A297002 -id:A297002 - cp's OEIS frontend

A066207 All primes that divide n are of the form prime(2k), where prime(k) is k-th prime.

1, 3, 7, 9, 13, 19, 21, 27, 29, 37, 39, 43, 49, 53, 57, 61, 63, 71, 79, 81, 87, 89, 91, 101, 107, 111, 113, 117, 129, 131, 133, 139, 147, 151, 159, 163, 169, 171, 173, 181, 183, 189, 193, 199, 203, 213, 223, 229, 237, 239, 243, 247, 251, 259, 261, 263, 267, 271, 273

Offset: 1

Leroy Quet, Dec 16 2001

nonn

The partitions into even parts, encoded by their Heinz numbers. We define the Heinz number of a partition p = [p_1, p_2, ..., p_r] as Product(p_j-th prime, j=1..r) (concept used by Alois P. Heinz in A215366 as an "encoding" of a partition). For example, for the partition [1, 1, 2, 4, 10] we get 2*2*3*7*29 = 2436. Example: 63 ( = 3*3*7) is in the sequence because it is the Heinz number of the partition [2, 2, 4]. - Emeric Deutsch, May 19 2015
Numbers divisible only by primes with even indices (A031215). - Michael De Vlieger, Dec 11 2017
Numbers that are not divisible by any prime with an odd index (A031368). - Antti Karttunen, Jul 18 2020
For every positive integer m there exists a unique ordered pair of positive integers (j,k) such that m = a(j)*A066208(k). - Christopher Scussel, Jul 01 2023

			39 is included because 3 * 13 = prime(2) * prime(6) and 2 and 6 are both even.

Antti Karttunen, Table of n, a(n) for n = 1..20000 (original first 1000 terms from Harry J. Smith)

Cf. A031215, A031368, A066208, A215366.
Cf. A297002 (a permutation).
Numbers in the odd bisection of A336321.

Select[Range[273], AllTrue[PrimePi@ FactorInteger[#][[All, 1]], EvenQ] &] (* Michael De Vlieger, Dec 11 2017, range adjusted by Antti Karttunen, Jul 18 2020 *)

{ n=0; for (m=2, 10^9, f=factor(m); b=1; for(i=1, matsize(f)[1], if (primepi(f[i, 1])%2, b=0; break)); if (b, write("b066207.txt", n++, " ", m); if (n==1000, return)) ) } \\ Harry J. Smith, Feb 06 2010

isA066207(n) = (!#select(p -> (primepi(p)%2), factor(n)[,1])); \\ Antti Karttunen, Jul 18 2020

Offset changed from 0 to 1 by Harry J. Smith, Feb 06 2010
a(53)-a(58) from Harry J. Smith, Feb 06 2010
a(1) = 1 inserted (and the indexing of the rest of terms changed) by Antti Karttunen, Jul 18 2020

A353397 Replace prime(k) with prime(2^k) in the prime factorization of n.

Original entry on oeis.org

1, 3, 7, 9, 19, 21, 53, 27, 49, 57, 131, 63, 311, 159, 133, 81, 719, 147, 1619, 171, 371, 393, 3671, 189, 361, 933, 343, 477, 8161, 399, 17863, 243, 917, 2157, 1007, 441, 38873, 4857, 2177, 513, 84017, 1113, 180503, 1179, 931, 11013, 386093, 567, 2809, 1083

Offset: 1

Gus Wiseman, May 17 2022

			The terms together with their prime indices begin:
      1: {}
      3: {2}
      7: {4}
      9: {2,2}
     19: {8}
     21: {2,4}
     53: {16}
     27: {2,2,2}
     49: {4,4}
     57: {2,8}
    131: {32}
     63: {2,2,4}

Amiram Eldar, Table of n, a(n) for n = 1..397 (calculated using the b-file at A033844)

