A318400 -id:A318400 - cp's OEIS frontend

A322911 Numbers whose prime indices are all powers of the same squarefree number.

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, 38, 40, 41, 42, 43, 44, 46, 47, 48, 49, 50, 52, 53, 54, 56, 57, 58, 59, 62, 63, 64, 67, 68, 72, 73, 76, 79, 80, 81, 82, 83, 84, 86, 88, 92

Offset: 1

Gus Wiseman, Dec 30 2018

nonn

The complement is {15, 30, 33, 35, 37, 39, 45, ...}. First differs from A318991 at a(33) = 38, A318991(33) = 37.

A multiset multisystem is a finite multiset of finite multisets. 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 multiset multisystem with MM-number n is formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. For example, the prime indices of 78 are {1,2,6}, so the multiset multisystem with MM-number 78 is {{},{1},{1,2}}. The dual of a multiset multisystem has, for each vertex, one block consisting of the indices (or positions) of the parts containing that vertex, counted with multiplicity. For example, the dual of {{1,2},{2,2}} is {{1},{1,2,2}}. The sequence lists all MM-numbers of multiset multisystems whose dual is constant, i.e. of the form {x,x,x,...,x} for some multiset x.

			The prime indices of 756 are {1,1,2,2,2,4}, which are all powers of 2, so 756 belongs to the sequence.
The prime indices of 841 are {10,10}, which are all powers of 10, so 841 belongs to the sequence.
The prime indices of 2645 are {3,9,9}, which are all powers of 3, so 2645 belongs to the sequence.
The prime indices of 3178 are {1,4,49}, which are all powers of squarefree numbers but not of the same squarefree number, so 3178 does not belong to the sequence.
The prime indices of 30599 are {12,144}, which are all powers of the same number 12, but this number is not squarefree, so 30599 does not belong to the sequence.
The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k). The sequence of all integer partitions whose Heinz numbers belong to the sequence begins: (3,2), (3,2,1), (5,2), (4,3), (6,2), (3,2,2), (7,2), (5,3), (3,2,1,1), (6,3), (5,2,1), (9,2), (4,3,1), (3,3,2), (5,4), (6,2,1), (7,3), (10,2), (3,2,2,1), (6,4), (11,2), (8,3), (5,2,2).

Cf. A000688, A000961, A001597, A005117, A023893, A052410, A056239, A072720, A072774, A302242, A302593, A318400, A322847, A322901, A322912.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
powsqfQ[n_]:=SameQ@@Last/@FactorInteger[n];
sqfker[n_]:=Times@@First/@FactorInteger[n];
Select[Range[100],And[And@@powsqfQ/@primeMS[#],SameQ@@sqfker/@DeleteCases[primeMS[#],1]]&]

A325122 Sum of binary digits of the prime indices of n, minus Omega(n).

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Mar 29 2019

nonn

The sum of binary digits of an integer is the number of 1's in its binary representation. 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.

Positions of zeros are A318400.
Cf. A000120, A018900, A019565, A048881, A051438, A070939, A112798.
Cf. A325103, A325104, A325106, A325118.
Other totally additive sequences: A056239, A302242, A318994, A318995, A325033, A325034, A325120, A325121.

Table[Sum[pr[[2]]*(DigitCount[PrimePi[pr[[1]]],2,1]-1),{pr,If[n==1,{},FactorInteger[n]]}],{n,100}]

Totally additive with a(prime(n)) = A048881(n).

A374717 For n a power of 2, a(n) = n. Otherwise let k = n - 2^j (> 0) where 2^j is the greatest power of 2 not exceeding n, then a(n) = least novel m*a(k); m a term in A033844.

Original entry on oeis.org

1, 2, 3, 4, 7, 6, 9, 8, 19, 14, 21, 12, 49, 18, 27, 16, 53, 38, 57, 28, 133, 42, 63, 24, 361, 98, 147, 36, 343, 54, 81, 32, 131, 106, 159, 76, 371, 114, 171, 56, 1007, 266, 399, 84, 931, 126, 189, 48, 2809, 722, 1083, 196, 2527, 294, 441, 72, 6859, 686, 1029, 108, 2401, 162, 243, 64, 311

Offset: 1

David James Sycamore, Jul 17 2024

Definition developed from the Doudna recursion A005940, see Example. Conjectured to be a permutation of A318400, numbers whose prime factorization consists only of primes with indices 2^k (terms in A033844).
From David James Sycamore, Aug 09 2024: (Start)
The even bisection when divided by 2 returns the sequence. The odd bisection when transformed by replacing all factors prime(2^k) in a(2*n-1) with prime(2^(k-1)) also returns the sequence (similar to properties of A005940). The sequence is fixed on numbers of the form 2^n or 3*2^n (A029744), since by the definition points at 2^n are already named as such, and if n = 3*2^r then the powers of 2 adjacent to n are 2^r < 2^(r+1) < 3*2^r < 2^(r+2), from which, by the definition we find k = 2^(r+2) - 3*2^r = 2^r(3 - 2) = 2^r, which is a fixed point so a(k) = 2^r, and a(n) is least novel m*a(k); m a term in A033844. Since 2^r is already a term the required m is 3, so a(n) = 3*2^r = n (compare with fixed points of A005940). (End)
Definition analogous to the Name of A005940: Let c_i = number of 1's in binary expansion of n-1 that have i 0's to their right, and let p(j) = j-th prime. Then a(n) = Product_i p(2^i)^c_i. - Michael De Vlieger, Aug 09 2024

