A045970 -id:A045970 - cp's OEIS frontend

A242378 Square array read by antidiagonals: to obtain A(i,j), replace each prime factor prime(k) in prime factorization of j with prime(k+i).

0, 1, 0, 2, 1, 0, 3, 3, 1, 0, 4, 5, 5, 1, 0, 5, 9, 7, 7, 1, 0, 6, 7, 25, 11, 11, 1, 0, 7, 15, 11, 49, 13, 13, 1, 0, 8, 11, 35, 13, 121, 17, 17, 1, 0, 9, 27, 13, 77, 17, 169, 19, 19, 1, 0, 10, 25, 125, 17, 143, 19, 289, 23, 23, 1, 0, 11, 21, 49, 343, 19, 221, 23, 361, 29, 29, 1, 0

Offset: 0

Antti Karttunen, May 12 2014

Each row i is a multiplicative function, being in essence "the i-th power" of A003961, i.e., A(i,j) = A003961^i (j). Zeroth power gives an identity function, A001477, which occurs as the row zero.
The terms in the same column have the same prime signature.
The array is read by antidiagonals: A(0,0), A(0,1), A(1,0), A(0,2), A(1,1), A(2,0), ... .

			The top-left corner of the array:
  0,   1,   2,   3,   4,   5,   6,   7,   8, ...
  0,   1,   3,   5,   9,   7,  15,  11,  27, ...
  0,   1,   5,   7,  25,  11,  35,  13, 125, ...
  0,   1,   7,  11,  49,  13,  77,  17, 343, ...
  0,   1,  11,  13, 121,  17, 143,  19,1331, ...
  0,   1,  13,  17, 169,  19, 221,  23,2197, ...
...
A(2,6) = A003961(A003961(6)) = p_{1+2} * p_{2+2} = p_3 * p_4 = 5 * 7 = 35, because 6 = 2*3 = p_1 * p_2.

Antti Karttunen, Table of n, a(n) for n = 0..10439; Antidiagonals n = 0..143, flattened

Taking every second column from column 2 onward gives array A246278 which is a permutation of natural numbers larger than 1.
Transpose: A242379.
Row 0: A001477, Row 1: A003961 (from 1 onward), Row 2: A357852 (from 1 onward), Row 3: A045968 (from 7 onward), Row 4: A045970 (from 11 onward).
Column 2: A000040 (primes), Column 3: A065091 (odd primes), Column 4: A001248 (squares of primes), Column 6: A006094 (products of two successive primes), Column 8: A030078 (cubes of primes).
Excluding column 0, a subtable of A297845.
Permutations whose formulas refer to this array: A122111, A241909, A242415, A242419, A246676, A246678, A246684.

A(0,j) = j, A(i,0) = 0, A(i > 0, j > 0) = A003961(A(i-1,j)).

For j > 0, A(i,j) = A297845(A000040(i+1),j) = A297845(j,A000040(i+1)). - Peter Munn, Sep 02 2025

A306697 Square array T(n, k) read by antidiagonals, n > 0 and k > 0: T(n, k) is obtained by applying a Minkowski sum to sets related to the Fermi-Dirac factorizations of n and of k (see Comments for precise definition).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 5, 4, 1, 1, 5, 9, 9, 5, 1, 1, 6, 7, 16, 7, 6, 1, 1, 7, 15, 25, 25, 15, 7, 1, 1, 8, 11, 36, 11, 36, 11, 8, 1, 1, 9, 27, 49, 35, 35, 49, 27, 9, 1, 1, 10, 25, 64, 13, 30, 13, 64, 25, 10, 1, 1, 11, 21, 81, 125, 77, 77, 125, 81