These are the positions of first appearances in A353394.
A001222 counts prime factors with multiplicity, distinct A001221.
A003963 gives product of prime indices, counted by A339095.
A033844 lists primes indexed by powers of 2.
A056239 adds up prime indices, row sums of A112798 and A296150.
A124010 gives prime signature, sorted A118914, product A005361.
A181819 gives prime shadow, firsts A181821, relatively prime A325131.
Equivalent sequence with prime(2*k) instead of prime(2^k): A297002.
Cf. A003586, A005117, A130091, A182850, A289509, A324850, A325755, A353393, A353395, A353399.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Times@@Prime/@(2^primeMS[n]),{n,100}]

a(n) = my(f=factor(n)); for(k=1, #f~, f[k,1] = prime(2^primepi(f[k,1]))); factorback(f); \\ Michel Marcus, May 20 2022

from math import prod
from sympy import prime, primepi, factorint
def A353397(n): return prod(prime(2**primepi(p))**e for p, e in factorint(n).items()) # Chai Wah Wu, May 20 2022

If n = prime(e_1)...prime(e_k), then a(n) = prime(2^(e_1))...prime(2^(e_k)).

Sum_{n>=1} 1/a(n) = 1/Product_{k>=1} (1 - 1/prime(2^k)) = 1.90812936178871496289... . - Amiram Eldar, Dec 09 2022

A357975 Divide all prime indices by 2, round down, and take the number with those prime indices, assuming prime(0) = 1.

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 3, 1, 4, 2, 3, 2, 5, 3, 4, 1, 5, 4, 7, 2, 6, 3, 7, 2, 4, 5, 8, 3, 11, 4, 11, 1, 6, 5, 6, 4, 13, 7, 10, 2, 13, 6, 17, 3, 8, 7, 17, 2, 9, 4, 10, 5, 19, 8, 6, 3, 14, 11, 19, 4, 23, 11, 12, 1, 10, 6, 23, 5, 14, 6, 29, 4, 29, 13, 8, 7, 9, 10, 31

Offset: 1

Gus Wiseman, Oct 23 2022

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
Also the Heinz number of the part-wise half (rounded down) of the partition with Heinz number n, where the Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
Each n appears A000005(n) times at odd positions (infinitely many at even). To see this, note that our transformation does not distinguish between A066207 and A066208.

			The prime indices of n = 1501500 are {1,1,2,3,3,3,4,5,6}, so the prime indices of a(n) are {1,1,1,1,2,2,3}; hence we have a(1501500) = 720.
The 6 odd positions of 2124 are: 63, 99, 105, 165, 175, 275, with prime indices:
   63: {2,2,4}
   99: {2,2,5}
  105: {2,3,4}
  165: {2,3,5}
  175: {3,3,4}
  275: {3,3,5}

SiYang Hu, Table of n, a(n) for n = 1..10000

Positions of 1's are A000079.
Positions of 2's are 3 and A164095.
Positions of first appearances are A297002, sorted A066207.
A004526 is floor(n/2), with an extra first zero.
A056239 adds up prime indices, row-sums of A112798.
A109763 lists primes of index floor(n/2).
Cf. A003961, A064988, A064989, A076610, A215366, A248601, A357980.

Table[Times@@(If[#1<=2,1,Prime[Floor[PrimePi[#1]/2]]^#2]&@@@FactorInteger[n]),{n,100}]

Completely multiplicative with a(prime(2k)) = prime(k) and a(prime(2k+1)) = prime(k). Cf. A297002.

a(prime(n)) = A109763(n-1).

A336321 a(n) = A122111(A225546(n)).

Original entry on oeis.org

1, 2, 3, 4, 7, 5, 19, 6, 9, 11, 53, 10, 131, 23, 13, 8, 311, 15, 719, 22, 29, 59, 1619, 14, 49, 137, 21, 46, 3671, 17, 8161, 12, 61, 313, 37, 25, 17863, 727, 139, 26, 38873, 31, 84017, 118, 39, 1621, 180503, 20, 361, 77, 317, 274, 386093, 33, 71, 58, 733, 3673, 821641, 34, 1742537, 8167, 87, 18, 151, 67, 3681131, 626, 1627, 41, 7754077, 35, 16290047

Offset: 1

Antti Karttunen and Peter Munn, Jul 17 2020

nonn

A122111 and A225546 are both self-inverse permutations of the positive integers based on prime factorizations, and they share further common properties. For instance, they map the prime numbers to powers of 2: A122111 maps the k-th prime to 2^k, whereas A225546 maps it to 2^2^(k-1).

