[n],PrimePi[FactorInteger[n][[-1,1]]]],{n,100}]

A061395(n) = if(1==n, 0, primepi(vecmax(factor(n)[, 1])));
A325225(n) = min(bigomega(n), A061395(n)); \\ Antti Karttunen, Apr 14 2019

a(n) = min(A001222(n), A061395(n)).

More terms from Antti Karttunen, Apr 14 2019

A325166 Size of the internal portion of the integer partition with Heinz number n.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 2, 0, 0, 2, 0, 1, 2, 1, 0, 1, 2, 1, 2, 1, 0, 3, 0, 0, 2, 1, 3, 2, 0, 1, 2, 1, 0, 3, 0, 1, 3, 1, 0, 1, 3, 3, 2, 1, 0, 3, 3, 1, 2, 1, 0, 3, 0, 1, 3, 0, 3, 3, 0, 1, 2, 4, 0, 2, 0, 1, 4, 1, 4, 3, 0, 1, 3, 1, 0, 3, 3, 1, 2, 1, 0, 4, 4, 1, 2, 1, 3, 1, 0, 4, 3, 3, 0, 3, 0, 1, 5

Offset: 1

Gus Wiseman, Apr 05 2019

nonn

The internal portion of an integer partition consists of all squares in the Young diagram that have a square both directly below and directly to the right.

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

			The partition with Heinz number 7865 is (6,5,5,3), with diagram
  o o o o o o
  o o o o o
  o o o o o
  o o o
with internal portion
  o o o o o
  o o o o
  o o o
of size 12, so a(7865) = 12.

Antti Karttunen, Table of n, a(n) for n = 1..20000
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Positions of zeros are A174090.

Cf. A001221, A001222, A052126, A056239, A061395, A064989, A065770, A112798, A252464, A257990, A297113, A325133, A325135, A325167, A325169.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[If[n==1,0,Total[primeMS[n]]-Max[primeMS[n]]-Length[primeMS[n]]+Length[Union[primeMS[n]]]],{n,100}]

A056239(n) = { my(f); if(1==n, 0, f=factor(n); sum(i=1, #f~, f[i,2] * primepi(f[i,1]))); }
A061395(n) = if(1==n, 0, primepi(vecmax(factor(n)[, 1])));
A325166(n) = (A056239(n) - A061395(n) - bigomega(n) + omega(n)); \\ Antti Karttunen, Apr 14 2019

a(n) = A056239(n) - A061395(n) - A001222(n) + A001221(n).

a(n) = A056239(n) - A297113(n).

More terms from Antti Karttunen, Apr 14 2019

A372437 (Least binary index of n) minus (least prime index of n).

Original entry on oeis.org

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

Offset: 2

Gus Wiseman, May 06 2024

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.
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.
Is 0 the only integer not appearing in the data?

Positions of first appearances are A174090.
For sum instead of minimum we have A372428, zeros A372427.
For maximum instead of minimum we have A372442, zeros A372436.
For length instead of minimum we have A372441, zeros A071814.
A003963 gives product of prime indices.
A019565 gives Heinz number of binary indices, adjoint A048675.
A029837 gives greatest binary index, least A001511.
A048793 lists binary indices, length A000120, reverse A272020, sum A029931.
A061395 gives greatest prime index, least A055396.
A070939 gives length of binary expansion.
A112798 lists prime indices, length A001222, reverse A296150, sum A056239.
Cf. A000720, A014499, A061712, A243055, A304818, A355536, A359495, A359402, A372429-A372432, A372588-A372591.

bix[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Min[bix[n]]-Min[prix[n]],{n,2,100}]

a(2n) = A001511(n).
a(2n + 1) = -A038802(n).
a(n) = A001511(n) - A055396(n).

cp's OEIS Frontend

A065770 Number of prime cascades to reach 1, where a prime cascade (A065769) is multiplicative with a(p(m)^k) = p(m-1) * p(m)^(k-1).

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A325169 Origin-to-boundary graph-distance of the Young diagram of the integer partition with Heinz number n.

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A297113 a(1) = 0, a(2) = 1, after which, a(n) = a(n/2) if n is of the form 4k+2, and otherwise a(n) = 1+a(A252463(n)).

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A257541 The rank of the partition with Heinz number n.

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A372441 Number of binary indices (binary weight) of n minus number of prime indices (bigomega) of n.

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A111774 Numbers that can be written as a sum of at least three consecutive positive integers.

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A372442 (Greatest binary index of n) minus (greatest prime index of n).

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