A122111 -id:A122111 - cp's OEIS frontend

A323173 Sum of divisors computed for conjugated prime factorization: a(n) = A000203(A122111(n)).

1, 3, 7, 4, 15, 12, 31, 6, 13, 28, 63, 18, 127, 60, 39, 8, 255, 24, 511, 42, 91, 124, 1023, 24, 40, 252, 31, 90, 2047, 72, 4095, 12, 195, 508, 120, 32, 8191, 1020, 403, 56, 16383, 168, 32767, 186, 93, 2044, 65535, 36, 121, 78, 819, 378, 131071, 48, 280, 120, 1651, 4092, 262143, 96, 524287, 8188, 217, 14, 600, 360, 1048575, 762, 3315, 234

Offset: 1

Antti Karttunen, Jan 10 2019

nonn

Cf. A000203, A038712, A122111, A322819, A323174.

Cf. also A323243.

A122111[n_] := Product[Prime[Sum[If[j < i, 0, 1], {j, #}]], {i, Max[#]}]&[ Flatten[Table[Table[PrimePi[f[[1]]], {f[[2]]}], {f, FactorInteger[n]}]]];
a[n_] := With[{k = A122111[n]}, DivisorSigma[1, k]];
Array[a, 70] (* Jean-François Alcover, Sep 23 2020 *)

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)));
A323173(n) = sigma(A122111(n));

a(n) = A000203(A122111(n)).
a(n) = 2*A122111(n) - A323174(n).
a(n) = A322819(n) * A038712(A122111(n)).

A336124 a(n) = A122111(n) mod 4.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Jul 15 2020

nonn

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

Cf. A010873, A064989, A122111, A336120.

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)));
A336124(n) = (A122111(n)%4);

a(n) = A010873(A122111(n)).

A322819 a(n) = A000593(A122111(n)).

Original entry on oeis.org

1, 1, 1, 4, 1, 4, 1, 6, 13, 4, 1, 6, 1, 4, 13, 8, 1, 24, 1, 6, 13, 4, 1, 8, 40, 4, 31, 6, 1, 24, 1, 12, 13, 4, 40, 32, 1, 4, 13, 8, 1, 24, 1, 6, 31, 4, 1, 12, 121, 78, 13, 6, 1, 48, 40, 8, 13, 4, 1, 32, 1, 4, 31, 14, 40, 24, 1, 6, 13, 78, 1, 48, 1, 4, 124, 6, 121, 24, 1, 12, 57, 4, 1, 32, 40, 4, 13, 8, 1, 48, 121, 6, 13, 4, 40, 14, 1, 240

Offset: 1

Antti Karttunen, Dec 27 2018

nonn

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

Cf. A000593, A122111, A322813.

A000593(n) = sigma(n>>valuation(n, 2));
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)));
A322819(n) = A000593(A122111(n));

a(n) = A000593(A122111(n)).

A322865 a(n) = A000265(A122111(n)).

Original entry on oeis.org

1, 1, 1, 3, 1, 3, 1, 5, 9, 3, 1, 5, 1, 3, 9, 7, 1, 15, 1, 5, 9, 3, 1, 7, 27, 3, 25, 5, 1, 15, 1, 11, 9, 3, 27, 21, 1, 3, 9, 7, 1, 15, 1, 5, 25, 3, 1, 11, 81, 45, 9, 5, 1, 35, 27, 7, 9, 3, 1, 21, 1, 3, 25, 13, 27, 15, 1, 5, 9, 45, 1, 33, 1, 3, 75, 5, 81, 15, 1, 11, 49, 3, 1, 21, 27, 3, 9, 7, 1, 35, 81, 5, 9, 3, 27, 13, 1, 135, 25, 63, 1, 15, 1, 7, 75

Offset: 1

Antti Karttunen, Dec 30 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences related to binary expansion of n
Index entries for sequences computed from indices in prime factorization

Cf. A000265, A122111, A322813, A322819, A322820, A322826, A322865, A322867.

Array[#/2^IntegerExponent[#, 2] &@ If[# < 3, 1, Block[{k = #, m = 0}, Times @@ Power @@@ Table[k -= m; k = DeleteCases[k, 0]; {Prime@ Length@ k, m = Min@ k}, Length@ Union@ k]] &@ Catenate[ConstantArray[PrimePi[#1], #2] & @@@ FactorInteger@ #]] &, 105] (* Michael De Vlieger, Dec 31 2018, after JungHwan Min at A122111 *)

A000265(n) = (n>>valuation(n, 2));
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)));
A322865(n) = A000265(A122111(n));

a(n) = A000265(A122111(n)).
a(n) = A122111(A322820(n)).
A000005(a(n)) = A322813(n).
A000203(a(n)) = A322819(n).
A122111(a(n)) = A322820(n).
A000120(a(n)) = A322867(n).

