A000028 -id:A000028 - cp's OEIS frontend

A332822 One part of a 3-way classification of the positive integers. Numbers n for which A048675(n) == 2 (mod 3).

3, 4, 7, 10, 13, 18, 19, 22, 24, 25, 29, 32, 34, 37, 42, 43, 45, 46, 53, 55, 56, 60, 61, 62, 71, 78, 79, 80, 81, 82, 85, 89, 94, 98, 99, 101, 104, 105, 107, 108, 113, 114, 115, 118, 121, 131, 132, 134, 139, 140, 144, 146, 150, 151, 152, 153, 155, 163, 166, 173, 174, 176, 181, 182, 187, 189, 192, 193, 194, 195, 199, 200, 204

Offset: 1

Antti Karttunen and Peter Munn, Feb 25 2020

nonn

The positive integers are partitioned between A332820, A332821 and this sequence.
For each prime p, the terms include exactly one of p and p^2. The primes alternate between this sequence and A332821. This sequence has the primes with even indexes, those in A031215.
The terms are the even numbers in A332820 halved. The terms are also the numbers m such that 5m is in A332820, and so on for alternate primes: 11, 17, 23 etc. Likewise, the terms are the numbers m such that 3m is in A332821, and so on for alternate primes: 7, 13, 19, 29 etc.
If we take each odd term of this sequence and replace each prime in its factorization by the next smaller prime, we get the same set of numbers as we get from halving the even terms of this sequence, and A332821 consists exactly of those numbers. The numbers that are one third of the terms that are multiples of 3 are in A332820, which consists exactly of those numbers. The numbers that are one fifth of the terms that are multiples of 5 constitute A332821, and for larger primes, an alternating pattern applies as described in the previous paragraph.
The product of any 2 terms of this sequence is in A332821, the product of any 3 terms is in A332820, and the product of a term of A332820 and a term of this sequence is in this sequence. So if a number k is present, k^2 is in A332821, k^3 is in A332820, and k^4 is in this sequence.
If k is an even number, exactly one of {k/2, k, 2k} is in the sequence (cf. A191257 / A067368 / A213258); and generally if k is a multiple of a prime p, exactly one of {k/p, k, k*p} is in the sequence.

Positions of terms valued -1 in A332823; equivalently, numbers in row 3k-1 of A277905 for some k >= 1.
Cf. A048675, A332820, A332821.
Comparable 2 or 3-way classifications: A000379/A000028, A001969/A000069, A003159/A036554, A005843/A005408, A028260/A026424, A191257/A067368/A213258, A325431/A325432, A329609/A329604/A332812.
Subsequences: intersection of A026478 and A066207, A031215 (prime terms), A033430\{0}, A117642\{0}, A169604, A244727\{0}, A244729\{0}, A338910 (semiprime terms).

Select[Range@ 204, Mod[Total@ #, 3] == 2 &@ Map[#[[-1]]*2^(PrimePi@ #[[1]] - 1) &, FactorInteger[#]] &] (* Michael De Vlieger, Mar 15 2020 *)

isA332822(n) =  { my(f = factor(n)); (2==((sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2)%3)); };

{a(n) : n >= 1} = {2 * A332821(k) : k >= 1} U {A003961(A332821(k)) : k >= 1}.

{a(n) : n >= 1} = {A332821(k)^2 : k >= 1} U {A331590(2, A332821(k)) : k >= 1}.

A133009 One defining property of the sequences {A, B} = {A000069, A001969} is that they are the unique pair of sets complementary with respect to the nonnegative integers such that q(n) = |{x : x, y in A, x < y, x + y = n}| = |{x : x, y in B, x < y, x + y = n}| for all n >= 0. The present sequence gives the values of q(n).

Original entry on oeis.org

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

Offset: 0

David W. Wilson, Dec 21 2007

nonn

David W. Wilson, Table of n, a(n) for n = 0..10000
Antoine Renard, Michel Rigo, and Markus A. Whiteland, q-Parikh Matrices and q-deformed binomial coefficients of words, arXiv:2402.05657 [cs.FL], 2024. See pp. 3, 12.
C. Sándor, Partitions of natural numbers and their representation functions, INTEGERS 4 (2004), #A18.
Jeffrey Shallit, Additive Number Theory via Automata and Logic, arXiv:2112.13627 [math.NT], 2021.

Cf. A000028, A000379, A000069, A001969, A133008.

A066427 Numbers with mu = 0 and infinitary MoebiusMu = -1; (sum of binary digits of prime exponents is odd).

