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.

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

A355833 Lexicographically earliest infinite sequence such that a(i) = a(j) => A342671(i) = A342671(j) and A348717(i) = A348717(j) for all i, j >= 1.

Original entry on oeis.org

1, 2, 3, 4, 3, 5, 3, 6, 4, 7, 3, 8, 3, 9, 10, 11, 3, 12, 3, 13, 14, 15, 3, 16, 4, 17, 18, 19, 3, 20, 3, 21, 22, 23, 10, 24, 3, 25, 26, 27, 3, 28, 3, 29, 8, 30, 3, 31, 4, 32, 33, 34, 3, 35, 14, 36, 37, 38, 3, 39, 3, 40, 41, 42, 43, 44, 3, 45, 46, 47, 3, 48, 3, 49, 50, 51, 10, 52, 3, 53, 11, 54, 3, 55, 26, 56, 57, 58, 3, 59, 14, 60, 61, 62, 33, 63, 3, 64, 65, 66
Offset: 1

Views

Author

Antti Karttunen, Jul 20 2022

Keywords

Comments

Restricted growth sequence transform of the ordered pair [A342671(n), A348717(n)].
Terms that occur in positions given by A349166 may occur only a finite number of times in this sequence. See also the array A355924.

Examples

			a(100) = a(3025) [= 66 as allotted by the rgs-transform] because 3025 = A003961(A003961(100)), therefore it is in the same column of the prime shift array A246278 as 100 is], and as A342671(100) = A342671(3025) = 7.
a(300) = a(21175) [= 200 as allotted by the rgs-transform], as 21175 = A003961(A003961(300)) and as A342671(300) = A342671(21175) = 7.
a(1215) = a(21875) [= 831 as allotted by the rgs-transform] because 21875 = A003961(1215), therefore it is in the same column of the prime shift array A246278 as 1215 is, and as A342671(1215) = A342671(21875) = 7.
a(2835) = a(48125) [= 1953 as allotted by the rgs-transform] because 48125 = A003961(2835) and as A342671(2835) = A342671(48125) = 11.

Links

Crossrefs

Cf. A000203, A003961, A246278, A342671, A348717, A349165, A349166, A355924.
Cf. also A305801, A305897, A355834, A355835.

Programs

  • PARI
    up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A342671(n) = gcd(sigma(n), A003961(n));
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));
Aux355833(n) = [A342671(n), A348717(n)];
v355833 = rgs_transform(vector(up_to,n,Aux355833(n)));
A355833(n) = v355833[n];

A355836 Lexicographically earliest infinite sequence such that a(i) = a(j) => A046523(i) = A046523(j) and A355442(i) = A355442(j) for all i, j >= 1.

Original entry on oeis.org

1, 2, 3, 4, 3, 5, 3, 6, 7, 8, 3, 9, 3, 8, 5, 10, 3, 11, 3, 12, 5, 8, 3, 13, 14, 8, 15, 16, 3, 17, 3, 18, 19, 8, 20, 21, 3, 8, 5, 22, 3, 23, 3, 12, 24, 8, 3, 25, 14, 26, 5, 16, 3, 27, 20, 28, 5, 8, 3, 29, 3, 8, 30, 31, 20, 23, 3, 12, 5, 32, 3, 33, 3, 8, 34, 16, 19, 23, 3, 35, 36, 8, 3, 37, 20, 8, 5, 38, 3, 29, 19, 12, 19, 8, 20, 39, 3, 12, 9, 40, 3, 23, 3, 28, 41
Offset: 1

Views

Author

Antti Karttunen, Jul 20 2022

Keywords

Comments

Restricted growth sequence transform of the ordered pair [A046523(n), A355442(n)].
For all i, j: A355835(i) = A355835(j) => a(i) = a(j).