Offset: 1

Rémy Sigrist, Mar 05 2019

For any m > 0:
- let F(m) be the set of distinct Fermi-Dirac primes (A050376) with product m,
- for any i >=0 0 and j >= 0, let f(prime(i+1)^(2^i)) be the lattice point with coordinates X=i and Y=j (where prime(k) denotes the k-th prime number),
- f establishes a bijection from the Fermi-Dirac primes to the lattice points with nonnegative coordinates,
- let P(m) = { f(p) | p in F(m) },
- P establishes a bijection from the nonnegative integers to the set, say L, of finite sets of lattice points with nonnegative coordinates,
- let Q be the inverse of P,
- for any n > 0 and k > 0:
    T(n, k) = Q(P(n) + P(k))
              where "+" denotes the Minkowski addition on L.
This sequence has similarities with A297845, and their data sections almost match; T(6, 6) = 30, however A297845(6, 6) = 90.
This sequence has similarities with A067138; here we work on dimension 2, there in dimension 1.
This sequence as a binary operation distributes over A059896, whereas A297845 distributes over multiplication (A003991) and A329329 distributes over A059897. See the comment in A329329 for further description of the relationship between these sequences. - Peter Munn, Dec 19 2019

			Array T(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   5    9    7   15   11    27    25    21    13    45
    4|  1   4   9   16   25   36   49    64    81   100   121   144
    5|  1   5   7   25   11   35   13   125    49    55    17   175
    6|  1   6  15   36   35   30   77   216   225   210   143   540
    7|  1   7  11   49   13   77   17   343   121    91    19   539
    8|  1   8  27   64  125  216  343   128   729  1000  1331  1728
    9|  1   9  25   81   49  225  121   729   625   441   169  2025
   10|  1  10  21  100   55  210   91  1000   441   110   187  2100
   11|  1  11  13  121   17  143   19  1331   169   187    23  1573
   12|  1  12  45  144  175  540  539  1728  2025  2100  1573   720

Rémy Sigrist, PARI program for A306697
OEIS Wiki, "Fermi-Dirac representation" of n
Eric Weisstein's World of Mathematics, Distributive
Wikipedia, Minkowski addition

Columns (some differing for term 1) and equivalently rows: A003961(3), A000290(4), A045966(5), A045968(7), A045970(11).
Cf. A000040, A001221, A019565, A064547, A225546, A329050.
Related binary operations: A067138, A059896, A297845/A003991, A329329/A059897.

PARI
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\\ See Links section.
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For any m > 0, n > 0, k > 0, i >= 0, j >= 0:
- T(n, k) = T(k, n) (T is commutative),
- T(m, T(n, k)) = T(T(m, n), k) (T is associative),
- T(n, 1) = 1 (1 is an absorbing element for T),
- T(n, 2) = n (2 is an identity element for T),
- T(n, 3) = A003961(n),
- T(n, 4) = n^2 (A000290),
- T(n, 5) = A357852(n),
- T(n, 7) = A045968(n) (when n > 1),
- T(n, 11) = A045970(n) (when n > 1),
- T(n, 2^(2^i)) = n^(2^i),
- T(2^i, 2^j) = 2^A067138(i, j),
- T(A019565(i), A019565(j)) = A019565(A067138(i, j)),
- T(A000040(n), A000040(k)) = A000040(n + k - 1),
- T(2^(2^i), 2^(2^j)) = 2^(2^(i + j)),
- A001221(T(n, k)) <= A001221(n) * A001221(k),
- A064547(T(n, k)) <= A064547(n) * A064547(k).
From Peter Munn, Dec 05 2019:(Start)
T(A329050(i_1, j_1), A329050(i_2, j_2)) = A329050(i_1+i_2, j_1+j_2).
Equivalently, T(prime(i_1 - 1)^(2^(j_1)), prime(i_2 - 1)^(2^(j_2))) = prime(i_1+i_2 - 1)^(2^(j_1+j_2)), where prime(i) = A000040(i).
T(A059896(i,j), k) = A059896(T(i,k), T(j,k)) (T distributes over A059896).
T(A019565(i), 2^j) = A019565(i)^j.
T(A225546(i), A225546(j)) = A225546(T(i,j)).
(End)

A045966 a(1)=3; if n = Product p_i^e_i, n > 1, then a(n) = Product p_{i+2}^e_i.