Original entry on oeis.org

4, 9, 16, 24, 25, 40, 49, 54, 56, 60, 72, 81, 84, 88, 90, 96, 104, 108, 121, 126, 128, 132, 135, 136, 140, 150, 152, 156, 160, 169, 180, 184, 189, 192, 198, 200, 204, 220, 224, 228, 232, 234, 240, 248, 250, 252, 256, 260, 276, 288, 289, 294, 296, 297, 300, 306

Offset: 1

Wouter Meeussen, Dec 27 2001

First differs from A378489 (the intersection of A000028 and A028260) by the inclusion of 72. - Peter Munn, Jul 13 2024

			54 is in this sequence because its prime decomposition is 2^1 * 3^3, it is not squarefree and the binary digits of "1" and "3" add up to 3, an odd number.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A008683, A064179, A066428, A378489.

iMoebiusMu[ n_ ] := Switch[ MoebiusMu[ n ], 1, 1, -1, -1, 0, If[ OddQ[ Plus@@(DigitCount[ Last[ Transpose[ FactorInteger[ n ] ] ], 2, 1 ]) ], -1, 1 ] ]; Select[ Range[ 400 ], MoebiusMu[ # ]===0 && iMoebiusMu[ # ]===-1 & ]

is(n)=my(f=factor(n)[,2]); #f && vecmax(f)>1 && vecsum(apply(hammingweight, f))%2 \\ Charles R Greathouse IV, Oct 15 2015

A268386 a(n) = A193231(A268387(n)).

Original entry on oeis.org

0, 1, 1, 3, 1, 0, 1, 2, 3, 0, 1, 2, 1, 0, 0, 5, 1, 2, 1, 2, 0, 0, 1, 3, 3, 0, 2, 2, 1, 1, 1, 4, 0, 0, 0, 0, 1, 0, 0, 3, 1, 1, 1, 2, 2, 0, 1, 4, 3, 2, 0, 2, 1, 3, 0, 3, 0, 0, 1, 3, 1, 0, 2, 6, 0, 1, 1, 2, 0, 1, 1, 1, 1, 0, 2, 2, 0, 1, 1, 4, 5, 0, 1, 3, 0, 0, 0, 3, 1, 3, 0, 2, 0, 0, 0, 5, 1, 2, 2, 0, 1, 1, 1, 3, 1, 0, 1, 1, 1, 1, 0, 4, 1, 1, 0, 2, 2, 0, 0, 2

Offset: 1

Antti Karttunen, Feb 10 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384

A003987, A048720, A059897, A193231, A268385, A268387 are used in definitions of this sequence.

Cf. A000028 (indices of odd numbers), A000379 (indices of even numbers), A268390 (indices of zeros).

f[n_] := Which[0 <= # <= 1, #, EvenQ@ #, BitXor[2 #, #] &[f[#/2]], True, BitXor[#, 2 # + 1] &[f[(# - 1)/2]]] &@ Abs@ n; {0}~Join~Table[f[BitXor @@ Map[Last, FactorInteger@ n]], {n, 2, 120}] (* Michael De Vlieger, Feb 12 2016, after Robert G. Wilson v at A048724 and A065621 *)

a268387(n) = {my(f = factor(n), b = 0); for (k=1, #f~, b = bitxor(b, f[k, 2]); ); b; }
a193231(n) = {my(x='x); subst(lift(Mod(1, 2)*subst(Pol(binary(n), x), x, 1+x)), x, 2)};
a(n) = a193231(a268387(n)); \\ Michel Marcus, May 09 2020

(define (A268386 n) (A193231 (A268387 n)))

The following two formulas are equivalent because A193231 distributes over bitwise XOR (A003987):
a(n) = A193231(A268387(n)) and
a(n) = A268387(A268385(n)).
a(2^k) = A193231(k). - Peter Munn, May 07 2020
From Peter Munn, Jun 02 2020: (Start)
Alternative definition, for n, k >= 1, where XOR denotes A003987:
a(prime(n)) = 1, where prime(n) = A000040(n);
a(n^2) = a(n) XOR (2 * a(n)) = A048720(a(n), 3);
a(A059897(n, k)) = a(n) XOR a(k).
(End)

A374467 Numbers that are products of an odd number of primes and the total number of 1-bits in the exponents of their prime factorization n is odd.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 30, 31, 37, 41, 42, 43, 47, 53, 59, 61, 66, 67, 70, 71, 72, 73, 78, 79, 83, 89, 97, 101, 102, 103, 105, 107, 108, 109, 110, 113, 114, 127, 128, 130, 131, 137, 138, 139, 149, 151, 154, 157, 163, 165, 167, 170, 173, 174, 179, 180, 181, 182, 186, 190, 191, 192, 193, 195, 197, 199

Offset: 1

Antti Karttunen, Jul 14 2024

nonn

Antti Karttunen, Table of n, a(n) for n = 1..12000

Cf. A001222, A064547, A374466 (characteristic function).
Intersection of A000028 and A026424.
Differs from A374595 first at n=43, where a(43) = 128, a term that is missing from A374595.

