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Number of nonfixed points y(i) != i, where y is the weakly increasing sequence of prime indices of n.
+10
10
0, 0, 1, 1, 1, 0, 1, 2, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 1, 1, 2, 1, 1, 3, 2, 1, 2, 2, 1, 0, 1, 4, 2, 1, 2, 3, 1, 1, 2, 3, 1, 1, 1, 2, 1, 1, 1, 4, 2, 1, 2, 2, 1, 2, 2, 2, 2, 1, 1, 3, 1, 1, 2, 5, 2, 1, 1, 2, 2, 2, 1, 4, 1, 1, 2, 2, 2, 1, 1, 4, 3, 1, 1, 2, 2, 1, 2, 3, 1, 2, 2, 2, 2, 1, 2, 5, 1, 2, 2, 2, 1, 1, 1, 3, 3
COMMENTS
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
EXAMPLE
The prime indices of 6500 are {1,1,3,3,3,6}, with nonfixed points at positions {2,4,5}, so a(6500) = 3.
MATHEMATICA
pnq[y_]:=Length[Select[Range[Length[y]], #!=y[[#]]&]];
Table[pnq[If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]]], {n, 100}]
PROG
(PARI) A352823(n) = { my(f=factor(n), i=0, c=0); for(k=1, #f~, while(f[k, 2], f[k, 2]--; i++; c += (i!=primepi(f[k, 1])))); (c); }; \\ Antti Karttunen, Apr 11 2022
CROSSREFS
* = unproved
Positions of first appearances are A077552. The complement triangle version is A238352. A122111 represents partition conjugation using Heinz numbers.
A238394 counts reversed partitions without a fixed point, ranked by A352830.
A238395 counts reversed partitions with a fixed point, ranked by A352872.
Cf. A065770, A093641, A114088, A252464, A257990, A325163, A325164, A325165, A325169, A342192, A352486- A352491, A352828, A352829.
1, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 262144, 524288, 1048576, 2097152, 4194304, 8388608, 16777216, 33554432, 67108864, 134217728, 268435456, 536870912, 1073741824, 2147483648, 4294967296, 8589934592
COMMENTS
Hankel transform is := 1,-48,0,0,0,0,0,0,0,...
FORMULA
a(0)=1, a(n) = 2^(n+2) for n>0.
PROG
(PARI) Vec((1+6*x)/(1-2*x) + O(x^50)) \\ Colin Barker, Mar 17 2016
1, 3, 1, 7, 7, 1, 15, 23, 11, 1, 31, 63, 47, 15, 1, 63, 159, 159, 79, 19, 1, 127, 383, 479, 319, 119, 23, 1, 255, 895, 1343, 1119, 559, 167, 27, 1, 511, 2047, 3583, 3583, 2239, 895, 223, 31, 1, 1023, 4607, 9215, 10751, 8063, 4031, 1343, 287, 35, 1
COMMENTS
Row sums = A132308: (1, 4, 15, 50, 157, 480, 1451, ...). Inverse binomial transform of A132307 = triangle A132309 (having row sums A077552).
EXAMPLE
First few rows of the triangle:
1;
3, 1;
7, 7, 1;
15, 23, 11, 1;
31, 63, 47, 15, 1;
63, 159, 159, 79, 19, 1;
...
1, 4, 15, 50, 157, 480, 1451, 4366, 13113, 39356, 118087, 354282, 1062869, 3188632, 9565923, 28697798, 86093425, 258280308, 774840959, 2324522914, 6973568781, 20920706384, 62762119195, 188286357630, 564859072937, 1694577218860
EXAMPLE
a(2) = 15 = sum of row 2 terms of triangle A132307: (7 + 7 + 1). a(2) = 15 = (1, 2, 1) dot (1, 3, 8) = (1 + 6 + 8); where A077552 = (1, 3, 8, 16, 32, 64, ...).
MATHEMATICA
Table[2*3^n-n-1, {n, 0, 30}] (* or *) LinearRecurrence[{5, -7, 3}, {1, 4, 15}, 30] (* Harvey P. Dale, Aug 06 2013 *) CoefficientList[Series[(1 - x + 2 x^2)/((1 - 3 x) (1 - x)^2), {x, 0, 50}], x] (* Vincenzo Librandi, Mar 27 2014 *)
EXTENSIONS
Wrong formula 2*n^3-n-1 removed by Rolf Pleisch, Oct 19 2010
1, 2, 1, 2, 5, 1, 2, 5, 8, 1, 2, 9, 9, 11, 1, 2, 9, 24, 14, 14, 1, 2, 13, 25, 50, 20, 17, 1, 2, 13, 48, 55, 90, 27, 20, 1, 2, 17, 49, 133, 105, 147, 35, 23, 1, 2, 17, 80, 140, 308, 182, 224, 44, 26, 1
COMMENTS
Row sums = A077552: (1, 3, 8, 16, 32, 64, 128, ...).
FORMULA
Inverse binomial transform of triangle A132307.
EXAMPLE
First few rows of the triangle:
1;
2, 1;
2, 5, 1;
2, 5, 8, 1;
2, 9, 9, 11, 1;
2, 9, 24, 14, 14, 1;
2, 13, 25, 50, 20, 17, 1;
2, 13, 48, 55, 90, 27, 20, 1;
...
Independence number of Keller graphs.
+10
2
4, 5, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 262144, 524288, 1048576, 2097152, 4194304, 8388608, 16777216, 33554432, 67108864, 134217728, 268435456, 536870912, 1073741824, 2147483648, 4294967296, 8589934592
REFERENCES
W. Jarnicki, W. Myrvold, P. Saltzman, S. Wagon, Properties, proved and conjectured, of Keller, queen, and Mycielski graphs, Ars Mathematica Contemporanea 13:2 (2017) 427-460.
FORMULA
a(n) = 2^n except a(1) = 4 and a(2) = 5.
EXAMPLE
For G(2), a maximum independent set is {03,10,12,13,23}.
MATHEMATICA
Join[{4, 5}, 2^Range[3, 10]]
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