A293216 -id:A293216 - cp's OEIS frontend

A293217 Restricted growth sequence transform of A293216, where A293216(n) = Product_{d|n, dA260443(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, 4, 8, 17, 18, 2, 19, 2, 20, 21, 22, 17, 23, 2, 13, 24, 25, 2, 26, 2, 27, 28, 29, 2, 30, 31, 32, 33, 34, 2, 35, 36, 37, 38, 39, 2, 40, 2, 41, 42, 43, 44, 45, 2, 46, 47, 48, 2, 49, 2, 27, 50, 51, 44, 52, 2, 53, 54, 55, 2, 56, 57, 58, 59, 60, 2, 61, 62, 63, 64, 65, 66, 67, 2, 68, 69

Offset: 1

Antti Karttunen, Oct 03 2017

nonn

For all i, j: a(i) = a(j) => A001065(i) = A001065(j).

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

Cf. A001065, A260443, A293216, A293215 (a variant).

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])); }
A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ This function from Michel Marcus
A260443(n) = if(n<2, n+1, if(n%2, A260443(n\2)*A260443(n\2+1), A003961(A260443(n\2))));
A293216(n) = { my(m=1); fordiv(n,d,if(d < n,m *= A260443(d))); m; };
write_to_bfile(1,rgs_transform(vector(16384,n,A293216(n))),"b293217.txt");

A091954 Number of odd proper divisors of n. That is, the number of odd divisors of n that are less than n.

Original entry on oeis.org

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

Offset: 1

Mohammad K. Azarian, Mar 12 2004

			The odd divisors of 15 that are less than 15 are 1, 3 and 5. Therefore there are three odd divisors of 15 that are less than 15.

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

Cf. A000005, A000035, A001620, A032741, A001227, A293214, A293216.
Sum of the k-th powers of the odd proper divisors of n for k=0..10: this sequence (k=0), A091570 (k=1), A351647 (k=2), A352031 (k=3), A352032 (k=4), A352033 (k=5), A352034 (k=6), A352035 (k=7), A352036 (k=8), A352037 (k=9), A352038 (k=10).
Sum of the k-th powers of the divisor complements of the odd proper divisors of n for k=0..10: this sequence (k=0), A352047 (k=1), A352048 (k=2), A352049 (k=3), A352050 (k=4), A352051 (k=5), A352052 (k=6), A352053 (k=7), A352054 (k=8), A352055 (k=9), A352056 (k=10).

Count[Most[Divisors[#]],?OddQ]&/@Range[100] (* _Harvey P. Dale, Sep 28 2012 *)
a[n_] := DivisorSigma[0, n/2^IntegerExponent[n, 2]] - Boole[OddQ[n]]; Array[a, 100] (* Amiram Eldar, Jun 11 2022 *)

A091954(n) = sumdiv(n,d,(dAntti Karttunen, Oct 04 2017

my(N=66, x='x+O('x^N)); concat(0, Vec(sum(k=2, N, x^k/(1-x^(2*k))))) \\ Seiichi Manyama, Jan 23 2021

From Antti Karttunen, Oct 04 2017: (Start)
a(n) = Sum_{d|n, dA000035(n).
a(n) = A001227(n) - A000035(n).
a(n) = A007814(A293214(n)) = A007814(A293216(n)).
(End)
G.f.: Sum_{k>=2} x^k/(1 - x^(2*k)). - Seiichi Manyama, Jan 23 2021
Sum_{k=1..n} a(k) ~ n*log(n)/2 + (gamma + log(2)/2 - 1)*n, where gamma is Euler's constant (A001620). - Amiram Eldar, Nov 26 2023

Corrected and extended by Harvey P. Dale, Sep 28 2012

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).

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).

A317837 a(n) = Sum_{d|n, dA002487(d).