			a(1) = 1, a(2) = 2 because both are powers of 2. a(3) = 3 since for n = 3, k = 1, a(1) = 1 and m = 3. a(4) = 4 because 4 is a power of 2
For a(5), k = 1, a(1) = 1 and therefore a(5) = 1*7 since 7 is least term in A033844 not already used.
Whereas the Name defines each individual term recursively, the following procedure describes a recursion for generating the first 2^k terms from the first 2^(k-1) terms: Let S(0) = {1}, and then S(k) = {2*S(k-1), S(k-1)}, where 2*S(k-1) means twice every term in S(k-1). Thus from
       S(0) = {1} we obtain:
       S(1) = {2,1},
       S(2) = {4,2,2,1},
       S(3) = {8,4,4,2,4,2,2,1},
       S(4) = {16,8,8,4,8,4,4,2,8,4,4,2,4,2,2,1} etc.
Convert these (indices) to primes as follows:
       P(0) = {2},
       P(1) = {3,2},
       P(2) = {7,3,3,2},
       P(3) = {19,7,7,3,7,3,3,2},
       P(4) = {53,19,19,7,19,7,7,3,19,7,7,3,7,3,3,2}, etc.
Set U(0) = 1 and U(k) = U(k-1)*P(k-1) prepended by U(k-1), thus:
       U(0) = {1},
       U(1) = {1,2},
       U(2) = {1,2,3,4},
       U(3) = {1,2,3,4,7,6,9,8},
       U(4) = {1,2,3,4,7,6,9,8,19,14,21,12,49,18,27,16}, etc.
Thus U(k) gives the first 2^k terms of the sequence because the primes in P(k) are the greatest prime factors of the corresponding terms.
From _Michael De Vlieger_, Aug 09 2024: (Start)
Using the alternative binary definition:
For n = 9, n-1 = 1000_2; c_3 = 1, hence a(9) = prime(2^3)^1 = 19.
For n = 10, n-1 = 1001_2; c_0 = 1, c_2 = 1; a(10) = prime(2^0)^1 * prime(2^2)^1 = 2*7 = 14.
For n = 11, n-1 = 1010_2; c_1 = 1, c_2 = 1; a(11) = prime(2^1)^1 * prime(2^2)^1 = 3*7 = 21.
For n = 12, n-1 = 1011_2; c_0 = 2, c_1 = 1; a(12) = prime(2^0)^2 * prime(2^1)^1 = 2^2*3 = 12.
For n = 13, n-1 = 1100_2; c_2 = 2; a(13) = prime(2^2)^2 = 7^2 = 49.
For n = 2^k + 2^(k-1) = 3*2^(k-1), n-1 = 2^(k+1) - 2^(k-1) - 1.
  c_0 = k-1, c_1 = 1, therefore we have fixed point a(3*2^(k-1)) = 3*2^(k-1). (End)

Michael De Vlieger, Table of n, a(n) for n = 1..16384
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^14, showing primes in red, perfect prime powers in gold, squarefree composites in green, and with blue and purple representing numbers neither squarefree nor prime powers, where purple represents powerful numbers that are not prime powers.
Michael De Vlieger, Fan style binary tree showing a(n), n = 1..8192 using essentially the same color function as immediately above, with exception that composite primorials appear in bright green.

Cf. A005940, A029744, A033844, A318400.

Block[{a, c, k, m, t, nn}, nn = 2^7; c[_] = False;
Do[Set[{m, k}, {1, n - 2^Floor[Log2[n]]}];
  If[k == 0,
    Set[{a[n], c[n]}, {n, True}],
    While[Set[t, Prime[2^m] a[k]]; c[t], m++];
      Set[{a[n], c[t]}, {t, True}]], {n, nn}];
Array[a, nn] ] (* Michael De Vlieger, Aug 06 2024 *)

a(2^k) = 2^k, a(3*2^k) = 3*2^k.

a(2^k-1) = 3^(k-1), k >= 1; a(2^k+1) = A033844(k+1); k >= 0.

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A322911 Numbers whose prime indices are all powers of the same squarefree number.

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A325122 Sum of binary digits of the prime indices of n, minus Omega(n).

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A374717 For n a power of 2, a(n) = n. Otherwise let k = n - 2^j (> 0) where 2^j is the greatest power of 2 not exceeding n, then a(n) = least novel m*a(k); m a term in A033844.

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