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.

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A249810 a(1) = 0, a(n) = A078898(A003961(n)).

Original entry on oeis.org

0, 1, 1, 2, 1, 3, 1, 5, 2, 4, 1, 8, 1, 6, 3, 14, 1, 13, 1, 11, 4, 7, 1, 23, 2, 9, 9, 17, 1, 18, 1, 41, 5, 10, 3, 38, 1, 12, 6, 32, 1, 28, 1, 20, 12, 15, 1, 68, 2, 25, 7, 26, 1, 63, 4, 50, 8, 16, 1, 53, 1, 19, 19, 122, 5, 33, 1, 29, 10, 39, 1, 113, 1, 21, 17, 35, 3, 43, 1, 95, 42, 22, 1, 83, 6, 24, 11, 59, 1, 88, 4, 44, 13, 27, 7, 203
Offset: 1

Views

Author

Antti Karttunen, Dec 08 2014

Keywords

Links

Crossrefs

Cf. A003961, A078898, A246277, A249817, A249818, A249820, A249821, A249822, A251721, A251722, A250469, A250470.

Programs

Formula

a(1) = 0, a(n) = A078898(A003961(n)).
a(1) = 0, a(n) = A078898(n) + A249820(n).

A249826 Permutation of natural numbers: a(n) = A078898(A003961(A003961(A003961(2*n)))).

Original entry on oeis.org

1, 2, 3, 14, 4, 21, 5, 92, 33, 25, 6, 144, 7, 32, 39, 641, 8, 226, 9, 170, 50, 36, 10, 1007, 46, 43, 355, 223, 11, 267, 12, 4482, 56, 55, 59, 1582, 13, 58, 68, 1190, 15, 350, 16, 249, 420, 70, 17, 7043, 78, 316, 86, 301, 18, 2485, 66, 1555, 91, 77, 19, 1869, 20, 81, 549, 31374, 80, 391, 22, 379, 109, 413, 23, 11068, 24, 88, 496, 406, 87, 473, 26, 8324, 3905, 99, 27
Offset: 1

Views

Author

Antti Karttunen, Dec 06 2014

Keywords

Links

Crossrefs

Inverse: A249825.
Row 4 of A249822.
Cf. A003961, A251722, A048673, A078898, A246278, A249824, A249746, A250476.

Programs

Formula

a(n) = A078898(A003961(A003961(A003961(2*n)))).
a(n) = A078898(A246278(4,n)).
As a composition of other permutations:
a(n) = A250476(A249824(n)).
a(n) = A250476(A249746(A048673(n))). [Composition of the first three rows of array A251722.]

A250478 Number of times prime(n) occurs as the least prime factor among numbers 1 .. prime(n)^4: a(n) = A078898(A030514(n)).

Original entry on oeis.org

8, 14, 42, 92, 305, 455, 944, 1238, 2085, 3995, 4710, 7757, 10273, 11558, 14742, 20701, 28019, 30444, 39680, 46534, 49856, 62350, 71394, 86977, 111352, 124421, 130649, 145076, 151939, 167759, 236113, 257098, 291830, 302611, 370060, 382610, 427214, 475078
Offset: 1

Views

Author

Antti Karttunen, Dec 14 2014

Keywords

Links

Crossrefs

Column 8 of A249822.
Cf. also A250474 (column 4), A250477 (column 6).
Cf. A002110, A030514, A030078, A078898.

Programs

  • PARI
    allocatemem(234567890);
A002110(n) = prod(i=1, n, prime(i));
A250478(n) = { my(p3); p3 = (prime(n)^3); sumdiv(A002110(n-1), d, (moebius(d)*(p3\d))); };
for(n=1, 23, print1(A250478(n),", "));
  • Scheme
    (define (A250478 n) (A078898 (A030514 n)))

Formula

a(n) = A078898(A030514(n)).
a(1) = 1, a(n) = sum_{d | A002110(n-1)} moebius(d) * floor(prime(n)^3 / d). [Follows when A030514, prime(n)^4 is substituted to the similar formula given for A078898. Here A002110(n) gives the product of the first n primes. Because the latter is always squarefree, one could use here also Liouville's lambda (A008836) instead of Moebius mu (A008683).]

