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 1-8 of 8 results.

A240521 a(n) = A050376(n)*A050376(n+1) where A050376(n) is the n-th number of the form p^(2^k) with p is prime and k >= 0.

Original entry on oeis.org

6, 12, 20, 35, 63, 99, 143, 208, 272, 323, 437, 575, 725, 899, 1147, 1517, 1763, 2021, 2303, 2597, 3127, 3599, 4087, 4757, 5183, 5767, 6399, 6723, 7387, 8633, 9797, 10403, 11021, 11663, 12317, 13673, 15367, 16637, 17947, 19043, 20711, 22499, 23707, 25591
Offset: 1

Views

Author

Vladimir Shevelev, Apr 07 2014

Keywords

Comments

Let m be an odd positive number. Let S_m denote the sequence {Product_{i=1..r} q_(n+t_i)}A050376%20and%20Sum">{n>=1}, where {q_i} is sequence A050376 and Sum{i=1..r} 2^(t_1 - t_i) is the binary representation of m, such that t_1 > t_2 > ... > t_r = 0. Note that {S_1, S_3, S_5, ...} is a partition of all integers > 1. Then S_1=A050376, which is obtained when we set r=1, t_1 = 0. [Formula made compatible with A240535 data by Peter Munn, Aug 10 2021]
This present sequence is S_3 in this partition. It is obtained when we set r=2, t_1=1, t_2=0.
S_m(n) = A052330(A030101(m)*2^(n-1)) = A329330(A050376(n), A052330(A030101(m))). - Peter Munn, Aug 10 2021
A minimal set of generators for A000379 as a group under A059897(.,.). - Peter Munn, Aug 11 2019

Links

Crossrefs

Positions of 3's in A240535.
Sequences for other parts of the partition described in the first comment: A050376 (S_1), A240522 (S_5), A240524 (S_7), A240536 (S_9), A241024 (S_11), A241025 (S_13).
Cf. A000379, A030101, A052330, A059897, A329330.

Programs

  • Python
    from sympy import primepi, integer_nthroot
def A240521(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return n+x-sum(primepi(integer_nthroot(x,1<Chai Wah Wu, Feb 18-19 2025

Formula

a(n) = A052330(3*2^(n-1)) = A329330(A050376(n), 6). - Peter Munn, Aug 10 2021

Extensions

More terms from Peter J. C. Moses, Apr 18 2014

A240522 S_5 sequence in partition of integers > 1 described in A240521.

Original entry on oeis.org

8, 15, 28, 45, 77, 117, 176, 221, 304, 391, 475, 667, 775, 1073, 1271, 1591, 1927, 2107, 2491, 2891, 3233, 3953, 4331, 4891, 5609, 5913, 6557, 7209, 8051, 8989, 9991, 10807, 11227, 12091, 13189, 14351, 15851, 17399, 18209, 20413, 20989, 23393, 24613, 26219
Offset: 1

Views

Author

Vladimir Shevelev, Apr 07 2014

Keywords

Comments

See case r=2, t_1=2, t_2=0 in comment in A240521.

Crossrefs

Positions of 5's in A240535.
Sequences for other parts of the partition: A050376 (S_1), A240521 (S_3), A240524 (S_7), A240536 (S_9), A241024 (S_11), A241025 (S_13).

Formula

a(n)=q_n*q_(n+2), where q_n is the n-th term of A050376.

Extensions

More terms from Peter J. C. Moses, Apr 18 2014
Name revised by Peter Munn, Oct 11 2021

A240524 S_7 sequence in partition of integers > 1 described in A240521.

Original entry on oeis.org

24, 60, 140, 315, 693, 1287, 2288, 3536, 5168, 7429, 10925, 16675, 22475, 33263, 47027, 65231, 82861, 99029, 122059, 153223, 190747, 241133, 290177, 347261, 409457, 467127, 531117, 598347, 716539, 871933, 1009091, 1113121, 1201289, 1317919, 1490357, 1736471
Offset: 1

Views

Author

Vladimir Shevelev, Apr 07 2014

Keywords

Comments

See case r=3, t_1=2, t_2=1, t_3=0 of comment in A240521.

