A122111 -id:A122111 - cp's OEIS frontend

A331596 Number of distinct prime factors of gcd(A122111(n), A241909(n)).

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

Offset: 1

Antti Karttunen, Jan 22 2020

nonn

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

Cf. A001221, A331596, A331597.

Array[PrimeNu@ If[# == 1, 1, GCD @@ {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] & @@@ #], Function[t, Times @@ Prime@ Accumulate[If[Length@ t < 2, {0}, Join[{1}, ConstantArray[0, Length@ t - 2], {-1}]] + ReplacePart[t, Map[#1 -> #2 & @@ # &, #]]]]@ ConstantArray[0, Transpose[#][[1, -1]]] &[# /. {p_, e_} /; p > 0 :> {PrimePi@ p, e}]} &@ FactorInteger[#]] &, 105] (* Michael De Vlieger, Jan 24 2020, after JungHwan Min at A122111. *)

A331596(n) = omega(gcd(A122111(n), A241909(n)));

a(n) = A001221(A331596(n)) = A001221(gcd(A122111(n), A241909(n))).

a(n) = A001222(A331597(n)).

A331598 a(n) = A122111(n) / gcd(A122111(n),A241909(n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 3, 4, 1, 2, 1, 8, 1, 1, 1, 1, 1, 4, 2, 16, 1, 2, 9, 32, 5, 8, 1, 2, 1, 1, 4, 64, 3, 3, 1, 128, 8, 4, 1, 4, 1, 16, 1, 256, 1, 2, 27, 1, 16, 32, 1, 5, 1, 8, 32, 512, 1, 6, 1, 1024, 2, 1, 2, 8, 1, 64, 64, 2, 1, 3, 1, 2048, 5, 128, 9, 16, 1, 4, 7, 4096, 1, 12, 4, 8192, 128, 16, 1, 10, 3, 256, 256, 16384, 8, 2, 1, 1, 4, 9, 1, 32, 1, 32, 1

Offset: 1

Antti Karttunen, Jan 22 2020

nonn

It appears that these and the terms of A331599 have the same prime signatures, that is, A046523(a(n)) = A046523(A331599(n)) seems to hold for all n. However, the sequences are not equivalence-class-wise same: a(6) = a(12) = 2, whereas A331599(6) = 3 and A331599(12) = 5.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences computed from indices in prime factorization

Cf. A122111, A241909, A241916, A331594, A331595, A331599.

Array[If[# == 1, 1, #1/GCD[#1, #2] & @@ {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] & @@@ #], Function[t, Times @@ Prime@ Accumulate[If[Length@ t < 2, {0}, Join[{1}, ConstantArray[0, Length@t - 2], {-1}]] + ReplacePart[t, Map[#1 -> #2 & @@ # &, #]]]]@ ConstantArray[0, Transpose[#][[1, -1]]] &[# /. {p_, e_} /; p > 0 :> {PrimePi@ p, e}]} &@ FactorInteger[#]] &, 90] (* Michael De Vlieger, Jan 25 2020, after JungHwan Min at A122111 *)

A331598(n) = { my(u=A122111(n)); u/gcd(u, A241909(n)); };

a(n) = A122111(n)/A331598(n) = A122111(n) / gcd(A122111(n),A241909(n)).

a(n) = A331599(A241916(n)).

A331599 a(n) = A241909(n) / gcd(A122111(n),A241909(n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 1, 2, 9, 1, 5, 1, 27, 1, 1, 1, 1, 1, 25, 3, 81, 1, 7, 4, 243, 2, 125, 1, 5, 1, 1, 9, 729, 2, 5, 1, 2187, 27, 49, 1, 25, 1, 625, 1, 6561, 1, 11, 8, 1, 81, 3125, 1, 3, 1, 343, 243, 19683, 1, 35, 1, 59049, 5, 1, 3, 125, 1, 15625, 729, 5, 1, 7, 1, 177147, 2, 78125, 4, 625, 1, 121, 2, 531441, 1, 245, 9, 1594323, 2187, 2401, 1, 21

Offset: 1

Antti Karttunen, Jan 22 2020

nonn

It appears that these and the terms of A331598 have the same prime signatures, that is, A046523(a(n)) = A046523(A331598(n)) seems to hold for all n.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences computed from indices in prime factorization

Cf. A122111, A241909, A241916, A331595, A331598.

