A050376 -id:A050376 - cp's OEIS frontend

A330684 Square roots of the positions of records in A050377, number of ways to factor n into "Fermi-Dirac primes" (A050376).

1, 2, 4, 8, 12, 16, 24, 32, 36, 48, 72, 96, 144, 288, 432, 576, 864, 1152, 1440, 1728, 2304, 2880, 4320, 4608, 5184, 5760, 6912, 7200, 8640, 10368, 11520, 14400, 20736, 23040, 25920, 28800, 34560, 41472, 43200, 51840, 57600, 82944, 86400, 100800, 103680, 115200, 129600, 172800, 207360, 230400, 259200, 345600, 388800, 403200

Offset: 1

Antti Karttunen, Dec 29 2019

nonn

Like terms in A330687, also these are all found in A025487.

Antti Karttunen, Table of n, a(n) for n = 1..1004 (computed from the b-file David A. Corneth computed for A330687)

Cf. A000196, A050376, A050377, A108951, A329900, A330687, A330689.

Subsequence of A025487.

PARI
```
A330684(n) = sqrtint(A330687(n));
```

a(n) = A000196(A330687(n)).
A108951(a(n)) = A000196(A330689(n)).
a(n) = A329900(A000196(A330689(n))).

A372332 The number of "Fermi-Dirac primes" (A050376) that are noninfinitary divisors of n.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 2, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 2, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 2, 2, 0, 0, 1, 0, 0, 0

Offset: 1

Amiram Eldar, Apr 28 2024

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A023416, A050376, A064547, A348341, A372331.

f[p_, e_] := DigitCount[e, 2, 0]; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = vecsum(apply(x -> logint(x, 2) + 1 - hammingweight(x), factor(n)[, 2]));

Additive with a(p^e) = A023416(e).
a(n) = log_2(A372331(n)).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{p prime} f(1/p) = 0.39726277693465233149..., where f(x) = Sum_{k>=0} x^(2^(k+1))/(1+x^(2^k)).

A109429 Rearrange terms of A050376 so that a(2^j)=2^(2^j) for j>=0.

Original entry on oeis.org

2, 4, 3, 16, 5, 7, 9, 256, 11, 13, 17, 19, 23, 25, 29, 65536, 31, 37, 41, 43, 47, 49, 53, 59, 61, 67, 71, 73, 79, 81, 83, 4294967296, 89, 97

Offset: 1

Thomas Ordowski, Aug 26 2005

nonn

A073904(2^n) is the product of the first n members of this sequence. Generalization: for any prime p, we may consider the analogous permutation of numbers of the form q^(p^k) such that a(p^j)=p^(p^j); then A073904(p^n)=(product of the first n members)^(p-1). - David Wasserman and Thomas Ordowski. Corrected by Thomas Ordowski, Jun 06 2015

			Numbers: 2, 3, 2^2, 5, 7, 3^2, 11, 13, 2^(2^2), 17, ..., 2^(2^3), ...
Permutation: 2, 2^2, 3, 2^(2^2), 5, 7, 3^2, 2^(2^3), 11, 13, 17, ...
If n=4 then A073904(16)=2*4*3*16=384.

Cf. A050376.

a(2^j)=2^(2^j). So a(1)=2 for j=0; a(2)=4 for j=1; a(4)=16 for j=2.

A073904(2^n)=2*4*3*...*a(n) for every n.

Definition edited by N. J. A. Sloane, Oct 27 2014

More terms from Thomas Ordowski, Jun 05 2015

A187043 Number of factors in expansion of infinitary perfect numbers (A007357) over distinct terms of A050376.

Original entry on oeis.org

2, 3, 3, 5, 6, 7, 7, 7, 8, 9, 8, 9, 9, 10, 10, 11, 11, 12, 13, 12, 13, 14, 14, 15, 16, 17, 18, 19, 19, 19, 20, 21, 21, 23, 22, 23

Offset: 1

Vladimir Shevelev, Mar 02 2011

nonn

All terms beyond a(17) should be regarded conjectural, because they reach beyond the known terms in A007357. - R. J. Mathar, Mar 30 2011

			Consider A007357(3)=90. It is a unique product 2*5*9 of distinct terms of A050376. Thus a(3)=3.