Crossrefs

Positions of 7's in A240535.
Sequences for other parts of the partition: A050376 (S_1), A240521 (S_3), A240522 (S_5), A240536 (S_9), A241024 (S_11), A241025 (S_13).

Formula

a(n) = q_n*q_(n+1)*q_(n+2), where q_n is the n-th term of A050376.

Extensions

More terms from Peter J. C. Moses, Apr 18 2014
Name revised by Peter Munn, Oct 11 2021

A240536 S_9 sequence in partition of integers > 1 described in A240521.

Original entry on oeis.org

10, 21, 36, 55, 91, 144, 187, 247, 368, 425, 551, 713, 925, 1189, 1333, 1739, 2009, 2279, 2773, 2989, 3551, 4189, 4453, 5293, 5751, 6059, 7031, 7857, 8383, 9167, 10379, 11009, 11639, 12947, 13843, 14803, 16577, 17653, 19519, 20687, 21823, 24287, 25217, 26533
Offset: 1

Views

Author

Vladimir Shevelev, Apr 07 2014

Keywords

Comments

See case r=2, t_1=3, t_2=0 in comment in A240521.

Crossrefs

Positions of 9's in A240535.
Sequences for other parts of the partition: A050376 (S_1), A240521 (S_3), A240522 (S_5), A240524 (S_7), A241024 (S_11), A241025 (S_13).

Formula

a(n) = A050376(n) * A050376(n+3).

Extensions

More terms from Peter J. C. Moses, Apr 18 2014
Name revised by Peter Munn, Oct 11 2021

A241024 S_11 sequence in partition of integers > 1 described in A240521.

Original entry on oeis.org

40, 105, 252, 495, 1001, 1872, 2992, 4199, 6992, 9775, 13775, 20677, 28675, 43993, 54653, 74777, 94423, 111671, 146969, 176351, 216611, 280663, 316163, 386389, 454329, 490779, 583573, 699273, 813151, 925867, 1069037, 1177963, 1268651, 1463011, 1675003, 1879981
Offset: 1

Views

Author

Vladimir Shevelev, Apr 15 2014

Keywords

Comments

See case r=3, t_1=3, t_2=2, t_3=0 in comment in A240521. [Subscripts made consistent by Peter Munn, Oct 11 2021]

Crossrefs

Positions of 11's in A240535.
Sequences for other parts of the partition: A050376 (S_1), A240521 (S_3), A240522 (S_5), A240524 (S_7), A240536 (S_9), A241025 (S_13).

Formula

a(n) = A050376(n) * A050376(n+2) * A050376(n+3).

Extensions

Name revised by Peter Munn, Oct 11 2021

A241025 S_13 sequence in partition of integers > 1 described in A240521.

Original entry on oeis.org

30, 84, 180, 385, 819, 1584, 2431, 3952, 6256, 8075, 12673, 17825, 26825, 36859, 49321, 71299, 86387, 107113, 135877, 158417, 209509, 255529, 298351, 375803, 419823, 478661, 569511, 652131, 746087, 889199, 1048279, 1133927, 1245373, 1411223, 1564259, 1791163
Offset: 1

Views

Author

Vladimir Shevelev, Apr 15 2014

Keywords

Comments

See case r=3, t_1=3, t_2=1, t_3=0 in comment in A240521. [Subscripts made consistent by Peter Munn, Oct 11 2021]

Crossrefs

Positions of 13's in A240535.
Sequences for other parts of the partition: A050376 (S_1), A240521 (S_3), A240522 (S_5), A240524 (S_7), A240536 (S_9), A241024 (S_11).

Formula

a(n) = A050376(n) * A050376(n+1) * A050376(n+3).

Extensions

Name revised by Peter Munn, Oct 11 2021

A302785 Index of the largest Fermi-Dirac factor of n, a(1) = 0 by convention: a(n) = A302778(A223491(n)).