Array[If[# == 1, 1, #2/GCD[#1, #2] & @@ {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] & @@@ #], Function[t, Times @@ Prime@ Accumulate[If[Length@ t < 2, {0}, Join[{1}, ConstantArray[0, Length@ t - 2], {-1}]] + ReplacePart[t, Map[#1 -> #2 & @@ # &, #]]]]@ ConstantArray[0, Transpose[#][[1, -1]]] &[# /. {p_, e_} /; p > 0 :> {PrimePi@ p, e}]} &@ FactorInteger[#]] &, 90] (* Michael De Vlieger, Jan 24 2020, after JungHwan Min at A122111 *)

A331599(n) = { my(u=A241909(n)); u/gcd(A122111(n), u); };

a(n) = A241909(n) / A331595(n) = A241909(n) / gcd(A122111(n),A241909(n)).

a(n) = A331598(A241916(n)).

A331731 Odd part of A331595(n), where A331595(n) = gcd(A122111(n), A241909(n)).

Original entry on oeis.org

1, 1, 1, 3, 1, 3, 1, 5, 3, 3, 1, 5, 1, 3, 9, 7, 1, 15, 1, 5, 9, 3, 1, 7, 3, 3, 5, 5, 1, 15, 1, 11, 9, 3, 9, 7, 1, 3, 9, 7, 1, 15, 1, 5, 25, 3, 1, 11, 3, 45, 9, 5, 1, 7, 27, 7, 9, 3, 1, 7, 1, 3, 25, 13, 27, 15, 1, 5, 9, 45, 1, 11, 1, 3, 15, 5, 9, 15, 1, 11, 7, 3, 1, 7, 27, 3, 9, 7, 1, 7, 27, 5, 9, 3, 27, 13, 1, 135, 25, 7, 1, 15, 1, 7, 75

Offset: 1

Antti Karttunen, Jan 25 2020

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences computed from indices in prime factorization

Cf. A122111, A241909, A331595, A331600.

Cf. also A322865, A331732.

PARI
```
A331731(n) = A000265(A331595(n));
```

a(n) = A000265(A331595(n)).

A336123 a(1) = 0, a(2) = 1, and for n > 2, a(n) = [A122111(n) == 1 (mod 4)] + a(A253553(n)).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Jul 17 2020

nonn

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

Cf. A000120, A000244, A001222, A122111, A292375, A336121, A336124, A336125.

\\ 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));
A336123(n) = if(n<=2,n-1,(1==A336124(n))+A336123(A253553(n)));

a(1) = 0, a(2) = 1, and for n > 2, a(n) = [A336124(n) == 1] + a(A253553(n)).
a(n) = A000120(A336125(n)).
For n > 1, a(n) = A292375(A122111(n)).
a(n) = A001222(n) - A336121(n).
For all n >= 0, a(3^n) = n.

A336315 The number of divisors in the conjugated prime factorization of n: a(n) = A000005(A122111(n)).