PAR
```
isA374467 = A374466;
```

{k | A001222(k) is odd and A064547(k) is odd}.

A374473 Numbers k such that bigomega(k) and A064547(k) are not both even, where A064547 is the count of 1-bits in the exponents of the prime factorization of n, and bigomega is the number of prime factors of n (with multiplicity).

Original entry on oeis.org

2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 16, 17, 18, 19, 20, 23, 24, 25, 27, 28, 29, 30, 31, 32, 37, 40, 41, 42, 43, 44, 45, 47, 48, 49, 50, 52, 53, 54, 56, 59, 60, 61, 63, 66, 67, 68, 70, 71, 72, 73, 75, 76, 78, 79, 80, 81, 83, 84, 88, 89, 90, 92, 96, 97, 98, 99, 101, 102, 103, 104, 105, 107, 108, 109, 110, 112, 113, 114

Offset: 1

Antti Karttunen, Jul 14 2024

nonn

Antti Karttunen, Table of n, a(n) for n = 1..12000

Cf. A001222, A064547, A374471, A374472 (complement).
Indices of odd terms in A374470.
Union of A000028 and A026424. Their intersection A374467 is a subsequence.

PARI
```
isA374473(n) = !A374471(n);
```

A374595 Products of an odd number of distinct primes and the square of a number in A268390.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 30, 31, 37, 41, 42, 43, 47, 53, 59, 61, 66, 67, 70, 71, 72, 73, 78, 79, 83, 89, 97, 101, 102, 103, 105, 107, 108, 109, 110, 113, 114, 127, 130, 131, 137, 138, 139, 149, 151, 154, 157, 163, 165, 167, 170, 173, 174, 179, 180, 181, 182, 186, 190, 191, 193, 195, 197, 199, 200, 211, 222

Offset: 1

Antti Karttunen and Peter Munn, Jul 13 2024

nonn

For any prime p: when trailing zeros are removed from the terms written in base p, every positive integer not divisible by p appears (in base p) exactly once. For example, 4 is written "11" in base 3, and "11" appears when the trailing zeros are removed from "11000", which is the term 108 written in base 3.
Numbers m such that when the exponents e_1 .. e_k in the canonical factorization m = p_1^e_1 * ... * p_k^e_k are bitwise-xored together, the result is 1.
As a set, membership is determined by prime signature. It is a coset of A268390 within the positive integers under the binary operation A059897(.,.). Using standard arithmetic operations and any given prime p, this set can be derived from A268390 by dividing by p all the terms of A268390 whose squarefree part is a multiple of p and multiplying by p all the other terms of A268390.

			30 = 2 * 3 * 5 is the product of an odd number of distinct primes times 1, and 1 is in A268390, so 30 is in the sequence.
72 = 2 * 6^2 is in the sequence since 2 is the product of 1 distinct prime and 6 is in A268390.
6 = 2 * 3 is the product of 2 (an even number) of distinct primes, so 6 is not in the sequence.

Index entries for sequences related to prime signature

Positions of 1's in A268387, cf. A268390 (positions of 0's).
Subsequence of A000028, A026424. Differs from their intersection, A374467, by not having 128, 192, 288, etc.
Cf. A030059 (squarefree terms), A374130 (characteristic function).
See the formula section for the relationships with A006519, A008836, A059897.

PARI
```
isA374595 = A374130;
```

For prime p, {a(n) : n >= 1} = {A059897(k,p) : k in A268390}.

{a(n) : n >= 1} = {k * 2^(lambda(A006519(k))): k in A268390}, where lambda is Liouville's function, A008836, and A006519(k) is the highest power of 2 dividing k.

A368407 a(n) is the least start of a run of exactly n consecutive integers with an odd number of Fermi-Dirac factors (A064547).

