A293231 -id:A293231 - cp's OEIS frontend

A293232 Restricted growth sequence transform of A293231, where A293231(n) = Product_{d|n, dA019565(A193231(d)).

1, 2, 2, 3, 2, 4, 2, 5, 6, 7, 2, 8, 2, 9, 10, 11, 2, 12, 2, 13, 14, 15, 2, 16, 17, 18, 19, 20, 2, 21, 2, 22, 23, 24, 25, 26, 2, 27, 28, 29, 2, 30, 2, 31, 32, 33, 2, 34, 7, 35, 36, 37, 2, 38, 39, 40, 41, 42, 2, 43, 2, 44, 45, 46, 23, 47, 2, 48, 49, 50, 2, 51, 2, 52, 53, 54, 55, 56, 2, 57, 58, 59, 2, 60, 61, 62, 63, 64, 2, 65, 66, 67, 68, 69, 70, 71, 2, 72

Offset: 1

Antti Karttunen, Oct 03 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A019565, A193231.
Cf. A290090.
Differs from related A293215 for the first time at n=55, where a(55) = 39, while A293215(55) = 28.

rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
write_to_bfile(start_offset,vec,bfilename) = { for(n=1, length(vec), write(bfilename, (n+start_offset)-1, " ", vec[n])); }
A019565(n) = {my(j,v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ This function from M. F. Hasler
A193231(n) = { my(x='x); subst(lift(Mod(1, 2)*subst(Pol(binary(n), x), x, 1+x)), x, 2) }; \\ This function from Franklin T. Adams-Watters
A293231(n) = { my(m=1); fordiv(n,d,if(d < n,m *= A019565(A193231(d)))); m; };
write_to_bfile(1,rgs_transform(vector(65537,n,A293231(n))),"b293232.txt");

A293214 a(n) = Product_{d|n, dA019565(d).

Original entry on oeis.org

1, 2, 2, 6, 2, 36, 2, 30, 12, 60, 2, 2700, 2, 180, 120, 210, 2, 7560, 2, 6300, 360, 252, 2, 661500, 20, 420, 168, 94500, 2, 23814000, 2, 2310, 504, 132, 600, 43659000, 2, 396, 840, 2425500, 2, 187110000, 2, 207900, 352800, 1980, 2, 560290500, 60, 194040, 264, 485100, 2, 115259760, 840, 254677500, 792, 4620, 2, 264737261250000, 2, 13860

Offset: 1

Antti Karttunen, Oct 03 2017

Antti Karttunen, Table of n, a(n) for n = 1..1024
Index entries for sequences related to binary expansion of n

Cf. A001065, A002110, A019565, A048675, A091954, A292257, A293215 (restricted growth sequence transform).

Cf. also A293216, A293221, A293222, A293225, A293231, A293442, A300830, A300831, A300832, A300833, A300834.

A019565(n) = {my(j,v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ From A019565
A293214(n) = { my(m=1); fordiv(n,d,if(d < n,m *= A019565(d))); m; };

a(n) = Product_{d|n, dA019565(d).
a(n) = A300830(n) * A300831(n) * A300832(n). - Antti Karttunen, Mar 16 2018
Other identities.
For n >= 0, a(2^n) = A002110(n).
For n >= 1:
A048675(a(n)) = A001065(n).
A001222(a(n)) = A292257(n).
A007814(a(n)) = A091954(n).
A087207(a(n)) = A218403(n).
A248663(a(n)) = A227320(n).

A293443 Multiplicative with a(p^e) = A019565(A193231(e)).

Original entry on oeis.org

1, 2, 2, 6, 2, 4, 2, 3, 6, 4, 2, 12, 2, 4, 4, 10, 2, 12, 2, 12, 4, 4, 2, 6, 6, 4, 3, 12, 2, 8, 2, 5, 4, 4, 4, 36, 2, 4, 4, 6, 2, 8, 2, 12, 12, 4, 2, 20, 6, 12, 4, 12, 2, 6, 4, 6, 4, 4, 2, 24, 2, 4, 12, 15, 4, 8, 2, 12, 4, 8, 2, 18, 2, 4, 12, 12, 4, 8, 2, 20, 10, 4, 2, 24, 4, 4, 4, 6, 2, 24, 4, 12, 4, 4, 4, 10, 2, 12, 12, 36, 2, 8, 2, 6, 8

Offset: 1

Antti Karttunen, Oct 31 2017

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences computed from exponents in factorization of n

Cf. A019565, A028234, A067029, A193231.

Cf. also A270428, A293442, A293231, A293439.

A019565(n) = {my(j,v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ This function from M. F. Hasler
A193231(n) = { my(x='x); subst(lift(Mod(1, 2)*subst(Pol(binary(n), x), x, 1+x)), x, 2) }; \\ And this from Franklin T. Adams-Watters
vecproduct(v) = { my(m=1); for(i=1,#v,m *= v[i]); m; };
A293443(n) = vecproduct(apply(e -> A019565(A193231(e)), factorint(n)[, 2]));

;; With memoization-macro definec.
(definec (A293443 n) (if (= 1 n) n (* (A019565 (A193231 (A067029 n))) (A293443 (A028234 n)))))

a(1) = 1; for n > 1, a(n) = A019565(A193231(A067029(n))) * a(A028234(n)).
For all n >= 1, A007814(a(n)) = A293439(n).
For all k in A270428, A007814(a(k)) = A001221(k).

A300834 a(n) = Product_{d|n, dA019565(A003714(d)), where A003714(n) is the n-th Fibbinary number.