Original entry on oeis.org

1, 2, 3, 2, 4, 4, 5, 2, 3, 6, 6, 4, 7, 8, 6, 2, 8, 4, 9, 6, 9, 10, 10, 4, 4, 12, 3, 8, 11, 8, 12, 2, 12, 14, 8, 4, 13, 16, 15, 6, 14, 12, 15, 10, 6, 18, 16, 4, 5, 6, 18, 12, 17, 4, 12, 8, 21, 20, 18, 8, 19, 22, 9, 2, 16, 16, 20, 14, 24, 12, 21, 4, 22, 24, 6, 16, 10, 20, 23, 6, 3, 26, 24, 12, 20, 28, 27, 10, 25, 8, 15, 18, 30, 30, 24

Offset: 1

Antti Karttunen, Jul 18 2020

nonn

Antti Karttunen, Table of n, a(n) for n = 1..12000
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65539
Index entries for sequences computed from indices in prime factorization

Cf. A000005, A034444, A122111, A286621, A323173, A322813, A322819, A336316.

A336315(n) = if(1==n,n,my(p=apply(primepi,factor(n)[,1]~),m=1+p[1]); for(i=2, #p, m *= (1+p[i]-p[i-1])); (m));

a(n) = A000005(A122111(n)).

a(n) = A336316(n) + A034444(n).

A336317 Numbers k such that A122111(k) [conjugated prime factorization of k] is one of Ore's Harmonic numbers (in A001599).

Original entry on oeis.org

1, 6, 40, 126, 154, 204, 1716, 1914, 2772, 8580, 11264, 12090, 12540, 50960, 62790, 64350, 77748, 83200, 104720, 152320, 186116, 193440, 331890, 382720, 432768, 518364, 648788, 684684, 753480, 817344, 895356, 1083852, 1113840, 1619352, 1675044, 1743588, 1759680, 1991340, 2060322, 2360484, 2492028, 2621080, 2932800

Offset: 1

Antti Karttunen, Jul 19 2020

nonn

Numbers k for which A336314(k) = A323173(k).
Sequence A122111(A001599(n)), n >= 1, sorted into ascending order. Positions of zeros in A323174 (corresponding to perfect numbers similarly mapped) is a subsequence.
Note that all terms after 1 seem to be present in A102750. This observation is equal to Ore's conjecture that there are no odd Harmonic numbers larger than one.
Also, all terms after 1 seem to be even, which would imply that apart from its initial 1, A001599 were a subsequence of A102750. However, this is false, as there are terms of A001599 not in A102750, for example 8011798098793361832960 found by David A. Corneth. Note that A122111(8011798098793361832960) = 96922193555635754403846044921625, which is thus an odd term of this sequence.

Antti Karttunen, Table of n, a(n) for n = 1..61
Index entries for sequences computed from indices in prime factorization

Cf. A001599, A088902, A102750, A122111, A323173, A323174, A336314, A336315, A336353, A336397 (the intersection with A001599).

isA001599(n) = !((sigma(n,0)*n)%sigma(n,1));
isA336317(n) = isA001599(A122111(n)); \\ Program for A122111 given under that entry.

\\ Standalone program:
isA336317(n) = if(1==n,1,my(f=factor(n),es=Vecrev(f[,2]),is=concat(apply(primepi,Vecrev(f[,1])),[0]),pri=0,d=1,s=1,x=1,p,e); for(i=1, #es, pri += es[i]; p = prime(pri); e = 1+is[i]-is[1+i]; d *= e; s *= ((p^e)-1)/(p-1); x *= (p^(e-1))); !((x*d)%s));

A354866 Dirichlet inverse of A122111.

Original entry on oeis.org

1, -2, -4, 1, -8, 10, -16, -1, 7, 20, -32, -10, -64, 40, 46, 2, -128, -27, -256, -20, 92, 80, -512, 14, 37, 160, -17, -40, -1024, -150, -2048, -3, 184, 320, 202, 53, -4096, 640, 368, 28, -8192, -300, -16384, -80, -146, 1280, -32768, -26, 175, -129, 736, -160, -65536, 85, 404, 56, 1472, 2560, -131072, 242, -262144

Offset: 1

Antti Karttunen, Jun 09 2022

sign

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences computed from indices in prime factorization

Cf. A122111, A354867, A354868 (parity), A354869 (positions of odd terms).