Original entry on oeis.org

3, 5, 7, 25, 11, 35, 13, 125, 49, 55, 17, 175, 19, 65, 77, 625, 23, 245, 29, 275, 91, 85, 31, 875, 121, 95, 343, 325, 37, 385, 41, 3125, 119, 115, 143, 1225, 43, 145, 133, 1375, 47, 455, 53, 425, 539, 155, 59, 4375, 169, 605, 161, 475, 61, 1715, 187, 1625, 203, 185, 67

Offset: 1

N. J. A. Sloane

If we had a(1) = 1 (instead of 3), then this would be fully multiplicative with a(prime(k)) = prime(k+2) (see A357852). - Antti Karttunen, Jan 10 2020

From a puzzle proposed by Marc LeBrun.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

See A027748, A124010 for factorization data for n.
Cf. A000040, A003961, A101300.
Sequences with similar definitions: A045967, A045968, A045970, A126272.
A059896 is used to express relationship between terms of this sequence.
A357852 is a slightly better version. - N. J. A. Sloane, Oct 29 2022

a045966 1 = 3
a045966 n = product $ zipWith (^)
            (map a101300 $ a027748_row n) (a124010_row n)
-- Reinhard Zumkeller, Jun 03 2013, Dec 23 2011

a[1] = 3; a[n_] := With[{f = FactorInteger[n]}, Times @@ (Prime[PrimePi[f[[All, 1]]]+2]^f[[All, 2]])]; Array[a, 60] (* Jean-François Alcover, Jun 19 2015 *)

A045966(n) = if(1==n,3,my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(1+nextprime(1+f[i, 1]))); factorback(f)); \\ Antti Karttunen, Jan 10 2020

From Peter Munn, Dec 27 2019, for n >= 2, k >= 2: (Start)
a(n) = A003961^2(n).
a(n^k) = a(n)^k.
a(A003961(n)) = A003961(a(n)).
a(A059896(n,k)) = A059896(a(n), a(k)).
(End)

More terms from David W. Wilson

A045968 a(1)=5; for n >= 2, if n = Product p_i^e_i, then a(n) = Product p_{i+3}^e_i.

Original entry on oeis.org

5, 7, 11, 49, 13, 77, 17, 343, 121, 91, 19, 539, 23, 119, 143, 2401, 29, 847, 31, 637, 187, 133, 37, 3773, 169, 161, 1331, 833, 41, 1001, 43, 16807, 209, 203, 221, 5929, 47, 217, 253, 4459, 53, 1309, 59, 931, 1573, 259, 61, 26411, 289, 1183, 319, 1127, 67, 9317, 247

Offset: 1

N. J. A. Sloane

			If n = 9 = 3^2, then a(n) = 11^2 = 121 (since 11 is the third prime after 3).

From a puzzle proposed by Marc LeBrun.

Cf. A045965, A045966, A045967, A045969, A045970, A045971, A045972, A045973.

f[p_, e_] := NextPrime[p, 3]^e; a[1] = 5; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 19 2023 *)

More terms from David W. Wilson

Erroneous linear recurrence deleted by Harvey P. Dale, May 07 2018

A045973 a(1)=10; if n = Product p_i^e_i, n > 1, then a(n) = Product p_{i+1}^e_i * Product p_{i+3}^e_i.

Original entry on oeis.org

10, 21, 55, 441, 91, 1155, 187, 9261, 3025, 1911, 247, 24255, 391, 3927, 5005, 194481, 551, 63525, 713, 40131, 10285, 5187, 1073, 509355, 8281, 8211, 166375, 82467, 1271, 105105, 1591, 4084101, 13585, 11571, 17017, 1334025, 1927, 14973, 21505, 842751

Offset: 1

N. J. A. Sloane

From a puzzle proposed by Marc LeBrun.

Cf. A045965, A045966, A045967, A045968, A045969, A045970, A045971, A045972.

f[p_, e_] := NextPrime[p]^e * NextPrime[p, 3]^e; a[1] = 10; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 19 2023 *)

Sum_{n>=1} 1/a(n) = -9/10 + Product_{k>=1} (1+1/(prime(k)*prime(k+4)-1)) = 0.2602421684... . - Amiram Eldar, Sep 19 2023

More terms from David W. Wilson

A045969 a(1)=6; if n = Product p_i^e_i, n>1, then a(n) = Product p_{i+1}^e_i * Product p_{i+2}^e_i.

