A327979 -id:A327979 - cp's OEIS frontend

A327295 Numbers k such that e(k) > 1 and k == e(k) (mod lambda(k)), where e(k) = A051903(k) is the maximal exponent in prime factorization of k.

4, 12, 16, 48, 80, 112, 132, 208, 240, 1104, 1456, 1892, 2128, 4144, 5852, 12208, 17292, 18544, 21424, 25456, 30160, 45904, 78736, 97552, 106384, 138864, 153596, 154960, 160528, 289772, 311920, 321904, 399212, 430652, 545584, 750064, 770704, 979916, 1037040, 1058512

Offset: 1

Thomas Ordowski, Dec 05 2019

nonn

The condition e(k) > 1 excludes primes and Carmichael numbers.
Numbers n such that e(k) > 1 and b^k == b^e(k) (mod k) for all b.
These are numbers k such that A276976(k) = e(k) > 1.
Are there infinitely many such numbers? Are all such numbers even?
A number k is a term if and only if k is e(k)-Knödel number with e(k) > 1. So they may have the name nonsquarefree e(k)-Knodel numbers k.
It seems that if k is in this sequence, then e(k) = A007814(k) and k/2^e(k) is squarefree.
Conjecture: there are no composite numbers m > 4 such that m == e(m) (mod phi(m)). By Lehmer's totient conjecture, there are no such squarefree numbers.
Problem: are there odd numbers n such that e(n) > 1 and n == e(n) (mod ord_{n}(2)), where ord_{n}(2) = A002326((n-1)/2)? These are odd numbers n such that 2^n == 2^e(n) (mod n) with e(n) > 1.
Numbers k for which A051903(k) > 1 and A219175(k) = A329885(k). - Antti Karttunen, Dec 11 2019

			The number 4 = 2^2 is a term, because e(4) = A051903(4) = 2 > 1 and 4 == 2 (mod lambda(4)), where lambda(4) = A002322(4) = 2.

Amiram Eldar, Table of n, a(n) for n = 1..500

Cf. A000010, A002322, A002326, A002997, A050990, A050992, A051903, A068494, A219175, A270096, A276976, A324050, A327979, A329885.

A proper subset of A013929.

Select[Range[10^5], (e = Max @@ Last /@ FactorInteger[#]) > 1 && Divisible[# -e, CarmichaelLambda[#]] &] (* Amiram Eldar, Dec 05 2019 *)

isok(n) = ! issquarefree(n) && (Mod(n, lcm(znstar(n)[2])) == vecmax(factor(n)[, 2])); \\ Michel Marcus, Dec 05 2019

More terms from Amiram Eldar, Dec 05 2019

A329895 Lexicographically earliest infinite sequence such that a(i) = a(j) => A219175(i) = A219175(j) and A289625(i) = A289625(j) for all i, j.

Original entry on oeis.org

1, 1, 2, 3, 4, 3, 5, 6, 7, 8, 9, 6, 10, 11, 12, 13, 14, 15, 16, 13, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 32, 38, 39, 40, 41, 42, 36, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 36, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 49, 68, 69, 70, 71, 72, 73, 74, 75, 65, 76, 77, 78, 79, 80, 70, 81, 82, 83

Offset: 1

Antti Karttunen, Dec 07 2019

nonn

Restricted growth sequence transform of the ordered pair [A219175(n), A289625(n)].
For all i, j:
  a(i) = a(j) => A327979(i) = A327979(j).

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

Cf. A002322, A219175, A289625, A327979, A329896.

up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A219175(n) = (n%lcm(znstar(n)[2]));
A289625(n) = { my(m=1,p=2,v=znstar(n)[2]); for(i=1,length(v),m *= p^v[i]; p = nextprime(p+1)); (m); };
Aux329895(n) = [A219175(n),A289625(n)];
v329895 = rgs_transform(vector(up_to, n, Aux329895(n)));
A329895(n) = v329895[n];

A329896 Lexicographically earliest infinite sequence such that a(i) = a(j) => A219175(i) = A219175(j) and A322592(i) = A322592(j) for all i, j.

Original entry on oeis.org

1, 1, 2, 3, 2, 3, 2, 4, 5, 6, 2, 4, 2, 7, 8, 9, 2, 10, 2, 9, 11, 12, 2, 13, 14, 15, 16, 17, 2, 18, 2, 19, 20, 21, 22, 23, 2, 24, 25, 26, 2, 23, 2, 27, 28, 29, 2, 26, 30, 31, 32, 33, 2, 34, 35, 36, 37, 38, 2, 26, 2, 39, 40, 41, 42, 43, 2, 44, 45, 46, 2, 47, 2, 48, 35, 49, 50, 51, 2, 52, 53, 54, 2, 47, 55, 56, 57, 58, 2, 51, 59, 60, 61, 62, 63, 64, 2, 65, 66, 67, 2, 68, 2, 69, 70

Offset: 1

Antti Karttunen, Dec 07 2019

nonn

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

Cf. A002322, A219175, A289625, A322592, A327979, A329895.

up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A219175(n) = (n%lcm(znstar(n)[2]));
A289625(n) = { my(m=1,p=2,v=znstar(n)[2]); for(i=1,length(v),m *= p^v[i]; p = nextprime(p+1)); (m); };
Aux329896(n) = if((n>2)&&isprime(n),0,[A219175(n),A289625(n)]);
v329896 = rgs_transform(vector(up_to, n, Aux329896(n)));
A329896(n) = v329896[n];

cp's OEIS Frontend

A327295 Numbers k such that e(k) > 1 and k == e(k) (mod lambda(k)), where e(k) = A051903(k) is the maximal exponent in prime factorization of k.

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A329895 Lexicographically earliest infinite sequence such that a(i) = a(j) => A219175(i) = A219175(j) and A289625(i) = A289625(j) for all i, j.

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A329896 Lexicographically earliest infinite sequence such that a(i) = a(j) => A219175(i) = A219175(j) and A322592(i) = A322592(j) for all i, j.

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