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 1-7 of 7 results.

A048103 Numbers not divisible by p^p for any prime p.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 18, 19, 21, 22, 23, 25, 26, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 45, 46, 47, 49, 50, 51, 53, 55, 57, 58, 59, 61, 62, 63, 65, 66, 67, 69, 70, 71, 73, 74, 75, 77, 78, 79, 82, 83, 85, 86, 87, 89, 90, 91, 93, 94, 95, 97, 98
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

If a(n) = Product p_i^e_i then p_i > e_i for all i.
Complement of A100716; A129251(a(n)) = 0. - Reinhard Zumkeller, Apr 07 2007
Density is 0.72199023441955... = Product_{p>=2} (1 - p^-p) where p runs over the primes. - Charles R Greathouse IV, Jan 25 2012
A027748(a(n),k) <= A124010(a(n),k), 1<=k<=A001221(a(n)). - Reinhard Zumkeller, Apr 28 2012
Range of A276086. Also numbers not divisible by m^m for any natural number m > 1. - Antti Karttunen, Nov 18 2024

Examples

			6 = 2^1 * 3^1 is OK but 12 = 2^2 * 3^1 is not.
625 = 5^4 is present because it is not divisible by 5^5.

Links

Crossrefs

Complement: A100716.
Positions of 0's in A129251, A342023, A376418, positions of 1's in A327936, A342007, A359550 (characteristic function).
Cf. A048102, A048104, A051674 (p^p), A054743, A054744, A377982 (a left inverse, partial sums of char. fun, see also A328402).
Cf. A276086 (permutation of this sequence, see also A376411, A376413).
Subsequences: A002110, A005117, A006862, A024451 (after its initial 0), A057588, A099308 (after its initial 0), A276092, A328387, A328832, A359547, A370114, A371083, A373848, A377871, A377992.
Disjoint union of {1}, A327934 and A358215.
Also A276078 is a subsequence, from which this differs for the first time at n=451 where a(451)=625, while that value is missing from A276078.

Programs

  • Haskell
    a048103 n = a048103_list !! (n-1)
a048103_list = filter (\x -> and $
   zipWith (>) (a027748_row x) (map toInteger $ a124010_row x)) [1..]
-- Reinhard Zumkeller, Apr 28 2012
  • Mathematica
    {1}~Join~Select[Range@ 120, Times @@ Boole@ Map[First@ # > Last@ # &, FactorInteger@ #] > 0 &] (* Michael De Vlieger, Aug 19 2016 *)
  • PARI
    isok(n) = my(f=factor(n)); for (i=1, #f~, if (f[i,1] <= f[i,2], return(0))); return(1); \\ Michel Marcus, Nov 13 2020
  • PARI
    A359550(n) = { my(pp); forprime(p=2, , pp = p^p; if(!(n%pp), return(0)); if(pp > n, return(1))); }; \\ (A359550 is the characteristic function for A048103) - Antti Karttunen, Nov 18 2024
  • Python
    from itertools import count, islice
from sympy import factorint
def A048103_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n:all(map(lambda d:d[1]A048103_list = list(islice(A048103_gen(),30)) # Chai Wah Wu, Jan 05 2023
  • Scheme
    ;; With Antti Karttunen's IntSeq-library.
(define A048103 (ZERO-POS 1 1 A129251))
;; Antti Karttunen, Aug 18 2016

Formula

a(n) ~ kn with k = 1/Product_{p>=2}(1 - p^-p) = Product_{p>=2}(1 + 1/(p^p - 1)) = 1.3850602852..., where the product is over all primes p. - Charles R Greathouse IV, Jan 25 2012
For n >= 1, A377982(a(n)) = n. - Antti Karttunen, Nov 18 2024

Extensions

More terms from James Sellers, Apr 22 2000

A328386 a(n) = A276086(n) mod n.

Original entry on oeis.org

0, 1, 0, 1, 3, 5, 3, 7, 3, 5, 2, 1, 11, 5, 0, 1, 8, 17, 3, 15, 15, 3, 19, 1, 0, 3, 24, 25, 27, 7, 14, 21, 9, 29, 21, 35, 33, 29, 15, 35, 15, 7, 6, 41, 15, 11, 1, 11, 35, 25, 48, 23, 9, 1, 5, 21, 30, 51, 44, 49, 37, 23, 42, 57, 37, 47, 21, 55, 21, 35, 8, 1, 41, 49, 0, 5, 28, 41, 5, 55, 57, 21, 26, 49, 50, 27, 6, 9, 73, 73, 49
Offset: 1

Views

Author

Antti Karttunen, Oct 15 2019

Keywords

Links

Crossrefs

Cf. A276086, A324198, A328382, A328387 (positions of zeros).

Programs

  • Mathematica
    Block[{b = MixedRadix[Reverse@ Prime@ Range@ 12]}, Array[Mod[Apply[Times, Power @@@ # &@ Transpose@ {Prime@ Range@ Length@ #, Reverse@ #}] &@ IntegerDigits[#, b], #] &, 91]] (* Michael De Vlieger, Oct 15 2019 *)
  • PARI
    A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A328386(n) = (A276086(n)%n);

Formula

a(n) = A276086(n) mod n.

A358221 Numbers k such that A003415(k) divides A276086(k), where A003415 is the arithmetic derivative, and A276086 is the primorial base exp-function.