Original entry on oeis.org

0, 1, 1, 2, 1, 4, 1, 3, 3, 5, 1, 7, 1, 5, 6, 4, 1, 10, 1, 9, 6, 7, 1, 10, 4, 7, 7, 9, 1, 16, 1, 5, 8, 7, 7, 17, 1, 9, 8, 13, 1, 20, 1, 13, 14, 9, 1, 13, 4, 15, 8, 13, 1, 22, 9, 13, 10, 9, 1, 26, 1, 7, 18, 6, 9, 22, 1, 13, 10, 23, 1, 24, 1, 13, 17, 17, 9, 26, 1, 17, 15, 13, 1, 34, 9, 15, 10, 19, 1, 40, 9, 17, 8, 11, 11

Offset: 1

Antti Karttunen, Aug 09 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences related to Stern's sequences

Cf. A002487, A293216, A317838, A317839, A317840, A317841, A317843.

A002487(n) = { my(a=1, b=0); while(n>0, if(bitand(n, 1), b+=a, a+=b); n>>=1); (b); }; \\ From A002487
A317837(n) = sumdiv(n,d,(dA002487(d));

a(n) = Sum_{d|n, dA002487(d).
a(n) = A317838(n) - A002487(n).
a(n) = A001222(A293216(n)).

A317942 a(n) = Product_{d|n, dA286378(d)-1).

Original entry on oeis.org

1, 2, 2, 4, 2, 12, 2, 8, 6, 20, 2, 72, 2, 28, 30, 16, 2, 396, 2, 200, 42, 52, 2, 432, 10, 68, 66, 392, 2, 17100, 2, 32, 78, 92, 70, 26136, 2, 116, 102, 2000, 2, 54684, 2, 1352, 6270, 148, 2, 2592, 14, 3700, 138, 2312, 2, 178596, 130, 5488, 174, 172, 2, 9747000, 2, 188, 14322, 64, 170, 322452, 2, 4232, 222, 289100, 2, 1724976, 2

Offset: 1

Antti Karttunen, Aug 12 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..8192
Index entries for sequences related to Stern's sequences

Cf. A286378, A317943 (restricted growth sequence transform), A317944.

Cf. also A293216, A305792.

\\ Needs also code from A286378:
A317942(n) = { my(m=1); fordiv(n,d,if(dA286378(d)-1))); (m); };

a(n) = Product_{d|n, dA008578(A286378(d)).

For all k >= 0, a(2^k) = 2^k.

A318470 Multiplicative with a(p^e) = A260443(e).

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 6, 3, 4, 2, 6, 2, 4, 4, 5, 2, 6, 2, 6, 4, 4, 2, 12, 3, 4, 6, 6, 2, 8, 2, 18, 4, 4, 4, 9, 2, 4, 4, 12, 2, 8, 2, 6, 6, 4, 2, 10, 3, 6, 4, 6, 2, 12, 4, 12, 4, 4, 2, 12, 2, 4, 6, 15, 4, 8, 2, 6, 4, 8, 2, 18, 2, 4, 6, 6, 4, 8, 2, 10, 5, 4, 2, 12, 4, 4, 4, 12, 2, 12, 4, 6, 4, 4, 4, 36, 2, 6, 6, 9, 2, 8, 2, 12, 8

Offset: 1

Antti Karttunen, Aug 30 2018

Cf. A260443, A318306, A318307.
Differs from A293442 for the first time at n=32, where a(32) = 18, while A293442(32) = 10.
Cf. also A293216, A293217, A293443, A318469.

A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
A260443(n) = if(n<2, n+1, if(n%2, A260443(n\2)*A260443(n\2+1), A003961(A260443(n\2))));
A318470(n) = factorback(apply(e -> A260443(e),factor(n)[,2]));

For all n >= 1, A001222(a(n)) = A318306(n).

cp's OEIS Frontend

A293217 Restricted growth sequence transform of A293216, where A293216(n) = Product_{d|n, dA260443(d).

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A091954 Number of odd proper divisors of n. That is, the number of odd divisors of n that are less than n.

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

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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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A317837 a(n) = Sum_{d|n, dA002487(d).

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A317942 a(n) = Product_{d|n, dA286378(d)-1).

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A318470 Multiplicative with a(p^e) = A260443(e).

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