A336120 a(n) = A292383(A122111(n)).

Original entry on oeis.org

0, 0, 0, 1, 0, 2, 0, 2, 0, 4, 0, 4, 0, 8, 0, 5, 0, 5, 0, 8, 0, 16, 0, 10, 1, 32, 0, 16, 0, 10, 0, 11, 0, 64, 2, 8, 0, 128, 0, 20, 0, 20, 0, 32, 0, 256, 0, 22, 0, 8, 0, 64, 0, 11, 4, 40, 0, 512, 0, 16, 0, 1024, 0, 22, 8, 40, 0, 128, 0, 16, 0, 20, 0, 2048, 1, 256, 0, 80, 0, 44, 0, 4096, 0, 32, 16, 8192, 0, 80, 0, 22, 0, 512, 0, 16384, 32, 44, 0, 17, 0, 17, 0

Offset: 1

Antti Karttunen, Jul 14 2020

nonn

Antti Karttunen, Table of n, a(n) for n = 1..8192
Index entries for sequences computed from indices in prime factorization

Cf. A064989, A122111, A252463, A253553, A253566, A292383, A336121, A336124, A336125, A336312.

Cf. A336119 (positions of zeros).

\\ Uses also code given in A336124:
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));
A336120(n) = if(1==n,0,(3==A336124(n))+(2*A336120(A253553(n))));

a(1) = 0, and for n > 1, a(n) = [A122111(n) == 3 (mod 4)] + 2*a(A253553(n)).
a(n) = A292383(A122111(n)).
a(n) = A253566(n) - A336125(n).
A000120(a(n)) = A336121(n).

A244982 Permutation of natural numbers: a(n) = A243285(A122111(2*n)).

Original entry on oeis.org

1, 2, 4, 3, 8, 6, 18, 5, 11, 14, 38, 10, 79, 30, 22, 7, 164, 15, 337, 20, 47, 64, 694, 16, 35, 134, 26, 43, 1419, 32, 2888, 9, 100, 279, 73, 24, 5850, 575, 208, 34, 11822, 67, 23836, 92, 56, 1177, 47975, 19, 112, 50, 428, 193, 96431, 42, 152, 71, 877, 2395, 193614, 52

Offset: 1

Antti Karttunen, Jul 20 2014

nonn

Index entries for sequences that are permutations of the natural numbers

Inverse: A244981.

Cf. A122111, A243285, A244983-A244984, A243505-A243506, A243065-A243066.

(define (A244982 n) (A243285 (A122111 (* 2 n))))

a(n) = A243285(A122111(2*n)).

A244984 Permutation of natural numbers: a(n) = A243283(A122111((2*n)-1)).

Original entry on oeis.org

1, 2, 3, 5, 4, 9, 14, 6, 23, 37, 10, 58, 8, 7, 90, 143, 15, 13, 225, 24, 355, 563, 12, 894, 17, 38, 1426, 20, 60, 2277, 3643, 19, 31, 5839, 96, 9398, 15155, 16, 27, 24518, 11, 39758, 50, 153, 64607, 42, 242, 80, 105250, 30, 171874, 281237, 26

Offset: 1

Antti Karttunen, Jul 21 2014

nonn

Index entries for sequences that are permutations of the natural numbers

Inverse: A244983.

Cf. A122111, A243283, A244981-A244982, A243505-A243506, A243065-A243066, A006254, A244986.

(define (A244984 n) (A243283 (A122111 (-1+ (* 2 n)))))

a(n) = A243283(A122111((2*n)-1)).
a(n) = A243283(A105560((2*n)-1) * A243505(n)).
For all n >= 1, a(A006254(n)) = A244986(n+1).

A334107 a(n) = A329697(A122111(n)).

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 0, 1, 2, 1, 0, 1, 0, 1, 2, 2, 0, 2, 0, 1, 2, 1, 0, 2, 3, 1, 2, 1, 0, 2, 0, 2, 2, 1, 3, 3, 0, 1, 2, 2, 0, 2, 0, 1, 2, 1, 0, 2, 4, 3, 2, 1, 0, 3, 3, 2, 2, 1, 0, 3, 0, 1, 2, 2, 3, 2, 0, 1, 2, 3, 0, 3, 0, 1, 3, 1, 4, 2, 0, 2, 4, 1, 0, 3, 3, 1, 2, 2, 0, 3, 4, 1, 2, 1, 3, 2, 0, 4, 2, 4, 0, 2, 0, 2, 3

Offset: 1

Antti Karttunen, Apr 29 2020

nonn

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

Cf. A001248, A008578 (positions of zeros), A064989, A105560, A122111, A322865, A329697, A334108, A334109, A334201.

