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A374097 a(n) = A196563(n)*A196564(n). +0
0
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,101
COMMENTS
More than the usual number of terms are shown in order to distinguish this sequence from A180160, from which it first differs at n = 100.
LINKS
Table of n, a(n) for n=0..110.
MATHEMATICA
A374097[n_] := #*(IntegerLength[n] - #) & [Total[Mod[IntegerDigits[n], 2]]];
Array[A374097, 120, 0]
CROSSREFS
Cf. A055642, A059708 (positions of zeros), A180160, A196563, A196564.
KEYWORD
nonn,base,easy,new
AUTHOR
Paolo Xausa, Jun 28 2024
STATUS
approved
A373917 Triangle read by rows: T(n,k) = k*10 mod n, with n >= 1, k >= 0. +0
0
0, 0, 0, 0, 1, 2, 0, 2, 0, 2, 0, 0, 0, 0, 0, 0, 4, 2, 0, 4, 2, 0, 3, 6, 2, 5, 1, 4, 0, 2, 4, 6, 0, 2, 4, 6, 0, 1, 2, 3, 4, 5, 6, 7, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 10, 8, 6, 4, 2, 0, 10, 8, 6, 4, 2, 0, 10, 7, 4, 1, 11, 8, 5, 2, 12, 9, 6, 3 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
1,6
COMMENTS
Each row n encodes a "division graph" used to determine m mod n (where m is an arbitrary nonnegative integer), using the method described in the Numberphile link (see also example).
LINKS
Table of n, a(n) for n=1..91.
James Grime and Brady Haran, Solving Seven, Numberphile YouTube video, 2024.
EXAMPLE
Triangle begins:
n\k| 0 1 2 3 4 5 6 7 8 9
---------------------------------
1 | 0;
2 | 0, 0;
3 | 0, 1, 2;
4 | 0, 2, 0, 2;
5 | 0, 0, 0, 0, 0;
6 | 0, 4, 2, 0, 4, 2;
7 | 0, 3, 6, 2, 5, 1, 4;
8 | 0, 2, 4, 6, 0, 2, 4, 6;
9 | 0, 1, 2, 3, 4, 5, 6, 7, 8;
10 | 0, 0, 0, 0, 0, 0, 0, 0, 0, 0;
...
Suppose m = 3714289 and you want to determine m mod 7 (the example shown in the video).
Start with the first digit of m (3) and calculate T(7,3 mod 7) = T(7,3) = 2.
Add it to the next digit of m (7) and calculate T(7,(2+7) mod 7) = T(7,2) = 6.
Add it to the next digit of m (1) and calculate T(7,(6+1) mod 7) = T(7,0) = 0.
Add it to the next digit of m (4) and calculate T(7,(0+4) mod 7) = T(7,4) = 5.
Add it to the next digit of m (2) and calculate T(7,(5+2) mod 7) = T(7,0) = 0.
Add it to the next digit of m (8) and calculate T(7,(0+8) mod 7) = T(7,1) = 3.
Add it to the final digit of m (9) and calculate (3+9) mod 7 = 5, which corresponds to 3714289 mod 7.
MATHEMATICA
Table[Mod[Range[0, 10*(n-1), 10], n], {n, 15}]
CROSSREFS
Cf. A051127, A106611 (number of distinct terms in each row).
KEYWORD
nonn,tabl,new
AUTHOR
Paolo Xausa, Jun 26 2024
STATUS
approved
A373889 Square array read by ascending antidiagonals: T(k,n) is the cardinality of {(E is a proper finite subset of the natural numbers) such that E = {} or w_k(E) < min(E) <= max(E) <= n}, where w_k(E) = Sum_{i in E, i <> k} 1, with n, k >= 1. +0
0
2, 1, 3, 1, 2, 4, 1, 2, 4, 6, 1, 2, 4, 7, 9, 1, 2, 3, 6, 11, 14, 1, 2, 3, 6, 10, 17, 22, 1, 2, 3, 5, 10, 17, 26, 35, 1, 2, 3, 5, 10, 16, 28, 40, 56, 1, 2, 3, 5, 8, 16, 26, 45, 62, 90, 1, 2, 3, 5, 8, 16, 26, 43, 71, 97, 145, 1, 2, 3, 5, 8, 13, 26, 42, 71, 111, 153, 234 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
1,1
LINKS
Table of n, a(n) for n=1..78.
