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 1-4 of 4 results.

A328242 Numbers k such that A003415(A276086(k)) is a squarefree number, where A003415 is the arithmetic derivative, and A276086 is the primorial base exp-function.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 14, 15, 17, 30, 32, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 46, 47, 60, 61, 62, 63, 65, 66, 67, 68, 69, 71, 73, 74, 75, 76, 77, 210, 211, 212, 213, 215, 217, 218, 219, 220, 221, 223, 225, 226, 227, 241, 242, 243, 245, 246, 247, 248, 249, 250, 251, 254, 255, 256, 257, 270, 273, 274, 275, 276, 277
Offset: 1

Views

Author

Antti Karttunen, Oct 10 2019

Keywords

Comments

Numbers n such that A276086(n) is either in A328234 or in A000040 (i.e., it is a prime, in which case n itself is a primorial, A002110).

Links

Crossrefs

Positions of 1's in A370130. Subsequence of A370132.
Cf. A000040, A002110, A003415, A005117, A008966, A276086, A327860, A328234.
Cf. A328241 (complement).
Cf. A328233, A328240 (subsequences).

Programs

  • PARI
    A327860(n) = { my(s=0, m=1, p=2, e); while(n, e = (n%p); m *= (p^e); s += (e/p); n = n\p; p = nextprime(1+p)); (s*m); };
isA328242(n) = issquarefree(A327860(n));

A370121 Triangle read by rows: T(n,k) = A002110(n) + A002110(k), 0 <= k <= n; sums of two primorials, not necessarily distinct.

Original entry on oeis.org

2, 3, 4, 7, 8, 12, 31, 32, 36, 60, 211, 212, 216, 240, 420, 2311, 2312, 2316, 2340, 2520, 4620, 30031, 30032, 30036, 30060, 30240, 32340, 60060, 510511, 510512, 510516, 510540, 510720, 512820, 540540, 1021020, 9699691, 9699692, 9699696, 9699720, 9699900, 9702000, 9729720, 10210200, 19399380, 223092871, 223092872
Offset: 0

Views

Author

Antti Karttunen, Feb 29 2024

Keywords

Comments

After the initial 2, numbers with either one 2 or two 1's in their primorial base representation (A049345), with all the other digits zeros.

Examples

			Triangle begins as:
        2;
        3,       4;
        7,       8,      12;
       31,      32,      36,      60;
      211,     212,     216,     240,     420;
     2311,    2312,    2316,    2340,    2520,    4620;
    30031,   30032,   30036,   30060,   30240,   32340,   60060;
   510511,  510512,  510516,  510540,  510720,  512820,  540540,  1021020;
  9699691, 9699692, 9699696, 9699720, 9699900, 9702000, 9729720, 10210200, 19399380;

Links

Crossrefs

Cf. A002110, A049345, A087112, A276086, A276150, A370129 (arithmetic derivative applied to this triangle).
Cf. A006862 (left edge), A088860 (right edge).
Cf. A177689 (same triangle without the right edge), A370134 (without the leftmost column).
Subsequence of A370132.
Cf. also A173786.

Programs

  • PARI
    A002110(n) = prod(i=1,n,prime(i));
A370121(n) = { my(c = (sqrtint(8*n + 1) - 1) \ 2); (A002110(c) + A002110(n - binomial(c + 1, 2))); };

Formula

For n >= 1, A276150(a(n)) = 2.
For n >= 1, A276086(a(n)) = A087112(1+n).

A370134 Triangle read by rows: T(n,k) = A002110(n) + A002110(k), 1 <= k <= n; sums of two primorials > 1, not necessarily distinct.

Original entry on oeis.org

4, 8, 12, 32, 36, 60, 212, 216, 240, 420, 2312, 2316, 2340, 2520, 4620, 30032, 30036, 30060, 30240, 32340, 60060, 510512, 510516, 510540, 510720, 512820, 540540, 1021020, 9699692, 9699696, 9699720, 9699900, 9702000, 9729720, 10210200, 19399380, 223092872, 223092876, 223092900, 223093080, 223095180, 223122900, 223603380
Offset: 1

Views

Author

Antti Karttunen, Mar 07 2024

Keywords

Examples

			Triangle begins as:
        4;
        8,      12;
       32,      36,      60;
      212,     216,     240,     420;
     2312,    2316,    2340,    2520,    4620;
    30032,   30036,   30060,   30240,   32340,   60060;
   510512,  510516,  510540,  510720,  512820,  540540, 1021020;
  9699692, 9699696, 9699720, 9699900, 9702000, 9729720, 10210200, 19399380;

Links

Crossrefs

A370121 without its leftmost column. Subsequence of A370132.
Cf. A002110, A049345, A087112, A276150, A370135.
Cf. A088860 (right edge).

Programs

  • Mathematica
    nn = 20; MapIndexed[Set[P[First[#2] - 1], #1] &, FoldList[Times, 1, Prime@ Range[nn + 1]]]; Table[(P[n] + P[k]), {n, nn}, {k, n}] (* Michael De Vlieger, Mar 08 2024 *)
  • PARI
    A002110(n) = prod(i=1,n,prime(i));
A370134(n) = { n--; my(c = (sqrtint(8*n + 1) - 1) \ 2); (A002110(1+c) + A002110(1+n - binomial(c + 1, 2))); };

Formula

For n >= 1, A276150(a(n)) = 2.

A370133 Numbers with no digit larger than 3 in primorial base, A049345.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99
Offset: 1

Views

Author

Antti Karttunen, Feb 11 2024

Keywords

Comments

Numbers k for which A328114(k) <= 3.
Numbers k such that A276086(k) is biquadratefree, A046100.

Links

Crossrefs

Cf. A046100, A049345, A276086, A328114.
Cf. A369639 (nonsquarefree numbers whose arithmetic derivative is in this sequence).
Cf. A370132, A276156 (subsequences).
Subsequence of A351576: a(n) differs from A351576(n-1) for the first time at n=97, where a(97) = 210, while A351576(96) = 120, a term not present here.

Programs

  • Mathematica
    q[n_] := Module[{k = n, p = 2, s = {}, r}, While[{k, r} = QuotientRemainder[k, p]; k != 0 || r != 0, AppendTo[s, r]; p = NextPrime[p]]; Count[s, ?(# > 3 &)] == 0]; Select[Range[0, 100], q] (* _Amiram Eldar, Mar 06 2024 *)
  • PARI
    ismaxprimobasedigit_at_most(n,k) = { my(s=0, p=2); while(n, if((n%p)>k, return(0)); n = n\p; p = nextprime(1+p)); (1); };
isA370133(n) = ismaxprimobasedigit_at_most(n,3);
Showing 1-4 of 4 results.

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