Cf. also A354186, A354349, A354351, A354359, A354826.

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));
memoA354866 = Map();
A354866(n) = if(1==n,1,my(v); if(mapisdefined(memoA354866,n,&v), v, v = -sumdiv(n,d,if(dA122111(n/d)*A354866(d),0)); mapput(memoA354866,n,v); (v)));

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, dA122111(n/d) * a(d).

a(n) = A354867(n) - A122111(n).

A369031 LCM-transform of permutation induced by partition conjugation via Heinz numbers (A122111).

Original entry on oeis.org

1, 2, 2, 3, 2, 1, 2, 5, 3, 1, 2, 1, 2, 1, 1, 7, 2, 1, 2, 1, 1, 1, 2, 1, 3, 1, 5, 1, 2, 1, 2, 11, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 3, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 13, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 7, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1

Offset: 1

Antti Karttunen, Jan 12 2024

nonn

See discussion at A368900.

From the reduced formula it follows that for all i, j >= 1: A101296(i) = A101296(j) => a(i) = a(j), that is, the value of each a(n) is completely determined by its prime signature. Note that the same does not hold for related A369032.

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

Cf. A014963, A101296, A122111, A368900, A369032, A369033.

up_to = 2^18;
LCMtransform(v) = { my(len = length(v), b = vector(len), g = vector(len)); b[1] = g[1] = 1; for(n=2,len, g[n] = lcm(g[n-1],v[n]); b[n] = g[n]/g[n-1]); (b); };
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));
v369031 = LCMtransform(vector(up_to,i,A122111(i)));
A369031(n) = v369031[n];

A369031(n) = if(isprime(n),2, my(e=ispower(n,,&n)); if(e && isprime(n), prime(e), 1));

a(n) = lcm {1..A122111(n)} / lcm {1..A122111(n-1)}.
a(n) = A014963(A122111(n)). [A122111 satisfies the property S given in A368900]
If n = p^k, p prime, k >= 1, then a(n) = A000040(k), otherwise a(n) = 1.

A253562 Inverse permutation to A253561: a(n) = A252752(A122111(n)).

Original entry on oeis.org

1, 2, 3, 4, 8, 5, 30, 7, 6, 17, 122, 12, 498, 68, 38, 11, 2018, 9, 8130, 47, 155, 278, 32642, 23, 13, 1130, 10, 192, 130818, 107, 523778, 16, 632, 4562, 353, 18, 2096130, 18338, 2558, 93, 8386562, 437, 33550338, 782, 302, 73538, 134209538, 57, 39, 24, 10298, 3162, 536854530, 14, 1433, 380, 41330, 294530, 2147450882, 212, 8589869058, 1178882, 1227, 22

Offset: 1

Antti Karttunen, Jan 03 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 1..512
Index entries for sequences that are permutations of the natural numbers

Inverse: A253561.

Cf. A122111, A252752.

(define (A253562 n) (A252752 (A122111 n)))

a(n) = A252752(A122111(n)).

cp's OEIS Frontend

A331596 Number of distinct prime factors of gcd(A122111(n), A241909(n)).

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A331598 a(n) = A122111(n) / gcd(A122111(n),A241909(n)).

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A331599 a(n) = A241909(n) / gcd(A122111(n),A241909(n)).

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A331731 Odd part of A331595(n), where A331595(n) = gcd(A122111(n), A241909(n)).

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A336123 a(1) = 0, a(2) = 1, and for n > 2, a(n) = [A122111(n) == 1 (mod 4)] + a(A253553(n)).

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A336315 The number of divisors in the conjugated prime factorization of n: a(n) = A000005(A122111(n)).

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A336317 Numbers k such that A122111(k) [conjugated prime factorization of k] is one of Ore's Harmonic numbers (in A001599).

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A354866 Dirichlet inverse of A122111.

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A369031 LCM-transform of permutation induced by partition conjugation via Heinz numbers (A122111).