Extensions

More terms from Jon E. Schoenfield, Dec 14 2014

A349631 Dirichlet convolution of A003961 with A346479, which is Dirichlet inverse of A250469.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 6, 0, -6, 0, 12, 0, -6, 0, 18, 0, 24, 0, 24, 0, -24, 0, 0, 0, -24, 60, 36, 0, 48, 0, 42, -20, -42, 0, -12, 0, -42, -10, 12, 0, 72, 0, 60, 60, -48, 0, -24, 0, 42, -30, 72, 0, -84, 0, 12, -30, -78, 0, -120, 0, -72, 120, 126, 0, 180, 0, 96, -30, 132, 0, -48, 0, -96, 60, 108, 0, 174, 0, -84, 120
Offset: 1

Views

Author

Antti Karttunen, Nov 27 2021

Keywords

Comments

Note that for n = 2..36, a(n) = -A349632(n).
Dirichlet convolution of this sequence with A347376 is A003972.

Links

Crossrefs

Cf. A003961, A250469, A346479, A349632 (Dirichlet inverse).
Cf. also A003972, A347376, A349381.
Cf. also arrays A083221, A246278, A249821, A249822 and permutations A250245, A250246.

Programs

  • PARI
    up_to = 20000;
A020639(n) = if(1==n,n,vecmin(factor(n)[, 1]));
ordinal_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), pt); for(i=1, length(invec), if(mapisdefined(om,invec[i]), pt = mapget(om, invec[i]), pt = 0); outvec[i] = (1+pt); mapput(om,invec[i],(1+pt))); outvec; };
v078898 = ordinal_transform(vector(up_to,n,A020639(n)));
A078898(n) = v078898[n];
A250469(n) = if(1==n,n,my(spn = nextprime(1+A020639(n)), c = A078898(n), k = 0); while(c, k++; if((1==k)||(A020639(k)>=spn),c -= 1)); (k*spn));
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(dA250469(n)));
A346479(n) = v346479[n];
A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
A349631(n) = sumdiv(n,d,A003961(d)*A346479(n/d));

Formula

a(n) = Sum_{d|n} A003961(d) * A346479(n/d).

A349632 Dirichlet convolution of A250469 with A346234, which is Dirichlet inverse of A003961.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, -6, 0, 6, 0, -12, 0, 6, 0, -18, 0, -24, 0, -24, 0, 24, 0, 0, 0, 24, -60, -36, 0, -48, 0, -42, 20, 42, 0, 12, 0, 42, 10, -12, 0, -72, 0, -60, -60, 48, 0, 24, 0, -42, 30, -72, 0, 84, 0, -12, 30, 78, 0, 120, 0, 72, -120, -90, 0, -180, 0, -96, 30, -132, 0, 48, 0, 96, -60, -108, 0, -174, 0, 12, -120
Offset: 1

Views

Author

Antti Karttunen, Nov 27 2021

Keywords

Comments

Note that for n = 2..36, a(n) = -A349631(n).
Dirichlet convolution of this sequence with A003972 is A347376.

Links

Crossrefs

Cf. A003961, A250469, A346234, A349631 (Dirichlet inverse).
Cf. also A003972, A347376, A349382.
Cf. also arrays A083221, A246278, A249821, A249822 and permutations A250245, A250246.

Programs

  • PARI
    up_to = 20000;
A020639(n) = if(1==n,n,vecmin(factor(n)[, 1]));
ordinal_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), pt); for(i=1, length(invec), if(mapisdefined(om,invec[i]), pt = mapget(om, invec[i]), pt = 0); outvec[i] = (1+pt); mapput(om,invec[i],(1+pt))); outvec; };
v078898 = ordinal_transform(vector(up_to,n,A020639(n)));
A078898(n) = v078898[n];
A250469(n) = if(1==n,n,my(spn = nextprime(1+A020639(n)), c = A078898(n), k = 0); while(c, k++; if((1==k)||(A020639(k)>=spn),c -= 1)); (k*spn));
A346234(n) = (moebius(n)*A003961(n));
A349632(n) = sumdiv(n,d,A250469(n/d)*A346234(d));

Formula

a(n) = Sum_{d|n} A250469(d) * A346234(n/d).
Previous Showing 11-15 of 15 results.

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