In composing these permutations, this sequence maps the squarefree numbers, as listed in A019565, to the prime numbers in increasing order; and the list of powers of 2 to the "normal" numbers (A055932), as listed in A057335.

			From _Peter Munn_, Jan 04 2021: (Start)
In this set of examples we consider [a(n)] as a function a(.) with an inverse, a^-1(.).
First, a table showing mapping of the powers of 2:
  n     a^-1(2^n) =    2^n =        a(2^n) =
        A001146(n-1)   A000079(n)   A057335(n)
  0             (1)         1            1
  1               2         2            2
  2               4         4            4
  3              16         8            6
  4             256        16            8
  5           65536        32           12
  6      4294967296        64           18
  ...
Next, a table showing mapping of the squarefree numbers, as listed in A019565 (a lexicographic ordering by prime factors):
  n   a^-1(A019565(n))   A019565(n)      a(A019565(n))   a^2(A019565(n))
      Cf. {A337533}      Cf. {A005117}   = prime(n)      = A033844(n-1)
  0              1               1             (1)               (1)
  1              2               2               2                 2
  2              3               3               3                 3
  3              8               6               5                 7
  4              6               5               7                19
  5             12              10              11                53
  6             18              15              13               131
  7            128              30              17               311
  8              5               7              19               719
  9             24              14              23              1619
  ...
As sets, the above columns are A337533, A005117, A008578, {1} U A033844.
Similarly, we get bijections between sets A000290\{0} -> {1} U A070003; and {1} U A335740 -> A005408 -> A066207.
(End)

A122111 composed with A225546.
Cf. A336322 (inverse permutation).
Other sequences used in a definition of this sequence: A000040, A000188, A019565, A248663, A253550, A253560.
Sequences used to express relationship between terms of this sequence: A003159, A003961, A297002, A334747.
Cf. A057335.
A mapping between the binary tree sequences A334866 and A253563.
Lists of sets (S_1, S_2, ... S_j) related by the bijection defined by the sequence: (A000290\{0}, {1} U A070003), ({1} U A001146, A000079, A055932), ({1} U A335740, A005408, A066207), (A337533, A005117, A008578, {1} U A033844).

a(n) = A122111(A225546(n)).
Alternative definition: (Start)
Write n = m^2 * A019565(j), where m = A000188(n), j = A248663(n).
a(1) = 1; otherwise for m = 1, a(n) = A000040(j), for m > 1, a(n) = A253550^j(A253560(a(m))).
(End)
a(A000040(m)) = A033844(m-1).
a(A001146(m)) = 2^(m+1).
a(2^n) = A057335(n).
a(n^2) = A253560(a(n)).
For n in A003159, a(2n) = b(a(n)), where b(1) = 2, b(n) = A253550(n), n >= 2.
More generally, a(A334747(n)) = b(a(n)).
a(A003961(n)) = A297002(a(n)).
a(A334866(m)) = A253563(m).

A350066 Symmetric square array A(n,k) = A122111(A122111(n) * A122111(k)), n >= 1, k >= 1, read by antidiagonals.

Original entry on oeis.org

1, 2, 2, 3, 3, 3, 4, 5, 5, 4, 5, 6, 7, 6, 5, 6, 7, 10, 10, 7, 6, 7, 10, 11, 9, 11, 10, 7, 8, 11, 14, 14, 14, 14, 11, 8, 9, 12, 13, 15, 13, 15, 13, 12, 9, 10, 15, 20, 22, 22, 22, 22, 20, 15, 10, 11, 14, 21, 18, 17, 21, 17, 18, 21, 14, 11, 12, 13, 22, 25, 28, 26, 26, 28, 25, 22, 13, 12, 13, 20, 17, 21, 33, 30, 19, 30, 33, 21, 17, 20, 13

