A240522 -id:A240522 - cp's OEIS frontend

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.

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

Vladimir Shevelev, Apr 07 2014

nonn

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

Eric Weisstein's World of Mathematics, Group.
Wikipedia, Generating set of a group.

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.

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

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

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

A240535 a(n)=m if n belongs to the S_m sequence described in A240521.

Original entry on oeis.org

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, 13, 1, 257, 33, 513, 3, 9, 1, 1025, 65, 11, 1, 25, 1, 17, 5, 2049, 1, 129, 1, 4097, 257, 33, 1, 49, 9, 21, 513, 8193, 1, 7, 1, 16385, 3, 65, 17, 97, 1, 129, 1025, 19, 1

Offset: 2

Vladimir Shevelev, Apr 07 2014

nonn

See comments in A240521.

			Let n = 30. We have a unique representation of 30 as a product of distinct terms of A050376: 30 = 2*3*5. We write all the terms of A050376 in the interval [2,5]: 2,3,4,5. Under the terms used in the representation of 30 we write 1, under other terms we write 0. After concatenation we obtain the binary number corresponding to 30: 1101. In decimal it is 13. So a(30) = 13.
Let n = 60 = 3*4*5. In the interval [3,5] the terms of A050376 are 3,4,5, all of which are used in the representation of 60. So we write 1 under all 3 terms and obtain the binary number 111. In decimal it is 7. So a(60)=7.

Positions of particular values: A050376 (1), A240521 (3), A240522 (5), A240524 (7), A240536 (9), A241024 (11), A241025 (13).

Terms corrected and more terms added, Peter J. C. Moses, Apr 18 2014

Name revised and other edits 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

Vladimir Shevelev, Apr 07 2014

nonn

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

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).

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

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

Vladimir Shevelev, Apr 07 2014

nonn

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

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).

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

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

Vladimir Shevelev, Apr 15 2014

nonn

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]

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).

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

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

Vladimir Shevelev, Apr 15 2014

nonn

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]

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).

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

Name revised by Peter Munn, Oct 11 2021

A329330 Multiplication operation of a ring over the positive integers that has A059897(.,.) as addition operation and is isomorphic to GF(2)[x] with polynomial x^i mapped to A050376(i+1). Square array read by descending antidiagonals: A(n,k), n >= 1, k >= 1.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 4, 4, 1, 1, 5, 5, 5, 5, 1, 1, 6, 7, 7, 7, 6, 1, 1, 7, 12, 9, 9, 12, 7, 1, 1, 8, 9, 20, 11, 20, 9, 8, 1, 1, 9, 15, 11, 35, 35, 11, 15, 9, 1, 1, 10, 11, 28, 13, 8, 13, 28, 11, 10, 1, 1, 11, 21, 13, 45, 63, 63, 45, 13, 21, 11, 1

Offset: 1

Peter Munn, Nov 10 2019

When creating A329329, the author realized it was isomorphic to multiplication in the GF(2)[x,y] polynomial ring. However, A329329 was unusual in having A059897(.,.) as additive operator, whereas the equivalent univariate polynomial ring, GF(2)[x], is more commonly mapped (to integers) with bitwise exclusive-or (A003987) representing polynomial addition (and A048720(.,.) representing polynomial multiplication). This sequence shows how multiplication in GF(2)[x] can look when mapped to integers with A059897(.,.) representing polynomial addition.
The group defined by the binary operation A059897(.,.) over the positive integers is commutative with all elements self-inverse, and isomorphic to the additive group of the polynomial ring GF(2)[x]. There is a unique isomorphism extending each bijective mapping between respective minimal generating sets. The lexicographically earliest minimal generating set for the A059897 group is A050376, often called the Fermi-Dirac primes. The most meaningful generating set for the additive group of GF(2)[x] is {x^i: i >= 0}.
Using f to denote the intended isomorphism from GF(2)[x], we specify f(x^i) = A050376(i+1). This maps minimal generating sets of the additive groups, so the definition of f is completed by specifying f(a+b) = A059897(f(a), f(b)). We then calculate the image under f of polynomial multiplication in GF(2)[x], giving us this sequence as the matching multiplication operation for an isomorphic ring over the positive integers. Using g to denote the inverse of f, A(n,k) = f(g(n) * g(k)).
Note that A050376 is closed with respect to A(.,.).
Recall that GF(2)[x] is more usually mapped to integers with A003987(.,.) as addition and A048720(.,.) as multiplication. With this usual mapping, under which A000079(i) is the image of x^i, A052330(.) is the relevant isomorphism from nonnegative integers under A048720(.,.) and A003987(.,.) to positive integers under A(.,.) and A059897(.,.), with A052331(.) its inverse.

