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.

A355442 a(n) = gcd(A003961(n), A276086(n)), where A003961 is fully multiplicative with a(p) = nextprime(p), and A276086 is primorial base exp-function.

Original entry on oeis.org

1, 3, 1, 9, 1, 5, 1, 3, 5, 3, 1, 5, 1, 3, 5, 9, 1, 25, 1, 3, 5, 3, 1, 5, 1, 3, 125, 9, 1, 7, 1, 3, 1, 3, 7, 5, 1, 3, 5, 63, 1, 5, 1, 3, 175, 3, 1, 5, 1, 21, 5, 9, 1, 125, 7, 3, 5, 3, 1, 7, 1, 3, 1, 9, 7, 5, 1, 3, 5, 21, 1, 25, 1, 3, 245, 9, 1, 5, 1, 21, 125, 3, 1, 5, 7, 3, 5, 9, 1, 7, 1, 3, 1, 3, 7, 5, 1, 3, 5, 441
Offset: 1

Views

Author

Antti Karttunen, Jul 13 2022

Keywords

Links

Crossrefs

Cf. A003961, A020639, A276086, A355001 [smallest prime factor of a(n)], A355456 [= gcd(sigma(n), a(n))], A355692 (Dirichlet inverse), A355820, A355821 (positions of 1's).
Cf. also A322361, A324198, A351459.

Programs

  • PARI
    A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A355442(n) = gcd(A003961(n), A276086(n));

Formula

a(n) = gcd(A003961(n), A276086(n)).

A355821 Numbers k for which A003961(k) and A276086(k) are relatively prime, where A003961 is fully multiplicative with a(p) = nextprime(p), and A276086 is primorial base exp-function.

Original entry on oeis.org

1, 3, 5, 7, 11, 13, 17, 19, 23, 25, 29, 31, 33, 37, 41, 43, 47, 49, 53, 59, 61, 63, 67, 71, 73, 77, 79, 83, 89, 91, 93, 97, 101, 103, 107, 109, 113, 119, 121, 123, 127, 131, 133, 137, 139, 143, 149, 151, 153, 157, 161, 163, 167, 169, 173, 179, 181, 183, 187, 191, 193, 197, 199, 203, 209, 211, 213, 215, 221, 223, 227
Offset: 1

Views

Author

Antti Karttunen, Jul 18 2022

Keywords

Links

Crossrefs

Positions of 1's in A355442 and in A355001.
Cf. A003961, A276086, A355820 (characteristic function), A355822 (complement).
Cf. also A324583.

Programs

  • PARI
    A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A355820(n) = (1==gcd(A003961(n), A276086(n)));
isA355821(n) = A355820(n);
  • Python
    from math import prod, gcd
from itertools import count, islice
from sympy import factorint, nextprime
def A355821_gen(startvalue=1): # generator of terms >= startvalue
    for n in count(max(startvalue,1)):
        k = prod(nextprime(p)**e for p, e in factorint(n).items())
        m, p, c = 1, 2, n
        while c:
            c, a = divmod(c,p)
            m *= p**a
            p = nextprime(p)
        if gcd(k,m) == 1:
            yield n
A355821_list = list(islice(A355821_gen(),30)) # Chai Wah Wu, Jul 18 2022

A284723 Smallest odd prime that is relatively prime to n.

Original entry on oeis.org

3, 3, 5, 3, 3, 5, 3, 3, 5, 3, 3, 5, 3, 3, 7, 3, 3, 5, 3, 3, 5, 3, 3, 5, 3, 3, 5, 3, 3, 7, 3, 3, 5, 3, 3, 5, 3, 3, 5, 3, 3, 5, 3, 3, 7, 3, 3, 5, 3, 3, 5, 3, 3, 5, 3, 3, 5, 3, 3, 7, 3, 3, 5, 3, 3, 5, 3, 3, 5, 3, 3, 5, 3, 3, 7, 3, 3, 5, 3, 3, 5, 3, 3, 5, 3, 3, 5, 3, 3, 7, 3, 3, 5, 3, 3, 5, 3, 3, 5, 3, 3
Offset: 1

Views

Author

N. J. A. Sloane, Apr 04 2017

Keywords

Comments

More than the usual number of terms are shown in order to distinguish this from A239278.
a(n) = smallest odd prime missing from rad(n).
If rad(m) = rad(n), a(m) = a(n) (cf. A007947). - Bob Selcoe, Apr 04 2017

Links

Crossrefs

Similar to but different from A239278.
Cf. A007947, A020639, A053669, A284721.
Even bisection of A355001.

Programs

  • Mathematica
    a[n_] := Module[{p = 3}, While[Divisible[n, p], p = NextPrime[p]]; p]; Array[a, 100] (* Amiram Eldar, Jul 25 2022 *)
  • PARI
    a(n) = my(p=3); while(gcd(n, p) != 1, p=nextprime(p+1)); p; \\ Michel Marcus, Apr 04 2017; corrected Jun 13 2022

Formula

a(n) = 3 unless n == 0 (mod 3).
For n>3, a(n) < 3*log(n).
a(n) = A355001(2*n). - Antti Karttunen, Jul 18 2022
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{p odd prime} ((p*(p-1)/Product_{q odd prime <= p} q)) = 3.8401019546... . - Amiram Eldar, Jul 25 2022

A355822 Numbers k such that A003961(k) and A276086(k) share a prime factor, where A003961 is fully multiplicative with a(p) = nextprime(p), and A276086 is primorial base exp-function.

Original entry on oeis.org

2, 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 26, 27, 28, 30, 32, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 64, 65, 66, 68, 69, 70, 72, 74, 75, 76, 78, 80, 81, 82, 84, 85, 86, 87, 88, 90, 92, 94, 95, 96, 98, 99, 100, 102, 104, 105, 106, 108, 110, 111, 112, 114, 115
Offset: 1

Views

Author

Antti Karttunen, Jul 18 2022

Keywords

Links

Crossrefs

Positions of terms > 1 in A355442 and in A355001.
Cf. A003961, A276086, A355002 (subsequence), A355820 (positions of zeros), A355821 (complement), A355835.
Cf. A005843 (even numbers, apart from 0, is a subsequence).
Cf. also A324584.

Programs

  • PARI
    A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A355820(n) = (1==gcd(A003961(n), A276086(n)));
isA355822(n) = !A355820(n);
  • Python
    from math import prod, gcd
from itertools import count, islice
from sympy import nextprime, factorint
def A355822_gen(startvalue=1): # generator of terms >= startvalue
    for n in count(max(startvalue,1)):
        k = prod(nextprime(p)**e for p, e in factorint(n).items())
        m, p, c = 1, 2, n
        while c:
            c, a = divmod(c,p)
            m *= p**a
            p = nextprime(p)
        if gcd(k,m) > 1:
            yield n
A355822_list = list(islice(A355822_gen(),30)) # Chai Wah Wu, Jul 18 2022
Showing 1-4 of 4 results.

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

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