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A376411
a(n) is the number of terms less than A276086(n) in the range of A276086, where A276086 is the primorial base exp-function.
4
0, 1, 2, 4, 6, 13, 3, 7, 11, 21, 32, 64, 18, 36, 54, 108, 162, 325, 90, 180, 271, 541, 812, 1624, 450, 902, 1354, 2707, 4061, 8122, 5, 10, 15, 30, 45, 91, 25, 50, 75, 151, 227, 454, 126, 253, 378, 758, 1137, 2274, 632, 1264, 1895, 3790, 5685, 11370, 3158, 6317, 9475, 18952, 28428, 56856, 35, 70, 106, 212, 318, 637
OFFSET
0,3
COMMENTS
Number of terms of A048103 that are less than A276086(n).
Permutation of nonnegative integers.
Troughs are at primorials, A002110, and the local maxima occur just before, at A057588.
FORMULA
a(n) = A377982(A276086(n))-1 = Sum_{i=1 .. A276086(n)-1} A359550(i).
For all n >= 1, a(A376413(n)) = n-1, and for all n >= 0, A376413(1+a(n)) = n.
a(i)/a(j) ~ A276086(i)/A276086(j), and particularly, a(2*n+1) ~ 2*a(2*n).
PROG
(PARI)
up_to = (2*210)-1; \\ Must be one of the terms of A343048.
A276085(n) = { my(f = factor(n), pr=1, i=1, s=0); for(k=1, #f~, while(i <= primepi(f[k, 1])-1, pr *= prime(i); i++); s += f[k, 2]*pr); (s); };
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A359550(n) = { my(pp); forprime(p=2, , pp = p^p; if(!(n%pp), return(0)); if(pp > n, return(1))); };
A376411list(up_to) = { my(size=up_to, v=vector(size), m=A276086(size), s=1, j); for(i=2, m, if(!(m%i), j=A276085(i); v[j] = s; print1("i=", i, " v[", j, "]=", s", "); ); s += A359550(i)); (v); };
v376411 = A376411list(up_to);
A376411(n) = if(!n, n, v376411[n]);
(PARI)
\\ For incremental computing, less efficient than above:
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A359550(n) = { my(pp); forprime(p=2, , pp = p^p; if(!(n%pp), return(0)); if(pp > n, return(1))); };
memoA376411 = Map(); \\ We use k=A276086(n) as our key. kvs will be a list of key-value-pairs sorted into descending order by the key. We search the largest key in it < k, and continue summing from that:
A376411(n) = if(n<=2, n, my(v, k=A276086(n)); if(mapisdefined(memoA376411, k, &v), v, my(kvs = vecsort(Mat(memoA376411)~, (x, y) -> sign(y[1]-x[1])), ss=si=0); for(i=1, #kvs, if(kvs[1, i]<k, si=kvs[1, i]; ss=kvs[2, i]; break)); v = ss + sum(i=si, k-1, A359550(i)); mapput(memoA376411, k, v); (v)));
CROSSREFS
Cf. A376413 (inverse permutation, but note the different offsets and ranges).
Cf. also A064273 (analogous permutation for base-2).
Sequence in context: A196444 A307617 A325278 * A110980 A376644 A376533
KEYWORD
nonn
AUTHOR
Antti Karttunen, Nov 13 2024
STATUS
approved