Offset: 1

Antti Karttunen, Dec 13 2021

A122111 is a self-inverse permutation, so this array represents a binary operation A(.,.) over the positive integers that is isomorphic to multiplication. Its primes are the positive powers of 2 (as defined by standard multiplication): 2, 4, 8, 16, 32, ... . The positive powers of 2, as defined by A(.,.), are the prime numbers as we usually understand them: 2, 3, 5, 7, 11, ... . - Peter Munn, Aug 04 2022

			The top left 15 X 15 corner of the array:
   1,  2,  3,  4,  5,  6,  7,  8,  9, 10, 11,  12, 13, 14,  15,
   2,  3,  5,  6,  7, 10, 11, 12, 15, 14, 13,  20, 17, 22,  21,
   3,  5,  7, 10, 11, 14, 13, 20, 21, 22, 17,  28, 19, 26,  33,
   4,  6, 10,  9, 14, 15, 22, 18, 25, 21, 26,  30, 34, 33,  35,
   5,  7, 11, 14, 13, 22, 17, 28, 33, 26, 19,  44, 23, 34,  39,
   6, 10, 14, 15, 22, 21, 26, 30, 35, 33, 34,  42, 38, 39,  55,
   7, 11, 13, 22, 17, 26, 19, 44, 39, 34, 23,  52, 29, 38,  51,
   8, 12, 20, 18, 28, 30, 44, 27, 50, 42, 52,  45, 68, 66,  70,
   9, 15, 21, 25, 33, 35, 39, 50, 49, 55, 51,  70, 57, 65,  77,
  10, 14, 22, 21, 26, 33, 34, 42, 55, 39, 38,  66, 46, 51,  65,
  11, 13, 17, 26, 19, 34, 23, 52, 51, 38, 29,  68, 31, 46,  57,
  12, 20, 28, 30, 44, 42, 52, 45, 70, 66, 68,  63, 76, 78, 110,
  13, 17, 19, 34, 23, 38, 29, 68, 57, 46, 31,  76, 37, 58,  69,
  14, 22, 26, 33, 34, 39, 38, 66, 65, 51, 46,  78, 58, 57,  85,
  15, 21, 33, 35, 39, 55, 51, 70, 77, 65, 57, 110, 69, 85,  91,

Index entries for sequences computed from indices in prime factorization

Cf. A122111, A297002 (main diagonal), A253550 (after its initial term, gives row 2 / column 2 from the second term onward).
See the formula section for the relationships with A003961, A061142.
Cf. also A003991, A129595, A331590.

up_to = 105;
A122111(n) = if(1==n,n,my(f=factor(n), es=Vecrev(f[,2]),is=concat(apply(primepi,Vecrev(f[,1])),[0]),pri=0,m=1); for(i=1, #es, pri += es[i]; m *= prime(pri)^(is[i]-is[1+i])); (m));
A350066sq(n,k) = A122111(A122111(n)*A122111(k));
A350066list(up_to) = { my(v = vector(up_to), i=0); for(a=1,oo, for(col=1,a, i++; if(i > up_to, return(v)); v[i] = A350066sq(col,(a-(col-1))))); (v); };
v350066 = A350066list(up_to);
A350066(n) = v350066[n]; \\ Antti Karttunen, Dec 13 2021

A(n, A061142(n)) = A003961(n). - Peter Munn, Aug 04 2022

A248601 Square array T(n, k) read by antidiagonals, n > 0 and k > 0: for any number n > 0, let f(n) be the function that associates k to the prime(k)-adic valuation of n (where prime(k) denotes the k-th prime); f establishes a bijection between the positive numbers and the arithmetic functions with nonnegative integer values and a finite number of nonzero values; let g be the inverse of f; T(n, k) = g(f(n) * f(k)) (where i * j denotes the Dirichlet convolution of i and j).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 7, 4, 1, 1, 5, 9, 9, 5, 1, 1, 6, 13, 16, 13, 6, 1, 1, 7, 21, 25, 25, 21, 7, 1, 1, 8, 19, 36, 23, 36, 19, 8, 1, 1, 9, 27, 49, 65, 65, 49, 27, 9, 1, 1, 10, 49, 64, 37, 126, 37, 64, 49, 10, 1, 1, 11, 39, 81, 125, 133, 133, 125

