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.

A250469 a(1) = 1; and for n > 1, a(n) = A078898(n)-th number k for which A055396(k) = A055396(n)+1, where A055396(n) is the index of smallest prime dividing n.

Original entry on oeis.org

1, 3, 5, 9, 7, 15, 11, 21, 25, 27, 13, 33, 17, 39, 35, 45, 19, 51, 23, 57, 55, 63, 29, 69, 49, 75, 65, 81, 31, 87, 37, 93, 85, 99, 77, 105, 41, 111, 95, 117, 43, 123, 47, 129, 115, 135, 53, 141, 121, 147, 125, 153, 59, 159, 91, 165, 145, 171, 61, 177, 67, 183, 155, 189, 119, 195, 71, 201, 175, 207, 73, 213, 79, 219, 185, 225, 143, 231, 83, 237, 205, 243, 89, 249, 133, 255
Offset: 1

Views

Author

Antti Karttunen, Dec 06 2014

Keywords

Comments

Permutation of odd numbers.
For n >= 2, a(n) = A078898(n)-th number k for which A055396(k) = A055396(n)+1. In other words, a(n) tells which number is located immediately below n in the sieve of Eratosthenes (see A083140, A083221) in the same column of the sieve that contains n.
A250471(n) = (a(n)+1)/2 is a permutation of natural numbers.
Coincides with A003961 in all terms which are primes. - M. F. Hasler, Sep 17 2016. Note: primes are a proper subset of A280693 which gives all n such that a(n) = A003961(n). - Antti Karttunen, Mar 08 2017

Links

Crossrefs

Cf. A000040, A003961, A016945, A046523, A055396, A078898, A083140, A083221, A084967, A249744, A249810, A249820, A249817, A249818, A250471, A266645, A280692, A280693, A283465.
Cf. A250470, A268674 (left inverses, the latter is simpler).

Programs

  • Mathematica
    a[1] = 1; a[n_] := If[PrimeQ[n], NextPrime[n], m1 = p1 = FactorInteger[n][[ 1, 1]]; For[k1 = 1, m1 <= n, m1 += p1; If[m1 == n, Break[]]; If[ FactorInteger[m1][[1, 1]] == p1, k1++]]; m2 = p2 = NextPrime[p1]; For[k2 = 1, True, m2 += p2, If[FactorInteger[m2][[1, 1]] == p2, k2++]; If[k1+2 == k2, Return[m2]]]]; Array[a, 100] (* Jean-François Alcover, Mar 08 2016 *)
g[n_] := If[n == 1, 0, PrimePi@ FactorInteger[n][[1, 1]]]; Function[s, MapIndexed[Lookup[s, g[First@ #2] + 1][[#1]] - Boole[First@ #2 == 1] &, #] &@ Map[Position[Lookup[s, g@#], #][[1, 1]] &, Range@ 120]]@ PositionIndex@ Array[g, 10^4] (* Michael De Vlieger, Mar 08 2017, Version 10 *)

Formula

a(1) = 1, a(n) = A083221(A055396(n)+1, A078898(n)).
a(n) = A249817(A003961(A249818(n))).
Other identities. For all n >= 1:
A250470(a(n)) = A268674(a(n)) = n. [A250470 and A268674 provide left inverses for this function.]
a(2n) = A016945(n-1). [Maps even numbers to the numbers of form 6n+3, in monotone order.]
a(A016945(n-1)) = A084967(n). [Which themselves are mapped to the terms of A084967, etc. Cf. the Example section of A083140.]
a(A000040(n)) = A000040(n+1). [Each prime is mapped to the next prime.]
For all n >= 2, A055396(a(n)) = A055396(n)+1. [A more general rule.]
A046523(a(n)) = A283465(n). - Antti Karttunen, Mar 08 2017

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).

A280703 a(n) = A003961(n) / A280702(n) = A003961(n) / gcd(A003961(n),A250469(n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 9, 1, 7, 1, 15, 1, 11, 1, 9, 1, 25, 1, 21, 1, 13, 1, 45, 1, 17, 25, 11, 1, 35, 1, 81, 13, 19, 1, 15, 1, 23, 17, 21, 1, 55, 1, 39, 35, 29, 1, 135, 1, 1, 19, 1, 1, 125, 1, 9, 23, 31, 1, 105, 1, 37, 55, 27, 1, 1, 1, 57, 29, 77, 1, 225, 1, 41, 49, 23, 1, 85, 1, 189, 125, 43, 1, 165, 1, 47, 31, 39, 1, 175, 1, 87, 37
Offset: 1

Views

Author

Antti Karttunen, Mar 08 2017

Keywords

Comments

If there are no such n that A250469(n) = k*A003961(n) for some integer k > 1, then A280693 gives the positions of ones in this sequence. Cf. also comment in A280704.

Links

Crossrefs

Cf. A003961, A250469, A280702, A280704, A280693.

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[p, p/GCD[Lookup[s, g@ First@ #2 + 1][[#1]] - Boole[First@ #2 == 1], p]]@ f@ First@ #2 &, #] &@ Map[Position[Lookup[s, g@ #], #][[1, 1]] &, Range@ 120]]@ PositionIndex@ Array[g, 10^4] (* Michael De Vlieger, Mar 08 2017, Version 10 *)
  • Scheme
    (define (A280703 n) (/ (A003961 n) (A280702 n)))

Formula

a(n) = A003961(n) / A280702(n) = A003961(n) / gcd(A003961(n),A250469(n)).
Showing 1-3 of 3 results.

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