A122111 -id:A122111 - cp's OEIS frontend

A253883 Permutation of natural numbers: a(n) = A243505(A122111(n)).

1, 2, 8, 4, 6, 16, 1024, 3, 64, 256, 20, 128, 1073741824, 16384, 18, 32, 240, 512, 3538944, 48, 524288, 288, 8640, 5, 32768, 4398046511104, 27, 2097152, 214990848, 65536, 660, 12, 162, 37778931862957161709568, 134217728, 4096

Offset: 1

Antti Karttunen, Jan 17 2015

nonn

Term a(37) has 310 decimal digits and its binary representation is 1028 bits long.

Index entries for sequences that are permutations of the natural numbers

Inverse: A253884.
Similar permutations: A253791, A253891, A122111, A243505.
Cf. also A253890.

(define (A253883 n) (A243505 (A122111 n)))

a(n) = A243505(A122111(n)).

A280491 a(n) = gcd(n,A122111(n)).

Original entry on oeis.org

1, 2, 1, 1, 1, 6, 1, 1, 9, 2, 1, 2, 1, 2, 3, 1, 1, 3, 1, 20, 3, 2, 1, 2, 1, 2, 1, 4, 1, 30, 1, 1, 3, 2, 1, 3, 1, 2, 3, 4, 1, 6, 1, 4, 5, 2, 1, 2, 1, 5, 3, 4, 1, 1, 1, 56, 3, 2, 1, 6, 1, 2, 1, 1, 1, 6, 1, 4, 3, 10, 1, 3, 1, 2, 75, 4, 1, 6, 1, 4, 1, 2, 1, 84, 1, 2, 3, 8, 1, 10, 1, 4, 3, 2, 1, 2, 1, 1, 1, 1, 1, 6, 1, 8, 15, 2, 1, 1, 1, 10, 3, 8, 1, 6, 1, 4, 1, 2

Offset: 1

Antti Karttunen, Jan 09 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..8192

Cf. A122111, A280490, also A280489.

Cf. A088902 (the fixed points, n for which a(n) = n).

(definec (A280491 n) (gcd n (A122111 n)))

a(n) = gcd(n,A122111(n)).
a(n) = n / A280490(n).
Other identities. For all n >= 1:
a(A122111(n)) = a(n).

A322863 Permutation of natural numbers: a(0) = 1; for n >= 1, a(n) = A005940(1+A122111(n)).

Original entry on oeis.org

1, 2, 3, 5, 4, 7, 9, 11, 6, 10, 25, 13, 15, 17, 49, 21, 8, 19, 16, 23, 35, 55, 121, 29, 27, 36, 169, 50, 77, 31, 81, 37, 12, 91, 289, 225, 30, 41, 361, 187, 125, 43, 625, 47, 143, 147, 529, 53, 45, 154, 90, 247, 221, 59, 28, 1225, 343, 391, 841, 61, 105, 67, 961, 605, 18, 5929, 2401, 71, 323, 551, 525, 73, 22, 79, 1369, 84, 437, 429, 14641, 83

Offset: 0

Antti Karttunen, Dec 30 2018

nonn

Note the indexing: the domain starts from 0, but the range excludes zero.

Inverse permutation: A322864.

Cf. A005940, A122111, A322866, A322867.

A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t };
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A122111(n) = if(1==n,n,prime(bigomega(n))*A122111(A064989(n)));
A322863(n) = if(!n,1,A005940(1+A122111(n)));

a(0) = 1; for n >= 1, a(n) = A005940(1+A122111(n)).
For all n >= 1:
a(prime(n)) = prime(1+n).
A001222(a(n)) = A322867(n).

A331170 a(n) = min(n, A122111(n)), where A122111 conjugates the prime factorization of n.

Original entry on oeis.org

1, 2, 3, 3, 5, 6, 7, 5, 9, 10, 11, 10, 13, 14, 15, 7, 17, 15, 19, 20, 21, 22, 23, 14, 25, 26, 25, 28, 29, 30, 31, 11, 33, 34, 35, 21, 37, 38, 39, 28, 41, 42, 43, 44, 45, 46, 47, 22, 49, 45, 51, 52, 53, 35, 55, 56, 57, 58, 59, 42, 61, 62, 63, 13, 65, 66, 67, 68, 69, 70, 71, 33, 73, 74, 75, 76, 77, 78, 79, 44, 49, 82, 83, 84, 85, 86, 87, 88, 89, 70, 91, 92, 93, 94, 95, 26, 97, 98, 99, 63

Offset: 1

Antti Karttunen, Jan 12 2020

nonn

For all i, j:
  a(i) = a(j) => A056239(i) = A056239(j),
  a(i) = a(j) => A243503(i) = A243503(j).

