A317837 -id:A317837 - cp's OEIS frontend

A323365 Sum of Stern's Diatomic sequence, A002487 and its Dirichlet inverse, A317843.

2, 0, 0, 1, 0, 4, 0, 1, 4, 6, 0, 2, 0, 6, 12, 1, 0, 4, 0, 3, 12, 10, 0, 2, 9, 10, 8, 3, 0, -4, 0, 1, 20, 10, 18, 4, 0, 14, 20, 3, 0, 4, 0, 5, 4, 14, 0, 2, 9, 5, 20, 5, 0, 8, 30, 3, 28, 14, 0, 4, 0, 10, 20, 1, 30, -8, 0, 5, 28, 0, 0, 4, 0, 22, -2, 7, 30, 0, 0, 3, 16, 22, 0, 8, 30, 26, 28, 5, 0, 20, 30, 7, 20, 18, 42, 2, 0, 9, 4, 7, 0, 4, 0, 5, 0

Offset: 1

Antti Karttunen, Jan 13 2019

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

Cf. A002487 (also a quadrisection of this sequence), A317843.

Cf. also A317837, A319687, A323885, A323887, A323894, A323896.

A002487(n) = { my(a=1, b=0); while(n>0, if(bitand(n, 1), b+=a, a+=b); n>>=1); (b); }; \\ From A002487
A317843(n) = if(1==n,1,-sumdiv(n,d,if(dA002487(n/d)*A317843(d),0)));
A323365(n) = (A002487(n) + A317843(n));

a(n) = A002487(n) + A317843(n).
From Antti Karttunen, Dec 08 2021: (Start)
a(1) = 2, and for n > 1, a(n) = -Sum_{d|n, 1A002487(d) * A317843(n/d).
a(4*n) = A002487(n).
(End)

A317839 Möbius transform of A002487, Stern's Diatomic sequence.

Original entry on oeis.org

1, 0, 1, 0, 2, 0, 2, 0, 2, 0, 4, 0, 4, 0, 0, 0, 4, 0, 6, 0, 4, 0, 6, 0, 4, 0, 4, 0, 6, 0, 4, 0, 0, 0, 4, 0, 10, 0, 4, 0, 10, 0, 12, 0, 6, 0, 8, 0, 6, 0, 6, 0, 12, 0, 4, 0, 2, 0, 10, 0, 8, 0, -4, 0, 0, 0, 10, 0, 6, 0, 12, 0, 14, 0, 10, 0, 10, 0, 12, 0, 6, 0, 18, 0, 14, 0, 10, 0, 16, 0, 12, 0, 10, 0, 2, 0, 10, 0, 8, 0, 18, 0, 16, 0, 4

Offset: 1

Antti Karttunen, Aug 09 2018

sign

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

Cf. A002487, A008683, A317837, A317838, 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
A317839(n) = sumdiv(n,d,moebius(n/d)*A002487(d));

a(n) = Sum_{d|n} A008683(n/d)*A002487(d).

a(n) = A000010(n) - A317841(n).

A317838 a(n) = Sum_{d|n} A002487(d).

Original entry on oeis.org

1, 2, 3, 3, 4, 6, 4, 4, 7, 8, 6, 9, 6, 8, 10, 5, 6, 14, 8, 12, 14, 12, 8, 12, 11, 12, 15, 12, 8, 20, 6, 6, 14, 12, 16, 21, 12, 16, 18, 16, 12, 28, 14, 18, 26, 16, 10, 15, 13, 22, 20, 18, 14, 30, 20, 16, 20, 16, 12, 30, 10, 12, 24, 7, 16, 28, 12, 18, 24, 32, 14, 28, 16, 24, 35, 24, 26, 36, 14, 20, 29, 24, 20, 42, 30, 28, 28, 24

Offset: 1

Antti Karttunen, Aug 09 2018

nonn

Inverse Möbius transform of A002487, Stern's Diatomic sequence.

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

Cf. A002487, A317837, 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
A317838(n) = sumdiv(n,d,A002487(d));

a(n) = Sum_{d|n} A002487(d).

a(n) = A317837(n) + A002487(n).

A317943 Filter sequence constructed from the coefficients of the Stern polynomials B(d,t) collected for each proper divisor d of n; Restricted growth sequence transform of A317942.