Hung Viet Chu and Zachary Louis Vasseur, Weighted Schreier-type Sets and the Fibonacci Sequence, arXiv:2405.19352 [math.CO], 2024. See p. 2, Table 1 and Theorem 1.2.
FORMULA
T(k,n) = A000045(n-k+2) + 1, for k = 1 and n >= k;
T(k,n) = 2*(Sum_{i=0..k-2} binomial(n-k,i)*A000045(k-i)) + 2*binomial(n-k,k-1) + Sum_{j=1..n-k} binomial(j,n-j), for k >= 2 and n >= k;
T(k,n) = A000045(n+1) otherwise.
T(n,n) = 2*A000045(n).
EXAMPLE
The array begins:
k\n| 1 2 3 4 5 6 7 8 9 10 ...
----------------------------------------------
1 | 2, 3, 4, 6, 9, 14, 22, 35, 56, 90, ... = A001611 (from n = 2).
2 | 1, 2, 4, 7, 11, 17, 26, 40, 62, 97, ...
3 | 1, 2, 4, 6, 10, 17, 28, 45, 71, 111, ...
4 | 1, 2, 3, 6, 10, 16, 26, 43, 71, 116, ...
5 | 1, 2, 3, 5, 10, 16, 26, 42, 68, 111, ...
6 | 1, 2, 3, 5, 8, 16, 26, 42, 68, 110, ...
7 | 1, 2, 3, 5, 8, 13, 26, 42, 68, 110, ...
8 | 1, 2, 3, 5, 8, 13, 21, 42, 68, 110, ...
9 | 1, 2, 3, 5, 8, 13, 21, 34, 68, 110, ...
10 | 1, 2, 3, 5, 8, 13, 21, 34, 55, 110, ...
...
MATHEMATICA
A373889[k_, n_] := Which[n < k, Fibonacci[n+1], k == 1, Fibonacci[n-k+2] + 1, True, 2*Sum[Binomial[n-k, i]*Fibonacci[k-i], {i, 0, k-2}] + 2*Binomial[n-k, k-1] + Sum[Binomial[j, n-j], {j, n-k}]];
Table[A373889[k-n+1, n], {k, 15}, {n, k}]
CROSSREFS
Cf. A000045, A001611, A373345, A373556, A373557.
KEYWORD
nonn,tabl,new
AUTHOR
Paolo Xausa, Jun 21 2024
STATUS
approved
A373196 Maximal coefficient (in absolute value) in the numerator of C({1..n},x). +0
1
1, 1, 2, 17, 444, 66559954, 14648786369948422, 791540878703169050660325841979096789557779, 1918013047695258943191946313451491492494186620117241479813740479213857275772347178176158 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
LINKS
Table of n, a(n) for n=0..8.
FORMULA
C({s},x) = Sum_{i in {s}} (C({s}-{i},x)*x^i)/(1 - Sum_{i in {s}} (x^i)) with C({},x) = 1.
EXAMPLE
C_x({1,2,3},x) = (-x^15 - 5*x^14 - 12*x^13 - 17*x^12 - 11*x^11 + 4*x^10 + 16*x^9 + 10*x^8 - 6*x^6)/(x^15 + 4*x^14 + 7*x^13 + 4*x^12 - 8*x^11 - 18*x^10 - 13*x^9 + 7*x^8 + 19*x^7 + 11*x^6 - 6*x^5 - 10*x^4 - 2*x^3 + 3*x^2 + 2*x - 1) with maximal coefficient abs(-17) in the numerator, so a(3) = 17.
PROG
(PARI)
C_x(s)={my(g=if(#s <1, 1, sum(i=1, #s, C_x(s[^i])*x^(s[i]))/(1-sum(i=1, #s, x^(s[i]))))); return(g)}
a(n)={vecmax(abs(Vec(numerator(C_x([1..n])))))}
CROSSREFS
Cf. A107429, A114536, A278567.