Cf. A050376, A007357

a(n) = A064547(A007357(n)). - R. J. Mathar, Mar 30 2011

A214625 Let n=r_1r_2...*r_k is Fermi-Dirac factorization of n (see comment). Set g(n) = n + k - 1 and g_i, i>=0 (g_0(n) = n, g_1=g), is i-th iteration of g. a(n) is the minimal i such that g_i(n) is in A050376.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 2, 1, 0, 0, 1, 0, 3, 2, 1, 0, 4, 0, 3, 2, 1, 0, 6, 0, 5, 4, 3, 2, 1, 0, 7, 6, 5, 0, 4, 0, 3, 2, 1, 0, 1, 0, 3, 2, 1, 0, 3, 3, 2, 2, 1, 0, 16, 0, 15, 14, 13, 12, 11, 0, 10, 9, 8, 0, 7, 0, 6, 5, 4, 3, 2, 0, 1, 0, 1, 0, 13, 13

Offset: 2

Vladimir Shevelev, Feb 16 2013

nonn

Recall that every n>=2 has a unique factorization over distinct numbers from A050376 which is called Fermi-Dirac factorization of n. The sequence is a dual to A213980.

Conjecture: a(n) exists for every n >= 2.

			Since 24 = 2*3*4, then g_1(24) = 24 + 3 - 1 = 26; analogously, g_1(26) = 26 +2 -1 = 27, g_1(27) = 27 + 2 - 1 = 28, g_1(28) = 28 + 2 - 1 = 29 is in A050376. We used 4 iterations, therefore, a(24) = 4.

Amiram Eldar, Table of n, a(n) for n = 2..10000

Cf. A050376, A213980.

f[1]=0; f[n_] := Plus @@ (DigitCount[Last /@ FactorInteger[n], 2, 1]); g[n_] := n + f[n] - 1; a[n_] := Length @ FixedPointList[g, n]; Array[a, 30] (* Amiram Eldar, Sep 17 2019 *)

a(63) corrected by Amiram Eldar, Sep 17 2019

A228776 Positions of even terms of A050376.

Original entry on oeis.org

1, 3, 9, 63, 6605, 203286826, 425656284238504569

Offset: 1

Vladimir Shevelev, Sep 04 2013

Cf. A050376, A153450 (pi(2^(2^(n-1)))).

a[1] = 1; a[n_] := a[n] = a[n - 1] + PrimePi[2^(2^(n - 1))]; Array[a, 6] (* Amiram Eldar, Dec 04 2018 *)

a(n) = if (n==1, 1, a(n-1) + primepi(2^(2^(n-1)))); \\ Michel Marcus, Dec 04 2018

from sympy import primepi
def A228776(n): return sum(primepi(1<<(1<Chai Wah Wu, Feb 18 2025

For n>=2, a(n) = a(n-1) + pi(2^(2^(n-1))), where pi(x) is the prime counting function.
For s>1, Product_{n>=1} (1 + A050376(a(n))^(-s)) = 2^s/(2^s-1).
A generalization. Let p be a prime. Let for n>=1 the sequence {a^(p)(n)} be sequence of places of terms of A050376 divisible by p. Then, for n>=2, a^(p)(n) = a^(p)(n-1) + pi(p^(2^(n-1))); for s>1, Product_{n>=1} (1 + A050376(a^(p)(n))^(-s)) = p^s/(p^s-1).

a(6) from Peter J. C. Moses

a(7) from Jinyuan Wang, Mar 03 2020

A228868 Sum of all numbers n>=2 such that in their Fermi-Dirac representation every A050376-factor does not exceed A050376(n).

Original entry on oeis.org

2, 11, 59, 359, 2879, 28799, 345599, 4838399, 82252799, 1480550399, 29611007999, 710664191999, 18477268991999, 554318069759999, 17738178232319999, 674050772828159999, 28310132458782719999, 1245645828186439679999, 59790999752949104639999

Offset: 1

Vladimir Shevelev and Peter J. C. Moses, Sep 06 2013

nonn

Or, the same, diminished on 1 the sum of Fermi-Dirac divisors of the number Product_{i=1..n} A050376(i). Note that the sequence of the first differences 2, 9, 48, 300, ... lists sums of all numbers such that the maximal A050376-factor in their Fermi-Dirac representation is A050376(n). Note also that the average of numbers n >= 2 with A050376-factors not exceeding A050376(n) is a(n)/(2^n-1). Thus the sequence of such averages begins 2, 11/3, 59/7, 359/15, ...

Prime terms are 2, 11, 59, 359, 2879, 345599, 4838399, ...

			a(3) = 2 + 3 + 2*3 = 11.

Peter J. C. Moses, Table of n, a(n) for n = 1..100

Cf. A050376.

a(n) = -1 + Product_{i=1..n} (A050376(i) + 1).