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences computed from indices in prime factorization

Cf. A056239, A064989, A122111, A243503.

A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A122111(n) = if(1==n,n,prime(bigomega(n))*A122111(A064989(n)));
A331170(n) = min(n, A122111(n));

A336119 Numbers k such that A122111(k) [the conjugated prime factorization of k] is a square or twice a square.

Original entry on oeis.org

1, 2, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 27, 29, 31, 33, 37, 39, 41, 43, 45, 47, 49, 51, 53, 57, 59, 61, 63, 67, 69, 71, 73, 77, 79, 81, 83, 87, 89, 91, 93, 97, 99, 101, 103, 107, 109, 111, 113, 117, 119, 123, 127, 129, 131, 133, 135, 137, 139, 141, 147, 149, 151, 153, 157, 159, 161, 163, 167, 169, 171, 173

Offset: 1

Antti Karttunen, Jul 14 2020

nonn

Sequence A122111(A028982(k)), k >= 1, sorted into ascending order.

Cf. A000040, A066207 (subsequences), A335909 (characteristic function).
Cf. A028982, A031215, A031368, A053866, A122111.
Positions of odd terms in A323173, positions of zeros in A336120 and A336121, positions of ones in A336312.

isA336119(n) = (A335909(n)); \\ See code in A335909.

A336121 a(1) = 0, and for n > 1, a(n) = [A122111(n) == 3 (mod 4)] + a(A253553(n)).

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 2, 0, 2, 0, 1, 0, 1, 0, 2, 1, 1, 0, 1, 0, 2, 0, 3, 0, 1, 1, 1, 0, 1, 0, 2, 0, 2, 0, 1, 0, 1, 0, 3, 0, 1, 0, 1, 0, 3, 1, 2, 0, 1, 0, 1, 0, 1, 0, 3, 1, 2, 0, 1, 0, 1, 0, 2, 0, 1, 1, 1, 0, 2, 0, 3, 0, 1, 0, 1, 1, 1, 0, 2, 0, 3, 0, 1, 0, 1, 1, 3, 0, 2, 0, 2, 0, 2, 0, 2, 1

Offset: 1

Antti Karttunen, Jul 17 2020

nonn

Positions for the first occurrence of each n, for n >= 0, are: 1, 4, 16, 32, 144, 512, 2048, 6912, 20736, 62208, ...

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000120, A001222, A122111, A253553, A292377, A292383, A336120, A336123, A336124.

Cf. A336119 (positions of zeros).

A253553(n) = if(n<=2,1,my(f=factor(n), k=#f~); if(f[k,2]>1,f[k,2]--,f[k,1] = precprime(f[k,1]-1)); factorback(f));
A336121(n) = if(1==n,0,(3==A336124(n))+A336121(A253553(n)));

a(1) = 0, and for n > 1, a(n) = [A336124(n) == 3] + a(A253553(n)).
a(n) = A000120(A336120(n)).
a(n) = A292377(A122111(n)).
a(n) = A001222(n) - A336123(n).

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

A253884 Permutation of natural numbers: a(n) = A122111(A243506(n)).

Original entry on oeis.org

1, 2, 8, 4, 24, 5, 8192, 3, 64, 512, 393216, 32, 7077888, 320, 384, 6, 3802951800684688204490109616128, 15, 300578991243264, 11, 1073741824, 5184, 1134000, 128, 100, 1664, 27, 864, 392318858461667547739736838950479151006397215279002157056000, 65536, 9822276308431282926640601754292746977280, 16, 1215

Offset: 1

Antti Karttunen, Jan 17 2015

nonn

Index entries for sequences that are permutations of the natural numbers

Inverse: A253883.

Similar permutations: A253792, A253892, A122111, A243506.

(define (A253884 n) (A122111 (A243506 n)))

a(n) = A122111(A243506(n)).

A323167 a(n) = A294898(A122111(n)).