KEYWORD
nonn,new
AUTHOR
John Tyler Rascoe, Jun 28 2024
STATUS
approved
A373793 First differences of A373792, halved. +0
0
4, 6, 6, 1, 5, 2, 4, 4, 13, 9, 7, 2, 4, 6, 2, 7, 6, 4, 6, 1, 3, 7, 2, 10, 10, 5, 2, 4, 11, 6, 10, 2, 4, 6, 5, 6, 6, 4, 7, 5, 3, 8, 9, 0, 8, 4, 3, 4, 6, 8, 5, 3, 2, 8, 6, 1, 6, 6, 6, 9, 2, 17, 4, 13, 5, 7, 2, 5, 9, 4, 5, 1, 6, 3, 4, 2, 9, 10, 1, 3, 4, 4, 2, 12, 4, 5, 5, 6, 7, 12, 6, 6, 3, 4, 8, 3, 4, 9, 5, 7 (list; graph; refs; listen; history; text; internal format)
OFFSET
38,1
COMMENTS
The offset is 38, because up to that point the terms are not integers.
It appears that in A373390 the primes up to and including 157 (which is A373390(335)) appear irregularly. The next prime is 163 = A373390(350). So there is a possibility that ignoring the first 347 or so terms of A373390 may make it easier to analyze. A formula or other explanation for the present sequence would be of great help.
LINKS
N. J. A. Sloane, Table of n, a(n) for n = 38..6266
EXAMPLE
The first 38 terms of the first differences of A373792, halved, are
0, 5, -5, 29/2, -15, 23/2, -23/2, 27, -19/2, -35/2, 41, -41/2, 3, 3, 61/2, -49/2, -65/2, 71/2, 67/2, 4, 4, 9, 3, -77/2, 97/2, 3, 7, -2, 2, 16, 3, 8, 1, 12, 0, -137/2, 161/2, 4,
and that final 4 is the leading term of the present sequence.
KEYWORD
nonn,new
AUTHOR
N. J. A. Sloane, Jun 29 2024
STATUS
approved
A374120 Numbers k such that A113177(k) and A276085(k) are both multiples of 3, where A113177 and A276085 are fully additive with a(p) = Fibonacci(p) and a(p) = p#/p, respectively. +0
0
1, 6, 8, 27, 35, 36, 48, 64, 77, 85, 91, 115, 125, 133, 155, 162, 187, 203, 205, 210, 216, 221, 235, 253, 259, 275, 280, 288, 299, 301, 323, 325, 341, 343, 355, 365, 371, 384, 395, 403, 413, 427, 437, 445, 451, 462, 469, 475, 485, 493, 510, 512, 515, 517, 533, 546, 565, 581, 589, 605, 611, 616, 629, 635, 667, 680 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
A multiplicative semigroup: if m and n are in the sequence, then so is m*n.
LINKS
Antti Karttunen, Table of n, a(n) for n = 1..12000
PROG
(PARI) isA374120 = A374119;
CROSSREFS
Intersection of A374052 and A339746.
Indices of multiples of 3 in A374112.
Cf. A113177, A276085, A374119 (characteristic function).
Cf. also A374114.
KEYWORD
nonn,new
AUTHOR
Antti Karttunen, Jun 29 2024
STATUS
approved
A374119 a(n) = 1 if A113177(n) and A276085(n) are both multiples of 3, otherwise 0, where A113177 and A276085 are fully additive with a(p) = Fibonacci(p) and a(p) = p#/p, respectively. +0
0
1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET
1
LINKS
Antti Karttunen, Table of n, a(n) for n = 1..100000
Index entries for characteristic functions
FORMULA
a(n) = A372573(n) * A374051(n).
a(n) = A079978(A374112(n)).
PROG
(PARI)
A113177(n) = if(n<=1, 0, my(f=factor(n)); sum(i=1, #f~, f[i, 2]*fibonacci(f[i, 1])));
A276085(n) = { my(f=factor(n)); sum(k=1, #f~, f[k, 2]*prod(i=1, primepi(f[k, 1]-1), prime(i))); };
A374119(n) = (!(A113177(n)%3) && !(A276085(n)%3));
CROSSREFS
Characteristic function of A374120.
Cf. A079978, A113177, A276085, A372573, A374051, A374112.
Cf. also A374113.