Original entry on oeis.org

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, 35, 36, 37, 38, 2, 39, 40, 41, 42, 43, 2, 44, 2, 45, 46, 47, 48, 49, 2, 50, 51, 52, 2, 53, 2, 54, 55, 56, 57, 58, 2, 59, 60, 61, 2, 62, 63, 64, 65, 66, 2, 67, 68, 69, 70, 71, 72, 73, 2, 74, 75, 76, 2, 77, 2, 78, 79, 80, 2, 81, 2, 82, 83, 84, 2, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 86

Offset: 1

Antti Karttunen, Aug 12 2018

nonn

For all i, j: a(i) = a(j) => A317837(i) = A317837(j).

			Proper divisors of 115 are 1, 5 and 23 and proper divisors of 125 are 1, 5 and 25. The divisors 1 and 5 occur in both, while for the Stern polynomials B(23,t) and B(25,t) (see A125184) the nonzero coefficients are {1, 2, 3, 1} and {1, 3, 2, 1}, that is, they are equal as multisets, thus A286378(23) = A286378(25). From this follows that a(115) = a(125).

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences related to Stern's sequences

Cf. A125184, A286378, A317837, A317942, A317945.
Cf. also A293217, A305793.
Differs from A305800 and A296073 for the first time at n=125, where a(125) = 86.

\\ Needs also code from A286378:
up_to = 65537;
A317942(n) = { my(m=1); fordiv(n,d,if(dA286378(d)-1))); (m); };
v317943 = rgs_transform(vector(up_to, n, A317942(n)));
A317943(n) = v317943[n];

A317840 Difference between Stern's Diatomic sequence (A002487) and its Möbius transform (A317839).

Original entry on oeis.org

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

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, A008683, A317837, A317838, A317839, A317841.

A002487(n) = { my(a=1, b=0); while(n>0, if(bitand(n, 1), b+=a, a+=b); n>>=1); (b); }; \\ From A002487
A317840(n) = -sumdiv(n,d,(dA002487(d));

a(n) = -Sum_{d|n, dA008683(n/d)*A002487(d).

a(n) = A002487(n) - A317839(n).

A319687 a(n) = A318509(n) - A002487(n).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, -2, 0, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, 2, -2, 0, 0, 0, 4, 0, 4, 0, 0, 2, 0, 0, 6, 0, 8, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 2, 0, 0, -2, -6, 0, -4, 0, 0, 0, -4, 0, -6, 0, 10, 0, 0, 0, 4, 2, 0, -2, 0, 0, 0

Offset: 1

Antti Karttunen, Oct 02 2018

All terms seem to be even. See the conjecture given in A261179.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences related to Stern's sequences

Cf. A002487, A261179, A317837, A318509, A323365.

A002487(n) = { my(a=1, b=0); while(n>0, if(bitand(n, 1), b+=a, a+=b); n>>=1); (b); }; \\ From A002487
A318509(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 1] = A002487(f[i, 1])); factorback(f); };
A319687(n) = (A318509(n) - A002487(n));

from math import prod
from functools import reduce
from sympy import factorint
def A319687(n): return prod(sum(reduce(lambda x,y:(x[0],x[0]+x[1]) if int(y) else (x[0]+x[1],x[1]),bin(p)[-1:2:-1],(1,0)))**e for p, e in factorint(n).items())-sum(reduce(lambda x,y:(x[0],x[0]+x[1]) if int(y) else (x[0]+x[1],x[1]),bin(n)[-1:2:-1],(1,0))) # Chai Wah Wu, May 18 2023

a(n) = A318509(n) - A002487(n).

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A323365 Sum of Stern's Diatomic sequence, A002487 and its Dirichlet inverse, A317843.

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A317839 Möbius transform of A002487, Stern's Diatomic sequence.

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A317838 a(n) = Sum_{d|n} A002487(d).

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A317943 Filter sequence constructed from the coefficients of the Stern polynomials B(d,t) collected for each proper divisor d of n; Restricted growth sequence transform of A317942.

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A317840 Difference between Stern's Diatomic sequence (A002487) and its Möbius transform (A317839).

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A319687 a(n) = A318509(n) - A002487(n).

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