Original entry on oeis.org

2, 3, 5, 6, 7, 9, 11, 13, 17, 19, 21, 23, 25, 26, 29, 31, 33, 37, 38, 41, 43, 46, 47, 49, 53, 59, 61, 65, 67, 71, 73, 77, 79, 83, 89, 94, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 141, 146, 149, 151, 157, 161, 163, 167, 173, 179, 181, 185, 191, 193, 197, 199, 201, 206, 207, 209, 211, 221, 223, 227, 229, 233
Offset: 1

Views

Author

Antti Karttunen, Nov 23 2022

Keywords

Links

Crossrefs

Union of A000040 and A358222.
Subsequence of A358215 (thus also of A048103) and of A358229.
Positions of zeros in A328382.
Cf. A003415, A276086, A358220 (characteristic function).
Cf. also A328387, A356311, A356312.

Programs

  • PARI
    A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A358220(n) = if(n<2,0,!(A276086(n)%A003415(n)));
isA358221(n) = A358220(n);

A351250 Numerator of n / A276086(n).

Original entry on oeis.org

0, 1, 2, 1, 4, 5, 6, 7, 8, 3, 2, 11, 12, 13, 14, 1, 16, 17, 18, 19, 4, 7, 22, 23, 24, 1, 26, 9, 28, 29, 30, 31, 32, 11, 34, 5, 36, 37, 38, 13, 8, 41, 6, 43, 44, 3, 46, 47, 48, 7, 2, 17, 52, 53, 54, 11, 8, 19, 58, 59, 60, 61, 62, 3, 64, 65, 66, 67, 68, 23, 2, 71, 72, 73, 74, 1, 76, 11, 78, 79, 16, 27, 82, 83, 12, 17
Offset: 0

Views

Author

Antti Karttunen, Feb 05 2022

Keywords

Links

Crossrefs

Cf. A276086, A324198, A328386, A328387 (positions of ones), A351251 (denominators).
Cf. A324583 (the positions of fixed points after the zero).
Cf. also A351230.

Programs

  • Mathematica
    Array[Block[{i, m, n = #, p}, m = i = 1; While[n > 0, p = Prime[i]; m *= p^Mod[n, p]; n = Quotient[n, p]; i++]; Numerator[#/m]] &, 86, 0] (* Michael De Vlieger, Feb 06 2022 *)
  • PARI
    A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A351250(n) = numerator(n/A276086(n));

Formula

a(n) = n / A324198(n) = n / gcd(n, A276086(n)).
a(n) = n / gcd(n, A328386(n)).

A358226 Numbers k such that A276086(k) mod k is an even number, where A276086 is the primorial base exp-function.

Original entry on oeis.org

1, 3, 11, 15, 17, 25, 27, 31, 43, 51, 57, 59, 63, 71, 75, 77, 83, 85, 87, 93, 95, 103, 105, 107, 109, 115, 119, 129, 133, 137, 139, 145, 147, 151, 157, 167, 169, 171, 173, 175, 177, 185, 189, 191, 199, 201, 207, 211, 213, 215, 217, 221, 223, 229, 233, 235, 237, 241, 257, 259, 263, 269, 281, 289, 299, 303, 305, 307
Offset: 1

Views

Author

Antti Karttunen, Nov 25 2022

Keywords

Comments

All terms are odd.

Links

Crossrefs

Cf. A276086, A328386, A328387 (subsequence), A358225 (complement).
Positions of zeros in A358224.
Cf. also A358229.

Programs

Formula

{k | A328386(k) == 0 (mod 2)}.

A383299 Numbers k such that A276086(k) is a multiple of A276086(A003415(k)), where A003415 is the arithmetic derivative, and A276086 is the primorial base exp-function.

Original entry on oeis.org

0, 1, 3, 4, 5, 7, 9, 11, 13, 15, 17, 19, 23, 27, 29, 31, 37, 41, 43, 45, 47, 51, 53, 59, 61, 67, 71, 73, 79, 83, 87, 89, 97, 101, 103, 107, 109, 113, 117, 119, 127, 131, 137, 139, 141, 147, 149, 151, 157, 161, 163, 165, 167, 171, 173, 177, 179, 181, 191, 193, 197, 199, 203, 207, 209, 211, 223, 227, 229, 233, 239
Offset: 1

Views

Author

Antti Karttunen, May 15 2025

Keywords

Comments

The sequence contains the intersection of A048103, A369650, and A328387. That is, {1, 15, 5005}, at least.

Examples

			5 is a term as A003415(5) = 1, and A276086(5) = 18 is a multiple of A276086(1) = 2, and ditto for all odd primes.
9 is a term as A003415(9) = 6, and A276086(9) = 30 is a multiple of A276086(6) = 5.
15 is a term as A003415(15) = 8, and A276086(15) = 150 is a multiple of A276086(8) = 15.
5005 is a term as A003415(5005) = 2556, and A276086(5005) = 39055266250 = 7803250 * A276086(2556) = 7803250 * 5005. See also A369650.
See also examples in A383300.

Links

Crossrefs

Cf. A003415, A048103, A276086, A327859, A328387, A383298 (characteristic function).
Cf. A006005, A051674, A383300, A383301 (subsequences).
Cf. also A369650.

Programs

A358231 Numbers k for which A276086(k) == 1 (mod k), where A276086 is the primorial base exp-function.

Original entry on oeis.org

2, 4, 12, 16, 24, 47, 54, 72, 120, 142, 144, 432, 540, 864, 972, 1049, 1260, 1916, 2628, 10152, 12798, 19024, 20304, 100565, 152668, 209760, 445362, 2071560, 2759034, 3344269, 85167240, 92667148, 111135679, 118344316, 162716506, 264678868, 599478496
Offset: 1

Views

Author

Antti Karttunen, Nov 24 2022

Keywords

Links

Crossrefs

Positions of 1's in A328386.
Cf. A276086, A328387.

Programs

  • PARI
    A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
isA358231(n) = (1==(A276086(n)%n));
Showing 1-7 of 7 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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