cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A280692 a(n) = A003961(n) - A250469(n).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 6, 0, -6, 0, 12, 0, -6, 0, 36, 0, 24, 0, 6, 0, -24, 0, 66, 0, -24, 60, 18, 0, 18, 0, 150, -20, -42, 0, 120, 0, -42, -10, 72, 0, 42, 0, -12, 60, -48, 0, 264, 0, 0, -30, 0, 0, 216, 0, 132, -30, -78, 0, 138, 0, -72, 120, 540, 0, 0, 0, -30, -30, 24, 0, 462, 0, -96, 60, -18, 0, 24, 0, 330, 420, -114, 0, 246
Offset: 1

Views

Author

Antti Karttunen, Mar 08 2017

Keywords

Links

Crossrefs

Cf. A003961, A250469, A249820, A280492, A280497, A280498, A280702.
Cf. A280693 (gives the positions of zeros).
Cf. also arrays A083221 and A246278.

Programs

  • Mathematica
    f[n_] := f[n] = Which[n == 1, 1, PrimeQ@ n, NextPrime@ n, True, Times @@ Replace[FactorInteger[n], {p_, e_} :> f[p]^e, 1]]; g[n_] := If[n == 1, 0, PrimePi@ FactorInteger[n][[1, 1]]]; Function[s, MapIndexed[ Function[{m, n}, f@ n - Lookup[s, g[n] + 1][[m]] + Boole[n == 1]][#1, First@ #2] &, #] &@ Map[Position[Lookup[s, g@ #], #][[1, 1]] &, Range@ 120]]@ PositionIndex@ Array[g, 10^4] (* Michael De Vlieger, Mar 09 2017, Version 10 *)
  • Scheme
    (define (A280692 n) (- (A003961 n) (A250469 n)))

Formula

a(n) = A003961(n) - A250469(n).

A300247 Restricted growth sequence transform of A286457(n), filter combining A078898(n) and A246277(n).

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 5, 3, 6, 2, 7, 2, 8, 4, 9, 2, 10, 2, 11, 12, 13, 2, 14, 3, 15, 16, 17, 2, 18, 2, 19, 20, 21, 4, 22, 2, 23, 24, 25, 2, 26, 2, 27, 28, 29, 2, 30, 3, 31, 32, 33, 2, 34, 12, 35, 36, 37, 2, 38, 2, 39, 40, 41, 42, 43, 2, 44, 45, 46, 2, 47, 2, 48, 49, 50, 4, 51, 2, 52, 53, 54, 2, 55, 56, 57, 58, 59, 2, 60, 12, 61, 62, 63, 64, 65, 2, 66
Offset: 1

Views

Author

Antti Karttunen, Mar 03 2018

Keywords

Comments

For all i, j:
a(i) = a(j) => A280492(i) = A280492(j).
a(i) = a(j) => A300248(i) = A300248(j).
The latter follows because A046523(n) = A046523(2*A246277(n)).

Examples

			a(65) = a(119) (= 42) because A078898(65) = A078898(119) = 5 (both numbers occur in column 5 of A083221) and because A246277(65) = A246277(119) = 7 (both numbers occur in column 7 of A246278). Note that 65 = 5*13 = prime(3)*prime(6) and 119 = 7*17 = prime(4)*prime(7) = A003961(65). A246277(n) contains complete information about the (relative) differences between prime indices in the prime factorization of n.

Links

Crossrefs

Cf. A078898, A246277, A286457.
Cf. also A300248, A280492.

Programs

  • PARI
    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])); }
A020639(n) = { if(1==n,n,vecmin(factor(n)[, 1])); };
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A078898(n) = { if(n<=1,n, my(spf=A020639(n),k=1,m=n/spf); while(m>1,if(A020639(m)>=spf,k++); m--); (k)); }; \\ Antti Karttunen, Mar 03 2018
A246277(n) = { if(1==n, 0, while((n%2), n = A064989(n)); (n/2)); };
A286457(n) = if(1==n,0,(1/2)*(2 + ((A078898(n)+A246277(n))^2) - A078898(n) - 3*A246277(n)));
write_to_bfile(1,rgs_transform(vector(65537,n,A286457(n))),"b300247.txt");

A286457 Compound filter: a(n) = P(A078898(n), A246277(n)), with a(1) = 0, where P(n,k) is sequence A000027 used as a pairing function.

Original entry on oeis.org

0, 1, 1, 5, 1, 13, 1, 25, 5, 41, 1, 61, 1, 85, 13, 113, 1, 145, 1, 181, 32, 221, 1, 265, 5, 313, 33, 365, 1, 421, 1, 481, 72, 545, 13, 613, 1, 685, 143, 761, 1, 841, 1, 925, 86, 1013, 1, 1105, 5, 1201, 219, 1301, 1, 1405, 32, 1513, 335, 1625, 1, 1741, 1, 1861, 201, 1985, 60, 2113, 1, 2245, 447, 2381, 1, 2521, 1, 2665, 223, 2813, 13, 2965, 1, 3121, 224, 3281
Offset: 1

Views

Author

Antti Karttunen, May 14 2017

Keywords

Links

Crossrefs

Cf. A000027, A078898, A246277, A280492.

Programs

Formula

a(1) = 0 and for n > 1, a(n) = (1/2)*(2 + ((A078898(n)+A246277(n))^2) - A078898(n) - 3*A246277(n)).
Showing 1-3 of 3 results.

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