A361058 -id:A361058 - cp's OEIS frontend

A361686 a(n) is the least totient divisor of A329872(n), where A329872 are nontotients (A005277) that are the product of two totients (A002202).

22, 22, 10, 46, 58, 46, 78, 82, 58, 46, 102, 22, 106, 82, 46, 138, 106, 82, 166, 172, 178, 190, 106, 226, 82, 166, 238, 172, 178, 22, 106, 262, 190, 282, 22, 106, 310, 316, 226, 82, 166, 238, 172, 346, 46, 178, 22, 358, 22, 10, 366, 106, 262, 382, 82, 388, 58, 22, 22, 46, 418

Offset: 1

Michel Marcus, Mar 29 2023

nonn

Let k be the least instance a(k) = m, then A329872(k) = m*A361058(m). For instance a(3)=10, and A329872(3) = 1100 = 10*110 = 10*A361058(10).

Can we get a(k)=30 or a(k)=52 (see A361058)?

			a(3)=10 because A329872(3)=1100 which can be expressed as 1*1100, 2*550, 4*275, 5*220, 10*110, ... where 10*110 is the first case where both factors are nontotients.

Michel Marcus, Table of n, a(n) for n = 1..7280

Cf. A002202, A005277, A329872, A361058.

is(n) = if(!istotient(n), my(v=divisors(n)); for(i=1, (1+#v)\2, if(istotient(v[i])&&istotient(n/v[i]), return(1))); 0); \\ A329872
lista(nn) = for (n=1, nn, if (is(n), my(d=divisors(n)); for (i=1, (1+#d)\2, if (istotient(d[i]) && istotient(n/d[i]), print1(d[i], ", "); break););););

A363371 a(n) is the least prime p for which (p-1)*phi(p^n) is a nontotient, where phi is the Euler totient function (A000010).

Original entry on oeis.org

23, 11, 23, 11, 23, 11, 47, 11, 11, 23, 47, 23, 23, 23, 47, 47, 103, 103, 103, 103, 103, 103, 167, 103, 103, 103, 103, 103, 103, 103, 103, 103, 103, 179, 103, 103, 103, 103, 103, 103, 103, 103, 127, 103, 103, 103, 103, 103, 103, 103, 103, 103, 103, 127, 127, 103, 127, 127, 127

Offset: 1

Michel Marcus, May 29 2023

nonn

Thus a(n) is the least prime p for which p-1=phi(p), a totient value, multiplied by phi(p^n), another totient value, gives a nontotient. There are several instances of these numbers in A361058.

Michel Marcus, Table of n, a(n) for n = 1..160

Cf. A000010, A002202 (totient values) A005277 (nontotients), A361058.

a(n) = my(p=2); while (istotient((p-1)*eulerphi(p^n)), p = nextprime(p+1)); p;

A361396 Integers k such that 28phi(29197^3*k) is not a totient number where phi is the totient function.

Original entry on oeis.org

1, 2, 3, 4, 6, 7517, 15034, 18059, 22551, 28019, 30068, 30983, 36118, 45102, 56038, 61966, 65267, 67427, 67499, 71387, 84057, 84947, 90677, 92949, 97187, 112076, 115469, 123932, 127487, 130534, 130787, 134854, 134998, 142774, 168114, 169067, 169894, 181354, 185898, 191579, 194374, 195801

Offset: 1

Michel Marcus, Mar 10 2023

nonn

For k=1, A000010(29*197^3) = 212983792 = A361058(28).

Math Overflow, The range of the Euler totient function and multiplication by 28, 2018.

Cf. A000010, A007617, A361058.

isok(k) = ! istotient(28*eulerphi(29*197^3*k));

cp's OEIS Frontend

A361686 a(n) is the least totient divisor of A329872(n), where A329872 are nontotients (A005277) that are the product of two totients (A002202).

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A363371 a(n) is the least prime p for which (p-1)*phi(p^n) is a nontotient, where phi is the Euler totient function (A000010).

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A361396 Integers k such that 28phi(29197^3*k) is not a totient number where phi is the totient function.

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cp's OEIS Frontend

A361686 a(n) is the least totient divisor of A329872(n), where A329872 are nontotients (A005277) that are the product of two totients (A002202).

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PARI

A363371 a(n) is the least prime p for which (p-1)*phi(p^n) is a nontotient, where phi is the Euler totient function (A000010).

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Author

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PARI

A361396 Integers k such that 28*phi(29*197^3*k) is not a totient number where phi is the totient function.

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PARI

A361396 Integers k such that 28phi(29197^3*k) is not a totient number where phi is the totient function.