Offset: 1

Rémy Sigrist, Dec 21 2017

For any n > 0, f(n) corresponds to the function k -> A249344(k, n).
For any n > 0 and m > 0, f(n * m) = f(n) + f(m).
Also, f(1) = A000004 and f(2) corresponds to k -> A000007(k-1).
The function f can be naturally extended to the set of positive rational numbers: if r = u/v (not necessarily in reduced form), then f(r) = f(u) - f(v); as such, f is an homomorphism from the multiplicative group of positive rational numbers to the additive group of arithmetic functions with integer values and a finite number of nonzero values.
For any arithmetic function with integer values and a finite number of nonzero values j, g(j) = Product_{k > 0} A000040(k)^j(k).
See A296857 for the main diagonal of T.

			Array T(n, k) begins:
  n\k|  1   2   3    4    5    6    7     8     9    10
  ---+-------------------------------------------------
    1|  1   1   1    1    1    1    1     1     1     1  -> A000012
    2|  1   2   3    4    5    6    7     8     9    10  -> A000027
    3|  1   3   7    9   13   21   19    27    49    39  -> A297002
    4|  1   4   9   16   25   36   49    64    81   100  -> A000290
    5|  1   5  13   25   23   65   37   125   169   115
    6|  1   6  21   36   65  126  133   216   441   390
    7|  1   7  19   49   37  133   53   343   361   259
    8|  1   8  27   64  125  216  343   512   729  1000  -> A000578
    9|  1   9  49   81  169  441  361   729  2401  1521
   10|  1  10  39  100  115  390  259  1000  1521  1150

Wikipedia, Dirichlet convolution.

Cf. A000004, A000007, A000012 (first row/column), A000027 (second row/column), A000040, A000290 (fourth row/column), A000578 (eighth row/column), A001221, A001222, A055396, A061395, A249344, A289508, A296857 (main diagonal), A297002 (third row/column).

T(n,k) = my(fn=factor(n), pn=apply(primepi,fn[,1]~), fk=factor(k), pk=apply(primepi,fk[,1]~)); prod(i=1, #pn, prod(j=1, #pk, prime(pn[i]*pk[j])^(fn[i,2]*fk[j,2])))

T is completely multiplicative in both parameters:
- for any n > 0
- and k > 0 with prime factorization Prod_{i > 0} prime(i)^e_i:
- T(prime(n), k) = T(k, prime(n)) = Prod_{i > 0} prime(n * i)^e_i.
For any m > 0, n > 0 and k > 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, 2^i) = n^i for any i >= 0,
- T(n, 4) = n^2 (A000290),
- T(n, 8) = n^3 (A000578),
- T(n, 3) = A297002(n),
- T(n, 3^i) = A297002(n)^i for any i >= 0,
- A001221(T(n, k)) <= A001221(n) * A001221(k),
- A001222(T(n, k)) = A001222(n) * A001222(k),
- A055396(T(n, k)) = A055396(n) * A055396(k),
- A061395(T(n, k)) = A061395(n) * A061395(k),
- A289508(T(n, k)) = A289508(n) * A289508(k),
- T(A000040(n), A000040(k)) = A000040(n * k),
- T(A000040(n)^i, A000040(k)^j) = A000040(n * k)^(i * j) for any i >= 0 and j >= 0.

A319525 Completely multiplicative with a(prime(k)) = prime(2*k - 1) (where prime(k) denotes the k-th prime).

Original entry on oeis.org

1, 2, 5, 4, 11, 10, 17, 8, 25, 22, 23, 20, 31, 34, 55, 16, 41, 50, 47, 44, 85, 46, 59, 40, 121, 62, 125, 68, 67, 110, 73, 32, 115, 82, 187, 100, 83, 94, 155, 88, 97, 170, 103, 92, 275, 118, 109, 80, 289, 242, 205, 124, 127, 250, 253, 136, 235, 134, 137, 220

Offset: 1

Rémy Sigrist, Sep 22 2018

This sequence is a permutation of A066208.