Examples

			a(6) = a(15) = a(21) = a(39) = a(51) = a(57) = a(69) = a(87) = a(111) = etc, for an infinite number of other indices k, because for all these k, A355442(k) = 5 and their prime signatures (A101296) are equal, as they are all squarefree semiprimes, A006881.
In contrast, powers of 2 (1, 2, 4, 8, 16, ..., A000079) obtain unique values in this sequence, and in general, for all proper prime powers k (A246547) for which A355442(k) > 1 [that are terms of A355822], the value a(k) is unique in this sequence.

Links

Crossrefs

Cf. A000079, A003961, A006881, A046523, A101296, A246547, A276086, A355442, A355822, A355835.
Cf. also A355000, A355830.

Programs

  • PARI
    up_to = 100000;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
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));
Aux355836(n) = [A046523(n), A355442(n)];
v355836 = rgs_transform(vector(up_to,n,Aux355836(n)));
A355836(n) = v355836[n];

A355926 Square array A(n,k) = A355442(A246278(n,k)), read by falling antidiagonals.

Original entry on oeis.org

3, 9, 1, 5, 5, 1, 3, 5, 1, 1, 3, 125, 7, 1, 1, 5, 5, 343, 1, 1, 1, 3, 175, 7, 11, 1, 1, 1, 9, 1, 49, 1, 1, 1, 1, 1, 25, 125, 7, 121, 1, 1, 1, 1, 1, 3, 245, 2401, 1, 1, 1, 1, 1, 1, 1, 3, 1, 77, 1, 1, 1, 1, 1, 1, 1, 1, 5, 5, 49, 11, 28561, 1, 1, 1, 1, 1, 1, 1, 3, 175, 7, 121, 1, 1, 1, 1, 1, 1, 1, 1, 1, 9, 5, 77, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1
Offset: 1

Views

Author

Antti Karttunen, Jul 22 2022

Keywords

Examples

			The top left corner of the array:
   n= 1  2  3    4  5    6  7      8    9   10 11     12 13   14  15       16
  2n= 2  4  6    8 10   12 14     16   18   20 22     24 26   28  30       32
----+--------------------------------------------------------------------------
  1 | 3, 9, 5,   3, 3,   5, 3,     9,  25,   3, 3,     5, 3,   9,  7,       3,
  2 | 1, 5, 5, 125, 5, 175, 1,   125, 245,   1, 5,   175, 5,   5, 35,       1,
  3 | 1, 1, 7, 343, 7,  49, 7,  2401,  77,  49, 7,    77, 7,  49, 77,   16807,
  4 | 1, 1, 1,  11, 1, 121, 1,     1,  11, 121, 1, 17303, 1, 121, 11,    1331,
  5 | 1, 1, 1,   1, 1,   1, 1, 28561,   1,   1, 1,  2197, 1,   1, 13,   28561,
  6 | 1, 1, 1,   1, 1,   1, 1,     1,   1,   1, 1,    17, 1,   1,  1, 1419857,
  7 | 1, 1, 1,   1, 1,   1, 1,     1,   1,   1, 1,     1, 1,   1,  1,     361,
  8 | 1, 1, 1,   1, 1,   1, 1,     1,   1,   1, 1,     1, 1,   1,  1,       1,
  9 | 1, 1, 1,   1, 1,   1, 1,     1,   1,   1, 1,     1, 1,   1,  1,       1,

Crossrefs

Cf. A003961, A246278, A276086, A355442, A355835.
Cf. also A355924, A355925 for similarly constructed arrays.

Programs

  • PARI
    up_to = 105;
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));
A246278sq(row,col) = if(1==row,2*col, my(f = factor(2*col)); for(i=1, #f~, f[i,1] = prime(primepi(f[i,1])+(row-1))); factorback(f));
A355926sq(row,col) = A355442(A246278sq(row,col));
A355926list(up_to) = { my(v = vector(up_to), i=0); for(a=1,oo, for(col=1,a, i++; if(i > up_to, return(v)); v[i] = A355926sq(col,(a-(col-1))))); (v); };
v355926 = A355926list(up_to);
A355926(n) = v355926[n];
Showing 1-4 of 4 results.

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