Original entry on oeis.org

6, 15, 35, 225, 77, 525, 143, 3375, 1225, 1155, 221, 7875, 323, 2145, 2695, 50625, 437, 18375, 667, 17325, 5005, 3315, 899, 118125, 5929, 4845, 42875, 32175, 1147, 40425, 1517, 759375, 7735, 6555, 11011, 275625, 1763, 10005, 11305, 259875, 2021, 75075

Offset: 1

N. J. A. Sloane

From a puzzle proposed by Marc LeBrun.

Cf. A045965, A045966, A045967, A045968, A045970, A045971, A045972, A045973.

f[p_, e_] := (NextPrime[p] * NextPrime[p, 2])^e; a[1] = 6; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 19 2023 *)

Sum_{n>=1} 1/a(n) = -5/6 + Product_{k>=2} (1+1/(prime(k)*prime(k+1)-1)) = 0.31383788... . - Amiram Eldar, Sep 19 2023

More terms from David W. Wilson

A045971 a(1)=8; if n = Product p_i^e_i, n > 1, then a(n) = Product p_{i+1}^{e_i+2}.

Original entry on oeis.org

8, 27, 125, 81, 343, 3375, 1331, 243, 625, 9261, 2197, 10125, 4913, 35937, 42875, 729, 6859, 16875, 12167, 27783, 166375, 59319, 24389, 30375, 2401, 132651, 3125, 107811, 29791, 1157625, 50653, 2187, 274625, 185193, 456533, 50625, 68921, 328509, 614125

Offset: 1

N. J. A. Sloane

From a puzzle proposed by Marc LeBrun.

Cf. A045965, A045966, A045967, A045968, A045969, A045970, A045972, A045973, A065483.

f[p_, e_] := NextPrime[p]^(e + 2); a[1] = 8; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 19 2023 *)

Sum_{n>=1} 1/a(n) = (4/5) * A065483 - 7/8 = 0.196827322859... . - Amiram Eldar, Sep 19 2023

More terms from David W. Wilson

A045972 a(1)=9; if n = Product p_i^e_i, n > 1, then a(n) = Product p_{i+2}^{e_i+1}.

Original entry on oeis.org

9, 25, 49, 125, 121, 1225, 169, 625, 343, 3025, 289, 6125, 361, 4225, 5929, 3125, 529, 8575, 841, 15125, 8281, 7225, 961, 30625, 1331, 9025, 2401, 21125, 1369, 148225, 1681, 15625, 14161, 13225, 20449, 42875, 1849, 21025, 17689, 75625, 2209, 207025

Offset: 1

N. J. A. Sloane

From a puzzle proposed by Marc LeBrun.

Cf. A045965, A045966, A045967, A045968, A045969, A045970, A045971, A045973, A082695.

f[p_, e_] := NextPrime[p, 2]^(e + 1); a[1] = 9; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 19 2023 *)

Sum_{n>=1} 1/a(n) = (4/7) * (zeta(2)*zeta(3)/zeta(6)) - 8/9 = 0.221737646437... . - Amiram Eldar, Sep 19 2023

More terms from David W. Wilson

cp's OEIS Frontend

A242378 Square array read by antidiagonals: to obtain A(i,j), replace each prime factor prime(k) in prime factorization of j with prime(k+i).

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A306697 Square array T(n, k) read by antidiagonals, n > 0 and k > 0: T(n, k) is obtained by applying a Minkowski sum to sets related to the Fermi-Dirac factorizations of n and of k (see Comments for precise definition).

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A045966 a(1)=3; if n = Product p_i^e_i, n > 1, then a(n) = Product p_{i+2}^e_i.

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A045968 a(1)=5; for n >= 2, if n = Product p_i^e_i, then a(n) = Product p_{i+3}^e_i.

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A045973 a(1)=10; if n = Product p_i^e_i, n > 1, then a(n) = Product p_{i+1}^e_i * Product p_{i+3}^e_i.

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A045969 a(1)=6; if n = Product p_i^e_i, n>1, then a(n) = Product p_{i+1}^e_i * Product p_{i+2}^e_i.

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A045971 a(1)=8; if n = Product p_i^e_i, n > 1, then a(n) = Product p_{i+1}^{e_i+2}.

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A045972 a(1)=9; if n = Product p_i^e_i, n > 1, then a(n) = Product p_{i+2}^{e_i+1}.

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