Original entry on oeis.org

1, 2, 2, 6, 2, 30, 2, 60, 10, 42, 2, 4200, 2, 126, 70, 660, 2, 9240, 2, 13860, 210, 330, 2, 5082000, 14, 78, 220, 32760, 2, 3783780, 2, 42900, 550, 780, 294, 924924000, 2, 1092, 130, 41621580, 2, 3898440, 2, 112200, 60060, 306, 2, 28078050000, 42, 235620, 1300, 92820, 2, 200119920, 770, 128648520, 1820, 1122, 2, 424964656116000, 2, 3366

Offset: 1

Antti Karttunen, Mar 18 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..3194

Cf. A003714, A019565, A300835 (rgs-transform of this sequence), A300836.

Cf. also A293214, A293216, A293221, A293222, A293231.

A072649(n) = { my(m); if(n<1, 0, m=0; until(fibonacci(m)>n, m++); m-2); }; \\ From A072649
A003714(n) = { my(s=0,w); while(n>2, w = A072649(n); s += 2^(w-1); n -= fibonacci(w+1)); (s+n); }
A019565(n) = {my(j,v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ From A019565
A300834(n) = { my(m=1); fordiv(n,d,if(d < n,m *= A019565(A003714(d)))); m; };

a(n) = Product_{d|n, dA019565(A003714(d)).

For n >= 1, A001222(a(n)) = A300836(n).

A318834 a(n) = Product_{d|n, dA019565(phi(d)), where phi is the Euler totient function A000010.

Original entry on oeis.org

1, 2, 2, 4, 2, 12, 2, 12, 6, 20, 2, 108, 2, 60, 30, 60, 2, 540, 2, 300, 90, 84, 2, 2700, 10, 140, 90, 2700, 2, 6300, 2, 420, 126, 44, 150, 121500, 2, 132, 210, 10500, 2, 283500, 2, 5292, 3150, 660, 2, 132300, 30, 5500, 66, 14700, 2, 267300, 210, 472500, 198, 1540, 2, 4630500, 2, 4620, 47250, 4620, 350, 873180, 2, 1452, 990

Offset: 1

Antti Karttunen, Sep 04 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..4096

Cf. A000010, A019565, A318835 (rgs-transform).

Cf. also A293214, A293231, A300834.

A019565(n) = {my(j,v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ From A019565
A318834(n) = { my(m=1); fordiv(n,d,if(d < n,m *= A019565(eulerphi(d)))); m; };

a(n) = Product_{d|n, dA019565(A000010(d)).

A048675(a(n)) = A051953(n).

A290090 a(n) is the number of proper divisors of n that are odious (A000069).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Oct 03 2017

If n is odd and k >= 1, then a(2^k*n) = (k+1)*n+k if n is in A000069 and (k+1)*n if n is not in A000069. - Robert Israel, Oct 03 2017

			For n = 55 whose proper divisors are 1, 5 and 11 (in binary "1", "101" and "1011"), only 1 and 11 have an odd number of 1's in their binary representations, thus a(55) = 2.

Antti Karttunen, Table of n, a(n) for n = 1..16385
Index entries for sequences related to binary expansion of n

Cf. A000005, A000069, A010060, A227872, A292257, A293231.

f:= proc(n) nops(select(t -> convert(convert(t,base,2),`+`)::odd, numtheory:-divisors(n) minus {n})) end proc:
map(f, [$1..200]); # Robert Israel, Oct 03 2017

Table[DivisorSum[n, 1 &, And[OddQ@ DigitCount[#, 2, 1], # < n] &], {n, 120}] (* Michael De Vlieger, Oct 03 2017 *)

PARI
```
A290090(n) = sumdiv(n,d,(d
				
```

a(n) = Sum_{d|n, dA010060(d).
a(n) = A227872(n) - A010060(n).
a(n) = A007814(A293231(n)).
A000035(a(n)) = A000035(A292257(n)). [Parity-wise equivalent with A292257.]

A365810 Squareferee numbers ordered factorization-wise by Blue code: a(n) = A019565(A193231(n)).

Original entry on oeis.org

1, 2, 6, 3, 10, 5, 15, 30, 210, 105, 35, 70, 21, 42, 14, 7, 22, 11, 33, 66, 55, 110, 330, 165, 1155, 2310, 770, 385, 462, 231, 77, 154, 858, 429, 143, 286, 2145, 4290, 1430, 715, 5005, 10010, 30030, 15015, 2002, 1001, 3003, 6006, 39, 78, 26, 13, 390, 195, 65, 130, 910, 455, 1365, 2730, 91, 182, 546, 273, 1870, 935

Offset: 0

Antti Karttunen, Oct 06 2023

nonn

Antti Karttunen, Table of n, a(n) for n = 0..16383
Index entries for sequences related to binary expansion of n

Permutation of A005117.
Cf. A019565, A193231, A293231, A293443, A334205.
Cf. also A366263.

A019565(n) = { my(m=1, p=1); while(n>0, p = nextprime(1+p); if(n%2, m *= p); n >>= 1); (m); };
A193231(n) = { my(x='x); subst(lift(Mod(1, 2)*subst(Pol(binary(n), x), x, 1+x)), x, 2) };
A365810(n) = A019565(A193231(n));

a(n) = A334205(A019565(n)).

cp's OEIS Frontend

A293232 Restricted growth sequence transform of A293231, where A293231(n) = Product_{d|n, dA019565(A193231(d)).

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A293214 a(n) = Product_{d|n, dA019565(d).

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A293443 Multiplicative with a(p^e) = A019565(A193231(e)).

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A300834 a(n) = Product_{d|n, dA019565(A003714(d)), where A003714(n) is the n-th Fibbinary number.

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A318834 a(n) = Product_{d|n, dA019565(phi(d)), where phi is the Euler totient function A000010.

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A290090 a(n) is the number of proper divisors of n that are odious (A000069).

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A365810 Squareferee numbers ordered factorization-wise by Blue code: a(n) = A019565(A193231(n)).

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