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.

A364297 a(n) = A348717(A163511(n)).

Original entry on oeis.org

1, 2, 4, 2, 8, 4, 6, 2, 16, 8, 18, 4, 12, 6, 10, 2, 32, 16, 54, 8, 36, 18, 50, 4, 24, 12, 30, 6, 20, 10, 14, 2, 64, 32, 162, 16, 108, 54, 250, 8, 72, 36, 150, 18, 100, 50, 98, 4, 48, 24, 90, 12, 60, 30, 70, 6, 40, 20, 42, 10, 28, 14, 22, 2, 128, 64, 486, 32, 324, 162, 1250, 16, 216, 108, 750, 54, 500, 250, 686, 8, 144
Offset: 0

Views

Author

Antti Karttunen, Aug 15 2023

Keywords

Comments

For all i, j: a(i) = a(j) => A278531(i) = A278531(j).
As the underlying sequence A163511 can be represented as a binary tree, so can this be also:
1
|
...................2...................
4 2
8......../ \........4 6......../ \........2
/ \ / \ / \ / \
/ \ / \ / \ / \
/ \ / \ / \ / \
16 8 18 4 12 6 10 2
32 16 54 8 36 18 50 4 24 12 30 6 20 10 14 2
etc.
Each rightward leaning branch stays constant, because a(2n+1) = a(n).
Conjecture: Mersenne primes (A000668) gives all such odd numbers k for which a(k) = A348717(k). If true, then it immediately implies that map n -> A163511(n) [or equally: map n -> A243071(n)] has no other fixed points than those given by A007283. But see also A364959. - Edited Sep 03 2023

Links

Crossrefs

Cf. A000265, A000668, A007283, A163511, A243071, A348717, A364949, A364956, A365423.
Cf. also A245611, A245612, A278531, A286531, A364959, A364963.

Programs

  • PARI
    A163511(n) = if(!n,1,my(p=2, t=1); while(n>1, if(!(n%2), (t*=p), p=nextprime(1+p)); n >>= 1); (t*p));
A348717(n) = if(1==n, 1, my(f = factor(n), k = primepi(f[1, 1])-1); for (i=1, #f~, f[i, 1] = prime(primepi(f[i, 1])-k)); factorback(f));
A364297(n) = A348717(A163511(n));

Formula

a(0) = 1, a(1) = 2, a(2n) = A163511(2n) = 2*A163511(n), and for n > 0, a(2n+1) = a(n).

A364961 Odd numbers k such that A005940(k) is either k itself or its descendant in Doudna-tree, A005940.

Original entry on oeis.org

1, 3, 5, 25, 45, 49, 40131, 50575, 79625, 1486485, 1872507, 3403125
Offset: 1

Views

Author

Antti Karttunen, Aug 14 2023

Keywords

Comments

Odd numbers k such that A252464(k) is equal to A364569(k).
Apparently, A364960 without the even terms of A029747.
Note that 1, 25, 45, 49 are so far the only known integers x for which A005940(x) = A003961(x).

Examples

			Term (and its factorization)            A005940(term) (and its factorization)
      1                              -> 1
      3                              -> 3
      5                              -> 5
     25 = 5^2                        -> 49        = 7^2
     45 = 3^2 * 5                    -> 175       = 5^2 * 7
     49 = 7^2                        -> 121       = 11^2
  40131 = 3^2 * 7^3 * 13             -> 100847877 = 3 * 13^2 * 19^3 * 29
  50575 = 5^2 * 7 * 17^2             -> 22467159  = 3^3 * 11^2 * 13 * 23^2
  79625 = 5^3 * 7^2 * 13             -> 787365187 = 7 * 19^3 * 23^2 * 31
1486485 = 3^3 * 5 * 7 * 11^2 * 13    -> 25468143451205
                                        = 5 * 7 * 13 * 17^3 * 19 * 23 * 29^2 * 31
1872507 = 3 * 7 * 13 * 19^3          -> 240245795625
                                        = 3 * 5^4 * 11 * 17 * 23 * 31^3,
3403125 = 3^2 * 5^5 * 11^2           -> 2394659631669305
                                        = 5 * 7^3 * 11 * 13^2 * 17^5 * 23^2.
See also examples in A364959.

Crossrefs

Odd terms in A364960.
Cf. A003961, A005940, A029747, A156552, A252464, A364569, A364570.
Cf. also A364956 (odd terms there), A364959, A364962.

Programs

  • PARI
    Abincompreflen(n, m) = { my(x=binary(n), y=binary(m), u=min(#x, #y)); for(i=1, u, if(x[i]!=y[i], return(i-1))); (u); };
A156552(n) = {my(f = factor(n), p, p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res}; \\ From A156552
A364569(n) = Abincompreflen(A156552(n), (n-1));
A061395(n) = if(n>1, primepi(vecmax(factor(n)[, 1])), 0);
A252464(n) = if(1==n,0,(bigomega(n) + A061395(n) - 1));
isA364961(n) = ((n%2)&&(A252464(n)==A364569(n)));
  • PARI
    A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); (t); };
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)};
A252463(n) = if(!(n%2),n/2,A064989(n));
isA364961(n) = if(!(n%2),0,my(k=A005940(n)); while(k>n, k = A252463(k)); (k==n));

A364962 Odd numbers k such that A005941(k) is either k itself or its descendant in Doudna-tree, A005940.

Original entry on oeis.org

1, 3, 5, 11, 19, 23, 31, 37, 41, 43, 47, 53, 59, 61, 71, 73, 79, 83, 85, 89, 97, 101, 103, 107, 109, 113, 127
Offset: 1

Views

Author

Antti Karttunen, Aug 14 2023

Keywords

Comments

Questions: Is 85 the only composite in this sequence? (See also A364565). Are there any more terms after 127, or is the sequence finite?

Examples

			85 = 5*17 is a term, because A005941(85) = 133 = 7*19 = A003961(85), thus 133 is a left hand side child of 85 in the tree depicted in A005940, and therefore 85 is included in this sequence. (See also the last example in A364959).

Crossrefs

Cf. A005940, A005941, A029747, A156552, A252464.
Cf. also A364959, A364961.

Programs

  • PARI
    A005941(n) = { my(f=factor(n), p, p2=1, res=0); for(i=1, #f~, p = 1 << (primepi(f[i, 1])-1); res += (p * p2 * (2^(f[i, 2])-1)); p2 <<= f[i, 2]); (1+res) }; \\ (After David A. Corneth's program for A156552)
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)};
A252463(n) = if(!(n%2),n/2,A064989(n));
isA364962(n) = if(!(n%2),0,my(k=A005941(n)); while(k>n, k = A252463(k)); (k==n));
Showing 1-3 of 3 results.

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