Map[Length@ NestWhileList[# - #/FactorInteger[#][[-1, 1]] &, #, # != 2^IntegerExponent[#, 2] &] - 1 &, Array[Times @@ Table[Prime[LengthWhile[#1, # >= j &] /. 0 -> 1], {j, #2}] & @@ {#, Max[#]} &@ PrimePi@ Flatten[ConstantArray[#1, {#2}] & @@@ FactorInteger@ #] &, 105] ] (* Michael De Vlieger, May 14 2020, after Robert G. Wilson v at A329697 *)

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)));
A329697(n) = if(!bitand(n,n-1),0,1+A329697(n-(n/vecmax(factor(n)[, 1]))));
A334107(n) = A329697(A122111(n));

a(n) = A329697(A122111(n)) = A329697(A322865(n)).
a(n) = A329697(A105560(n)) + a(A064989(n)).
For n >= 1, a(A001248(n)) = n, and these seem to be also the first occurrences of each n.

A244981 Permutation of natural numbers: a(n) = A122111(A102750(n)) / 2.

Original entry on oeis.org

1, 2, 4, 3, 8, 6, 16, 5, 32, 12, 9, 64, 128, 10, 18, 24, 256, 7, 48, 20, 512, 15, 1024, 36, 96, 27, 2048, 192, 72, 14, 4096, 30, 8192, 40, 25, 384, 16384, 11, 144, 80, 32768, 54, 28, 288, 768, 65536, 21, 131072, 1536, 50, 108, 60, 262144, 160, 576, 45, 524288, 1048576, 3072, 320, 81, 120, 2097152, 22, 6144, 4194304, 42

Offset: 1

Antti Karttunen, Jul 20 2014

nonn

Antti Karttunen, Table of n, a(n) for n = 1..1024
Index entries for sequences that are permutations of the natural numbers

Inverse: A244982.

Cf. A122111, A102750, A244983-A244984, A243505-A243506, A243065-A243066.

(define (A244981 n) (/ (A122111 (A102750 n)) 2))

a(n) = A122111(A102750(n)) / 2.

A244983 Permutation of natural numbers: a(1) = 1, a(n) = (1 + A122111(A070003(n-1))) / 2.

Original entry on oeis.org

1, 2, 3, 5, 4, 8, 14, 13, 6, 11, 41, 23, 18, 7, 17, 38, 25, 68, 32, 28, 122, 63, 9, 20, 113, 53, 39, 365, 95, 50, 33, 74, 203, 61, 188, 88, 10, 26, 1094, 158, 83, 46, 608, 313, 3281, 338, 123, 149, 59, 43, 221, 116, 284, 72, 263, 138, 1013, 12, 9842, 29, 1823, 248, 98, 563, 172, 60

Offset: 1

Antti Karttunen, Jul 21 2014

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10001
Index entries for sequences that are permutations of the natural numbers

Inverse: A244984.
Cf. A070003, A006254, A244986.
Related or similar permutations: A122111, A244981-A244982, A243505-A243506, A243065-A243066.

(define (A244983 n) (if (= 1 n) 1 (/ (+ 1 (A122111 (A070003 (- n 1)))) 2)))

a(1) = 1, a(n) = (1 + A122111(A070003(n-1))) / 2.

For all n >= 1, a(A244986(n+1)) = A006254(n).

cp's OEIS Frontend

A323173 Sum of divisors computed for conjugated prime factorization: a(n) = A000203(A122111(n)).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A336124 a(n) = A122111(n) mod 4.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A322819 a(n) = A000593(A122111(n)).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A322865 a(n) = A000265(A122111(n)).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A336120 a(n) = A292383(A122111(n)).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A244982 Permutation of natural numbers: a(n) = A243285(A122111(2*n)).

Views

Author

Keywords

Links

Crossrefs

Programs

Scheme

Formula

A244984 Permutation of natural numbers: a(n) = A243283(A122111((2*n)-1)).

Views

Author

Keywords

Links

Crossrefs

Programs

Scheme

Formula

A334107 a(n) = A329697(A122111(n)).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A244981 Permutation of natural numbers: a(n) = A122111(A102750(n)) / 2.

Views

Author

Keywords