			Square array A(n,k) begins:
  n\k |  1    2    3    4    5    6    7    8    9   10   11   12
  ----+----------------------------------------------------------
   1  |  1    1    1    1    1    1    1    1    1    1    1    1
   2  |  1    2    3    4    5    6    7    8    9   10   11   12
   3  |  1    3    4    5    7   12    9   15   11   21   13   20
   4  |  1    4    5    7    9   20   11   28   13   36   16   35
   5  |  1    5    7    9   11   35   13   45   16   55   17   63
   6  |  1    6   12   20   35    8   63  120   99  210  143   15
   7  |  1    7    9   11   13   63   16   77   17   91   19   99
   8  |  1    8   15   28   45  120   77   14  117  360  176  420
   9  |  1    9   11   13   16   99   17  117   19  144   23  143
  10  |  1   10   21   36   55  210   91  360  144   22  187  756
  11  |  1   11   13   16   17  143   19  176   23  187   25  208
  12  |  1   12   20   35   63   15   99  420  143  756  208   28

Eric Weisstein's World of Mathematics, Distributive
Eric Weisstein's World of Mathematics, Group
Eric Weisstein's World of Mathematics, Ring
Wikipedia, Generating set of a group
Wikipedia, Polynomial ring

Cf. A000079, A050376, A329329.
Distributes over A059897, and isomorphic to A048720 over A003987, with A052331 (inverse A052330) as isomorphism.
Row/column 3: A300841.
Row/column k sorted into increasing order: A003159 (k=3), A339690 (k=4), A000379 (k=6).
Subsequences of row/column k: A240521 (k=6), A240522 (k=8), A240536 (k=10), A240524 (k=24), A241025 (k=30), A241024 (k=40).

A(n,k) = A052330(A048720(A052331(n), A052331(k))), n >= 1, k >= 1.
A059897-based definition: (Start)
A(A050376(i), A050376(j)) = A050376(i+j-1).
A(A059897(n,k), m) = A059897(A(n,m), A(k,m)).
A(m, A059897(n,k)) = A059897(A(m,n), A(m,k)).
(End)
Derived identities: (Start)
A(n,1) = A(1,n) = 1.
A(n,2) = A(2,n) = n.
A(n,k) = A(k,n).
A(n, A(m,k)) = A(A(n,m), k).
(End)
A(A300841(n), k) = A(n, A300841(k)) = A300841(A(n,k)).
A(n,3) = A(3,n) = A300841(n).
A(n,4) = A(4,n) = A300841^2(n).
A(n,5) = A(5,n) = A300841^3(n).
A(A050376(m), 6) = A(6, A050376(m)) = A240521(m).
A(n,7) = A(7,n) = A300841^4(n).
A(A050376(m), 8) = A(8, A050376(m)) = A240522(m).
A(n,9) = A(9,n) = A300841^5(n).
A(A050376(m), 10) = A(10, A050376(m)) = A240536(m).
A(A050376(m), 12) = A(12, A050376(m)) = A300841(A240521(m)).
A(A050376(m), 24) = A(24, A050376(m)) = A240524(m).
A(A050376(m), 30) = A(30, A050376(m)) = A241025(m).
A(A050376(m), 40) = A(40, A050376(m)) = A241024(m).

cp's OEIS Frontend

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.

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A240535 a(n)=m if n belongs to the S_m sequence described in A240521.

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

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

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

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

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A329330 Multiplication operation of a ring over the positive integers that has A059897(.,.) as addition operation and is isomorphic to GF(2)[x] with polynomial x^i mapped to A050376(i+1). Square array read by descending antidiagonals: A(n,k), n >= 1, k >= 1.

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