A228891 Let A = A050376. Let Q be the smallest term of A more than 1 + Product_{i=1..n} A(i). a(n) = Q - Product_{i=1..n} A(i).

Original entry on oeis.org

2, 3, 5, 7, 13, 13, 17, 17, 17, 41, 59, 29, 41, 53, 37, 67, 79, 61, 89, 101, 139, 71, 67, 83, 151, 101, 89, 127, 163, 137, 101, 103, 131, 181, 139, 139, 181, 181, 139, 317, 191, 313, 163, 197, 199, 389, 191, 233, 229, 337, 239, 229, 347, 881, 239, 283, 487

Offset: 1

Vladimir Shevelev, Sep 07 2013

nonn

This sequence is a Fermi-Dirac analog of the Fortunate numbers (A005235).

			a(1) = 2, since 1 + Product_{i=1} A(i) = 1 + 2 = 3, the smallest term Q of A050376 more than 3 is 4 and a(1) = 4-2 = 2; let n=4, then 1 + Product_{i=1..4} A(i) = 1 + 2*3*4*5 = 121 and the smallest term Q of A050376 more than 121 is 127. So a(4) = 127 - 120 = 7.

Peter J. C. Moses, Table of n, a(n) for n = 1..1000

Cf. A005235, A050376.

More terms from Peter J. C. Moses, Sep 10 2013

A241270 Numbers with the property that in their factorization over distinct terms of A050376, the sums of prime and nonprime terms of A050376 are equal.

Original entry on oeis.org

126, 468, 624, 792, 880, 1056, 1150, 2900, 3264, 4606, 5824, 6375, 6624, 8320, 9856, 10388, 11375, 12798, 13650, 16400, 16704, 19250, 20925, 30135, 32625, 36720, 39150, 39900, 53784, 56446, 56925, 57000, 59500, 63455, 65520, 71400, 71500, 72471

Offset: 1

Vladimir Shevelev, Apr 18 2014

nonn

The corresponding sequence of the sum over the primes, which equals the sum over the nonprimes, is 9, 13, 16, 13, 16, 16, 25, 29, 20, 49, 20, 25, 25, 20, 20, 53, 25, 81, 25, 41, 29, 25, 34, 49, 34, 25, 34, 29, 85, 169, 34, 29, 29, 49, 25, 29, 29, 49, ... - Wolfdieter Lang, Apr 25 2014

			126 and 468 are in the sequence since the factorizations are 2*7*9 and 4*9*13 respectively, and 2+7=9, 4+9=13.

V. S. Shevelev, Multiplicative functions in the Fermi-Dirac arithmetic, Izvestia Vuzov of the North-Caucasus region, Nature sciences 4 (1996), 28-43 [Russian].

Peter J. C. Moses, Table of n, a(n) for n = 1..2000
S. Litsyn and V. S. Shevelev, On factorization of integers with restrictions on the exponent, INTEGERS: Electronic Journal of Combinatorial Number Theory, 7 (2007), #A33, 1-36.

Cf. A187039, A187042, A177329, A177333, A177334.

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

New extension from Wolfdieter Lang, Apr 25 2014

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

A330684 Square roots of the positions of records in A050377, number of ways to factor n into "Fermi-Dirac primes" (A050376).

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A372332 The number of "Fermi-Dirac primes" (A050376) that are noninfinitary divisors of n.

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A109429 Rearrange terms of A050376 so that a(2^j)=2^(2^j) for j>=0.

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A187043 Number of factors in expansion of infinitary perfect numbers (A007357) over distinct terms of A050376.

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A214625 Let n=r_1*r_2*...*r_k is Fermi-Dirac factorization of n (see comment). Set g(n) = n + k - 1 and g_i, i>=0 (g_0(n) = n, g_1=g), is i-th iteration of g. a(n) is the minimal i such that g_i(n) is in A050376.

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A228776 Positions of even terms of A050376.

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A228868 Sum of all numbers n>=2 such that in their Fermi-Dirac representation every A050376-factor does not exceed A050376(n).

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A228891 Let A = A050376. Let Q be the smallest term of A more than 1 + Product_{i=1..n} A(i). a(n) = Q - Product_{i=1..n} A(i).

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A241270 Numbers with the property that in their factorization over distinct terms of A050376, the sums of prime and nonprime terms of A050376 are equal.

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A214625 Let n=r_1r_2...*r_k is Fermi-Dirac factorization of n (see comment). Set g(n) = n + k - 1 and g_i, i>=0 (g_0(n) = n, g_1=g), is i-th iteration of g. a(n) is the minimal i such that g_i(n) is in A050376.