KEYWORD
nonn,new
AUTHOR
Antti Karttunen, Jun 29 2024
STATUS
approved
A374114 Numbers k such that A113177(k) and A276085(k) are both even, where A113177 and A276085 are fully additive with a(p) = Fibonacci(p) and a(p) = p#/p, respectively. +0
0
1, 3, 4, 9, 12, 16, 25, 27, 35, 36, 48, 49, 55, 64, 65, 75, 77, 81, 85, 91, 95, 100, 105, 108, 115, 119, 121, 133, 140, 143, 144, 145, 147, 155, 161, 165, 169, 185, 187, 192, 195, 196, 203, 205, 209, 215, 217, 220, 221, 225, 231, 235, 243, 247, 253, 255, 256, 259, 260, 265, 273, 285, 287, 289, 295, 299, 300, 301, 305 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
Numbers whose 2-adic valuation (A007814) is even, and the number of the prime factors (with multiplicity, A001222) and the 3-adic valuation (A007949) have the same parity.
A multiplicative semigroup: if m and n are in the sequence, then so is m*n.
LINKS
Antti Karttunen, Table of n, a(n) for n = 1..12000
PROG
(PARI) isA374114 = A374113;
CROSSREFS
Intersection of A003159 and A373586.
Indices of even terms in A374112.
Cf. A001222, A007814, A007949, A113177, A276085, A374113 (characteristic function), A374115 (complement).
KEYWORD
nonn,new
AUTHOR
Antti Karttunen, Jun 29 2024
STATUS
approved
A374115 Numbers k such that A113177(k) and A276085(k) are not both even, where A113177 and A276085 are fully additive with a(p) = Fibonacci(p) and a(p) = p#/p, respectively. +0
0
2, 5, 6, 7, 8, 10, 11, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 26, 28, 29, 30, 31, 32, 33, 34, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 50, 51, 52, 53, 54, 56, 57, 58, 59, 60, 61, 62, 63, 66, 67, 68, 69, 70, 71, 72, 73, 74, 76, 78, 79, 80, 82, 83, 84, 86, 87, 88, 89, 90, 92, 93, 94, 96, 97, 98, 99, 101, 102 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
LINKS
Antti Karttunen, Table of n, a(n) for n = 1..12000
PROG
(PARI) isA374115(n) = !A374113(n);
CROSSREFS
Cf. A001222, A007814, A007949, A113177, A276085, A374113, A374114 (complement).
Union of A036554 and A373587.
Indices of odd terms in A374112.
KEYWORD
nonn,new
AUTHOR
Antti Karttunen, Jun 29 2024
STATUS
approved
A374113 a(n) = 1 if A113177(n) and A276085(n) are both even, otherwise 0, where A113177 and A276085 are fully additive with a(p) = Fibonacci(p) and a(p) = p#/p, respectively. +0
0
1, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET
1
COMMENTS
a(n) = 1 if the 2-adic valuation of n is even, and the number of its prime factors (with multiplicity, A001222) and its 3-adic valuation (A007949) have the same parity, otherwise 0.
LINKS
Antti Karttunen, Table of n, a(n) for n = 1..100000
Index entries for characteristic functions
FORMULA
a(n) = A035263(n) * A373585(n).
a(n) = A059841(A374112(n)).
PROG
(PARI)
A113177(n) = if(n<=1, 0, my(f=factor(n)); sum(i=1, #f~, f[i, 2]*fibonacci(f[i, 1])));
A276085(n) = { my(f=factor(n)); sum(k=1, #f~, f[k, 2]*prod(i=1, primepi(f[k, 1]-1), prime(i))); };
A374113(n) = (!(A113177(n)%2) && !(A276085(n)%2));
(PARI) A374113(n) = (!(valuation(n, 2)%2) && !((bigomega(n)-valuation(n, 3))%2));
CROSSREFS
Characteristic function of A374114, whose complement A374115 gives the indices of 0's.
Cf. A001222, A007949, A035263, A113177, A059841, A276085, A373585, A374112.
KEYWORD
nonn,new
AUTHOR
Antti Karttunen, Jun 29 2024
STATUS
approved
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Last modified June 29 19:38 EDT 2024. Contains 373855 sequences. (Running on oeis4.)