See A297002 for a similar sequence.

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences computed from indices in prime factorization

Cf. A000079, A066208, A297002.

Cf. also A166651.

a(n) = my (f=factor(n)); prod(i=1, #f~, prime(2*primepi(f[i, 1]) - 1)^f[i, 2])

a(n) = A064989(A297002(n)).

a(n) >= n with equality iff n is a power of 2 (A000079).

A348067 Matula-Goebel tree number of tree n with a new leaf added below each existing vertex.

Original entry on oeis.org

2, 6, 26, 18, 202, 78, 122, 54, 338, 606, 2462, 234, 794, 366, 2626, 162, 1346, 1014, 502, 1818, 1586, 7386, 4546, 702, 20402, 2382, 4394, 1098, 8914, 7878, 43954, 486, 32006, 4038, 12322, 3042, 2962, 1506, 10322, 5454, 12178, 4758, 4946, 22158, 34138, 13638

Offset: 1

Kevin Ryde, Oct 01 2021

nonn

k times nested a(a(...a(1))) = A076146(k+1) is the Matula-Goebel number of the binomial tree order k constructed by an "expansion" method starting from a singleton and successively adding a new leaf under every vertex.

			tree n=6   tree a(6) = 78
  R             R___        root R
  | \           |\  \
  A  B          A @  B      new vertices
  |             |\    \     "@" below each
  C             C @    @    existing
                 \
                  @

Cf. A027746 (prime factors), A076146 (binomial tree).

Cf. A297002 (add leaves under children of the root).

a(n) = my(f=factor(n)); 2*factorback([prime(self()(primepi(p))) | p<-f[,1]], f[,2]);

a(n) = 2 * Product_{i=1..k} prime(a(primepi(p[i]))), where n = p[1]*...*p[k] is the prime factorization of n with multiplicity (A027746).

A352001 Square array A(n, k), n, k >= 1, read by antidiagonals upwards; A(n, k) = Product_{ i >= 1 } prime(k*i)^e_i where n = Product_{ i >= 1 } prime(i)^e_i (where prime(i) denotes the i-th prime number).

Original entry on oeis.org

1, 2, 1, 3, 3, 1, 4, 7, 5, 1, 5, 9, 13, 7, 1, 6, 13, 25, 19, 11, 1, 7, 21, 23, 49, 29, 13, 1, 8, 19, 65, 37, 121, 37, 17, 1, 9, 27, 37, 133, 47, 169, 43, 19, 1, 10, 49, 125, 53, 319, 61, 289, 53, 23, 1, 11, 39, 169, 343, 71, 481, 73, 361, 61, 29, 1

Offset: 1

Rémy Sigrist, Feb 27 2022

In other words, in prime factorization of n, replace prime(i) by prime(k*i).

For any k >= 1, n -> A(n, k) is completely multiplicative.

			Square array A(n, k) begins:
  n\k|   1   2    3    4     5     6     7     8      9     10
  ------------------------------------------------------------
    1|   1   1    1    1     1     1     1     1      1      1
    2|   2   3    5    7    11    13    17    19     23     29
    3|   3   7   13   19    29    37    43    53     61     71
    4|   4   9   25   49   121   169   289   361    529    841
    5|   5  13   23   37    47    61    73    89    103    113
    6|   6  21   65  133   319   481   731  1007   1403   2059
    7|   7  19   37   53    71    89   107   131    151    173
    8|   8  27  125  343  1331  2197  4913  6859  12167  24389
    9|   9  49  169  361   841  1369  1849  2809   3721   5041
   10|  10  39  115  259   517   793  1241  1691   2369   3277

Michael De Vlieger, Table of n, a(n) for n = 1..11325 (antidiagonals n = 1..150 flattened)

Cf. A000040, A001248, A031215, A051128, A297002, A351995.

Main diagonal gives A352028.

