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.

A364570 a(n) = A252464(n) - A364569(n), where A364569(n) is the length of the common prefix in the binary expansions of A156552(n) and n-1 [= A156552(A005940(n))].

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 3, 0, 2, 0, 3, 0, 5, 4, 3, 0, 2, 3, 5, 0, 3, 4, 7, 0, 0, 6, 2, 5, 9, 4, 10, 0, 2, 3, 3, 4, 9, 6, 4, 0, 11, 4, 12, 5, 0, 8, 13, 0, 0, 1, 7, 7, 15, 3, 5, 6, 8, 10, 16, 5, 17, 11, 5, 0, 3, 3, 14, 4, 6, 4, 16, 5, 18, 10, 4, 7, 4, 5, 19, 0, 4, 12, 21, 5, 6, 13, 9, 6, 22, 1, 5, 9, 10, 14, 7, 0, 24, 1, 6
Offset: 1

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Author

Antti Karttunen, Aug 14 2023

Keywords

Links

Crossrefs

Cf. A005940, A156552, A252464, A364569, A364570, A364960 (positions of 0's).
Cf. also A347381.

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));
A364570(n) = (A252464(n)-A364569(n));

A364956 Numbers k such that A163511(k) is either k itself or its descendant in Doudna-tree, A005940 (or equally, in A163511-tree).

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 64, 96, 128, 192, 256, 384, 512, 768, 1024, 1536, 2048, 3072, 4096, 6144, 8192, 12288, 16384, 24576, 32768, 49152, 65536, 98304, 131072, 196608, 262144, 341887, 393216, 524288, 683774, 786432, 1048576, 1572864, 2097152, 2495625, 3145728, 4194304, 4991250, 6291456
Offset: 1

Views

Author

Antti Karttunen, Sep 02 2023

Keywords

Comments

Numbers k such that A252464(k) = A364954(k), where A364954(n) is the length of the common prefix in the binary expansions of A156552(n) and A156552(A163511(n)).

Examples

			For n = 341887, A156552(n) = 1736, "11011001000" in binary, and A163511(n) = 1830711541, with A156552(A163511(n)) = 444544, "1101100100010000000" in binary, and as the former binary expansion is a prefix of the latter, 341887 is included in this sequence. In this case, 1830711541 = A003961^7(2*341887), where A003961^7 indicates a prime shift by seven steps towards larger primes.
For n = 683774 = 2*341887, A156552(n) = 3473 = "110110010001", and A163511(n) = 3661423082 = 2*1830711541, with A156552(A163511(n)) = 889089, "11011001000100000001", and as the former binary expansion is a prefix of the latter, 683774 is included in this sequence.
For n = 1367548 = 4*341887, A156552(n) = 6947, "1101100100011" in binary, and A163511(n) = 7322846164 = 2*3661423082 with A156552(A163511(n)) = 1778179, "110110010001000000011" in binary, as the former binary expansion is NOT a prefix of the latter, 1367548 is NOT included in this sequence.

Crossrefs

Positions of 0's in A364955.
Cf. A005940, A156552, A163511.
Cf. A029744 (subsequence).
Cf. also A364960.
Cf. A252464, A364954.

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));
Showing 1-3 of 3 results.

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