Original entry on oeis.org

0, 0, 0, 0, 0, -2, 0, 2, 3, -6, 0, 0, 0, -14, -5, 3, 0, 2, 0, -4, -21, -30, 0, 1, 10, -62, 16, -12, 0, -16, 0, 7, -53, -126, -16, 7, 0, -254, -117, -3, 0, -52, 0, -28, 4, -510, 0, 5, 38, 8, -245, -60, 0, 19, -68, -11, -501, -1022, 0, -15, 0, -2046, -20, 9, -172, -124, 0, -124, -1013, -58, 0, 16, 0, -4094, 22, -252, -42, -268, 0, 1, 38

Offset: 1

Antti Karttunen, Jan 10 2019

sign

Cf. A000203, A005187, A294898, A122111, A322867, A323174, A323168.

Cf. also A295296, A323248.

A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A122111(n) = if(1==n,n,prime(bigomega(n))*A122111(A064989(n)));
A005187(n) = { my(s=n); while(n>>=1, s+=n); s; };
A294898(n) = (A005187(n)-sigma(n));
A323167(n) = A294898(A122111(n));
\\ Or alternatively as:
A322867(n) = hammingweight(A122111(n));
A323174(n) = { my(k=A122111(n)); ((2*k)-sigma(k)); }
A323167(n) = (A323174(n) - A322867(n));

a(n) = A294898(A122111(n)).

a(n) = A323174(n) - A322867(n).

A329607 a(n) = A007947(A122111(n)).

Original entry on oeis.org

1, 2, 2, 3, 2, 6, 2, 5, 3, 6, 2, 10, 2, 6, 6, 7, 2, 15, 2, 10, 6, 6, 2, 14, 3, 6, 5, 10, 2, 30, 2, 11, 6, 6, 6, 21, 2, 6, 6, 14, 2, 30, 2, 10, 10, 6, 2, 22, 3, 15, 6, 10, 2, 35, 6, 14, 6, 6, 2, 42, 2, 6, 10, 13, 6, 30, 2, 10, 6, 30, 2, 33, 2, 6, 15, 10, 6, 30, 2, 22, 7, 6, 2, 42, 6, 6, 6, 14, 2, 70, 6, 10, 6, 6, 6, 26, 2, 15, 10, 21, 2, 30, 2, 14, 30

Offset: 1

Antti Karttunen, Nov 17 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences computed from indices in prime factorization

Cf. A007947, A064989, A071364, A077462, A108951, A122111, A181819, A181821, A329600.

Block[{f}, f[1] = 1; f[n_] := Module[{l = #, m = 0}, Times @@ Power @@@ Table[l -= m; l = DeleteCases[l, 0]; {Prime@ Length@ l, m = Min@ l}, Length@ Union@ l]] &@ Catenate[ConstantArray[PrimePi[#1], #2] & @@@ FactorInteger@ n]; Array[If[# < 1, 0, Sum[EulerPhi[d] Abs@ MoebiusMu[d], {d, Divisors[#]}]] &@ f[#] &, 105]] (* Michael De Vlieger, Nov 18 2019, after JungHwan Min at A122111. *)

A007947(n) = factorback(factorint(n)[, 1]);
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A122111(n) = if(1==n,n,prime(bigomega(n))*A122111(A064989(n)));
A329607(n) = A007947(A122111(n));

a(n) = A007947(A122111(n)) = A007947(A181819(A108951(n))).
a(n) = A122111(A071364(n)).
A181821(a(n)) = A329600(n).

cp's OEIS Frontend

A253883 Permutation of natural numbers: a(n) = A243505(A122111(n)).

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A280491 a(n) = gcd(n,A122111(n)).

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A322863 Permutation of natural numbers: a(0) = 1; for n >= 1, a(n) = A005940(1+A122111(n)).

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A331170 a(n) = min(n, A122111(n)), where A122111 conjugates the prime factorization of n.

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A336119 Numbers k such that A122111(k) [the conjugated prime factorization of k] is a square or twice a square.

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A336121 a(1) = 0, and for n > 1, a(n) = [A122111(n) == 3 (mod 4)] + a(A253553(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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A253884 Permutation of natural numbers: a(n) = A122111(A243506(n)).

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A323167 a(n) = A294898(A122111(n)).

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