A:= (n, k)-> mul(ithprime(k*numtheory[pi](i[1]))^i[2], i=ifactors(n)[2]):
seq(seq(A(d+1-k, k), k=1..d), d=1..12);  # Alois P. Heinz, Feb 28 2022

Table[If[# == 1, 1, Times @@ Map[Prime[#3*PrimePi[#1]]^#2 & @@ Flatten[{#1, k}] &, FactorInteger[#]]] &[n - k + 1], {n, 11}, {k, n}] // Flatten (* Michael De Vlieger, Feb 28 2022 *)

A(n,k) = { my (f=factor(n)); prod (i=1, #f~, prime(k * primepi(f[i,1])) ^ f[i,2]) }

A(A(n, k), k') = A(n, k*k').
A(n, 1) = n.
A(n, 2) = A297002(n).
A(1, k) = 1.
A(2, k) = prime(k) (the k-th prime number).
A(3, k) = prime(2*k) = A031215(k).
A(4, k) = A001248(k).

A319523 Square array T(n, k) (n >= 1, k >= 1) read by antidiagonals upwards: T(n, k) is the unique positive integer m such that A319521(m) = n and A319522(m) = k.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 4, 15, 14, 9, 11, 12, 35, 18, 13, 10, 33, 28, 45, 26, 21, 17, 30, 77, 36, 65, 42, 19, 8, 51, 70, 99, 52, 105, 38, 27, 25, 24, 119, 90, 143, 84, 95, 54, 49, 22, 75, 56, 153, 130, 231, 76, 135, 98, 39, 23, 66, 175, 72, 221, 210, 209, 108, 245

Offset: 1

Rémy Sigrist, Sep 22 2018

			Array T(n, k) begins:
  n\k|    1    2    3    4    5    6    7    8    9   10   11   12
  ---+------------------------------------------------------------
    1|    1    3    7    9   13   21   19   27   49   39   29   63
    2|    2    6   14   18   26   42   38   54   98   78   58  126
    3|    5   15   35   45   65  105   95  135  245  195  145  315
    4|    4   12   28   36   52   84   76  108  196  156  116  252
    5|   11   33   77   99  143  231  209  297  539  429  319  693
    6|   10   30   70   90  130  210  190  270  490  390  290  630
    7|   17   51  119  153  221  357  323  459  833  663  493 1071
    8|    8   24   56   72  104  168  152  216  392  312  232  504
    9|   25   75  175  225  325  525  475  675 1225  975  725 1575
   10|   22   66  154  198  286  462  418  594 1078  858  638 1386

Cf. A001221, A001222, A061898, A275407 (main diagonal), A297002 (first row), A319521, A319522, A319525 (first column).

T(n,k) = my (fn=factor(n), fk=factor(k)); prod(i=1, #fn~, prime(2*primepi(fn[i,1])-1)^fn[i,2]) * prod(i=1, #fk~, prime(2*primepi(fk[i,1]))^fk[i,2])

T(n, k) = A061898(T(k, n)).
T(n, n) = A275407(n).
T(n, 1) = A319525(n).
T(1, k) = A297002(k).
T(n, k) = T(n, 1) * T(1, k) = A319525(n) * A297002(k).
A001221(T(n, k)) = A001221(n) + A001221(k).
A001222(T(n, k)) = A001222(n) + A001222(k).

cp's OEIS Frontend

A066207 All primes that divide n are of the form prime(2k), where prime(k) is k-th prime.

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A353397 Replace prime(k) with prime(2^k) in the prime factorization of n.

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A357975 Divide all prime indices by 2, round down, and take the number with those prime indices, assuming prime(0) = 1.

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A336321 a(n) = A122111(A225546(n)).

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A350066 Symmetric square array A(n,k) = A122111(A122111(n) * A122111(k)), n >= 1, k >= 1, read by antidiagonals.

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A319525 Completely multiplicative with a(prime(k)) = prime(2*k - 1) (where prime(k) denotes the k-th prime).

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A348067 Matula-Goebel tree number of tree n with a new leaf added below each existing vertex.

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