Original entry on oeis.org

7, 16, 23, 2, 101, 135, 429, 227, 5018, 2682, 1692, 819, 184170, 175683, 93597, 117032, 300773, 466065, 2012413, 2177318, 10267248, 39069334, 25085909, 3829812, 132899228, 316564239, 391031618, 53858226, 355227517, 1484451701, 18503618566, 370565617, 52188004547, 5708850192

Offset: 1

Amiram Eldar, Dec 23 2023

Cf. A000028, A064547, A368408.

Analogous to A275509.

q[n_] := OddQ[Sum[DigitCount[e, 2, 1], {e, FactorInteger[n][[;;, 2]]}]]; q[1] = False; seq[len_] := Module[{s = Table[0, {len}], n = 1, count = 0, n1, d}, While[count < len, n1 = n; If[q[n], While[q[++n1]]; d = n1 - n; If[d <= len && s[[d]] == 0, count++; s[[d]] = n]]; n = n1 + 1]; s]; seq[16]

is(n) = {my(e = factor(n)[, 2]); sum(i = 1, #e, hammingweight(e[i])) % 2;}
lista(len) = {my(s = vector(len), n = 1, count = 0, n1, d); while(count < len, n1 = n; if(is(n), n1++; while(is(n1), n1++); d = n1 - n; if(d <= len && s[d] == 0, count++; s[d] = n)); n = n1 + 1); s};

A130285 Number of divisors of A026416(n).

Original entry on oeis.org

1, 2, 2, 3, 2, 2, 3, 2, 2, 5, 2, 2, 2, 8, 3, 2, 8, 2, 2, 8, 2, 8, 2, 2, 3, 2, 8, 8, 2, 2, 8, 2, 8, 2, 2, 8, 2, 5, 2, 8, 2, 2, 2, 8, 2, 8, 8, 2, 2, 8, 2, 8, 3, 2, 8, 8, 2, 8, 8, 2, 8, 2, 2, 2, 8, 8, 2, 2, 8, 2, 3, 8, 2, 8, 2, 18, 2, 8, 8, 8, 8, 8, 2, 14, 2, 8, 2, 2, 2, 8, 2, 2, 2, 8, 8, 8, 2, 8, 2, 20, 2

Offset: 1

Giovanni Teofilatto, Aug 06 2007

Cf. A000028, A123193, A130451.

Definition clarified, comment updated and values corrected by R. J. Mathar, Apr 22 2010

A324211 Numbers k such that the sum of the binary digits of the exponents of the prime factorization of k is odd and k is a product of primorials.

Original entry on oeis.org

2, 4, 16, 24, 30, 60, 72, 96, 128, 180, 192, 240, 256, 288, 432, 576, 720, 840, 900, 1080, 1536, 1920, 2048, 2310, 2520, 2592, 3072, 3360, 3456, 3600, 3840, 4320, 4608, 4620, 5184, 5400, 5760, 6480, 6720, 6912, 8192, 8640, 9216, 10080, 11520, 12288, 12600, 13860, 15120

Offset: 1

David A. Corneth, Mar 20 2019

This sequence is a primitive sequence of A000028; it lists minimal terms in that sequence having their prime exponents.

			180 is a term as 180 = 2^2 * 3^2 * 5 which has exponents in binary 10_2, 10_2 and 1_2 respectively. The sum of binary digits of those exponents is (1 + 0) + (1 + 0) + 1 = 3 which is odd. Furthermore, 180 is a product of primorials; 180 = 30 * 6. Therefore, 180 is in the sequence.

Intersection of A000028 and A025487.

is(n) = {if(n==1, return(0)); my(f = factor(n));  f[#f~, 1] == prime(#f~) && vecsort(f[, 2],,4) == f[, 2] && sum(i=1, #f~, hammingweight(f[i, 2]))%2}

cp's OEIS Frontend

A332822 One part of a 3-way classification of the positive integers. Numbers n for which A048675(n) == 2 (mod 3).

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A066427 Numbers with mu = 0 and infinitary MoebiusMu = -1; (sum of binary digits of prime exponents is odd).

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A268386 a(n) = A193231(A268387(n)).

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A374467 Numbers that are products of an odd number of primes and the total number of 1-bits in the exponents of their prime factorization n is odd.

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A374473 Numbers k such that bigomega(k) and A064547(k) are not both even, where A064547 is the count of 1-bits in the exponents of the prime factorization of n, and bigomega is the number of prime factors of n (with multiplicity).

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A374595 Products of an odd number of distinct primes and the square of a number in A268390.

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A368407 a(n) is the least start of a run of exactly n consecutive integers with an odd number of Fermi-Dirac factors (A064547).

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A130285 Number of divisors of A026416(n).

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