Original entry on oeis.org

0, 1, 2, 3, 4, 2, 5, 3, 6, 4, 7, 3, 8, 5, 4, 9, 10, 6, 11, 4, 5, 7, 12, 3, 13, 8, 6, 5, 14, 4, 15, 9, 7, 10, 5, 6, 16, 11, 8, 4, 17, 5, 18, 7, 6, 12, 19, 9, 20, 13, 10, 8, 21, 6, 7, 5, 11, 14, 22, 4, 23, 15, 6, 9, 8, 7, 24, 10, 12, 5, 25, 6, 26, 16, 13, 11, 7, 8, 27, 9, 28, 17, 29, 5, 10, 18, 14, 7, 30, 6, 8, 12, 15, 19, 11, 9, 31, 20, 7, 13, 32, 10, 33, 8, 5
Offset: 1

Views

Author

Antti Karttunen, Apr 13 2018

Keywords

Links

Crossrefs

A left inverse of A050376.
Cf. A052331, A223491, A240535, A302778, A302786, A302788, A302789 (ordinal transform).
Cf. also A061395.

Programs

  • PARI
    up_to = 65537;
v050376 = vector(up_to);
A050376(n) = v050376[n];
ispow2(n) = (n && !bitand(n,n-1));
i = 0; for(n=1,oo,if(ispow2(isprimepower(n)), i++; v050376[i] = n); if(i == up_to,break));
A302785(n) = if(1==n,0, my(e); fordiv(n, d, if(ispow2(isprimepower(n/d)), e = vecsearch(v050376, n/d); if(e, return(e), print("v050376 too short!"); return(1/0)))));

Formula

a(n) = A302778(A223491(n)).
For n > 1, A050376(a(n)) = A223491(n).
For n >= 1, a(A050376(n)) = n.

A302787 a(1) = 0; for n > 1, a(n) = A000265(A052331(n)).

Original entry on oeis.org

0, 1, 1, 1, 1, 3, 1, 5, 1, 9, 1, 3, 1, 17, 5, 1, 1, 33, 1, 3, 9, 65, 1, 7, 1, 129, 17, 5, 1, 11, 1, 257, 33, 513, 3, 9, 1, 1025, 65, 13, 1, 19, 1, 17, 5, 2049, 1, 129, 1, 4097, 257, 33, 1, 35, 9, 21, 513, 8193, 1, 7, 1, 16385, 3, 65, 17, 67, 1, 129, 1025, 25, 1, 37, 1, 32769, 2049, 257, 5, 131, 1, 33, 1, 65537, 1, 11, 65, 131073, 4097, 69, 1, 41, 9
Offset: 1

Views

Author

Antti Karttunen, Apr 13 2018

Keywords

Comments

After n=1, differs from A240535 (which gives the same terms, but with mirrored binary expansion) for the first time at n=30, where a(30) = 11, while A240535(30) = 13. Note how 11 = "1011" and 13 = "1101" in binary.
For all i, j: a(i) = a(j) => A302791(i) = A302791(j).

Links

Crossrefs

Cf. A000265, A050376, A052331, A240535, A302786, A302788, A302789, A302790, A302791.
Cf. also A246277.

Programs

  • PARI
    up_to = 8192;
v050376 = vector(up_to);
ispow2(n) = (n && !bitand(n,n-1));
i = 0; for(n=1,oo,if(ispow2(isprimepower(n)), i++; v050376[i] = n); if(i == up_to,break));
A052331(n) = { my(s=0,e); while(n > 1, fordiv(n, d, if(((n/d)>1)&&ispow2(isprimepower(n/d)), e = vecsearch(v050376, n/d); if(!e, print("v050376 too short!"); return(1/0)); s += 2^(e-1); n = d; break))); (s); };
A000265(n) = (n/2^valuation(n, 2));
A302787(n) = if(1==n,0,A000265(A052331(n)));

Formula

a(1) = 0; for n > 1, a(n) = A000265(A052331(n)).
For n > 1, a(n) = A030101(A240535(n)).
For n >= 1, A069010(a(n)) = A302790(n).
Showing 1-8 of 8 results.

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