A371418 -id:A371418 - cp's OEIS frontend

A371421 Numbers whose aliquot-like sequence based on the largest aliquot divisor of the sum of divisors of n (A371418) terminates in a fixed point.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 16, 17, 18, 19, 20, 21, 22, 24, 25, 26, 27, 28, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 43, 45, 46, 47, 49, 50, 51, 52, 53, 55, 57, 58, 59, 61, 63, 64, 67, 68, 71, 73, 74, 79, 80, 81, 82, 89, 93, 96, 97, 98, 100, 101

Offset: 1

Amiram Eldar, Mar 23 2024

nonn

It is unknown whether 222 is a term of this sequence or not (see A371423).

			3 is a term because when we start with 3 and repeatedly apply the mapping x -> A371418(x), we get the sequence 3, 2, 1, 1, 1, ...
40 is a term because when we start with 40 and repeatedly apply the mapping x -> A371418(x), we get the sequence 40, 45, 39, 28, 28, 28, ...

Amiram Eldar, Table of n, a(n) for n = 1..113
Robert D. Carmichael, Empirical Results in the Theory of Numbers, The Mathematics Teacher, Vol. 14, No. 6 (1921), pp. 305-310; alternative link. See p. 309.

Cf. A371418, A371422, A371423.
A023194 is a subsequence.
Cf. A063769, A080907.

r[n_] := n/FactorInteger[n][[1, 1]]; f[n_] := r[DivisorSigma[1, n]]; q[n_] := Module[{m = NestWhileList[f, n, UnsameQ, All][[-1]]}, f[m] == m]; Select[Range[221], q]

A371422 Numbers whose aliquot-like sequence based on the largest aliquot divisor of the sum of divisors of n (A371418) terminates in a cycle of length 2.

Original entry on oeis.org

12, 14, 15, 23, 29, 42, 44, 48, 54, 56, 60, 62, 65, 66, 69, 70, 72, 75, 76, 77, 78, 83, 84, 85, 86, 87, 88, 90, 91, 92, 94, 95, 99, 102, 107, 108, 110, 111, 112, 114, 115, 117, 118, 119, 120, 123, 124, 125, 128, 129, 131, 132, 134, 135, 136, 137, 139, 140, 142

Offset: 1

Amiram Eldar, Mar 23 2024

nonn

It is unknown whether 222 is a term of this sequence or not (see A371423).

			12 is a term because when we start with 12 and repeatedly apply the mapping x -> A371418(x), we get the sequence 12, 14, 12, 14, ...
76 is a term because when we start with 76 and repeatedly apply the mapping x -> A371418(x), we get the sequence 76, 70, 72, 65, 42, 48, 62, 48, 62, ...

Amiram Eldar, Table of n, a(n) for n = 1..108
Robert D. Carmichael, Empirical Results in the Theory of Numbers, The Mathematics Teacher, Vol. 14, No. 6 (1921), pp. 305-310; alternative link. See p. 309.

Cf. A371418, A371421, A371423.

Similar sequences: A127655, A127660, A127665.

r[n_] := n/FactorInteger[n][[1, 1]]; f[n_] := r[DivisorSigma[1, n]];
q[n_] := Module[{m = NestWhileList[f, n, UnsameQ, All][[-1]], k}, k = f[m]; k != m && f[k] == m]; Select[Range[221], q]

A371423 Aliquot-like sequence based on the largest aliquot divisor of the sum of divisors of n (A371418) that starts with 222.

Original entry on oeis.org

222, 228, 280, 360, 585, 546, 672, 1008, 1612, 1568, 1197, 1040, 1302, 1536, 2046, 2304, 949, 518, 456, 600, 930, 1152, 1105, 756, 1120, 1512, 2400, 3906, 4992, 7140, 12096, 20320, 24192, 40800, 70308, 108416, 135660, 241920, 490560, 902208, 1235456, 1309440, 2354688

Offset: 1

Amiram Eldar, Mar 23 2024

nonn

222 is the least number k for which the repeated iterations of the mapping k -> A371418(k) seem to generate an unbounded sequence.

			a(1) = 222 by definition.
a(2) = A371418(a(1)) = A371418(222) = 228.
a(3) = A371418(a(2)) = A371418(228) = 280.

Amiram Eldar, Table of n, a(n) for n = 1..1389
Robert D. Carmichael, Empirical Results in the Theory of Numbers, The Mathematics Teacher, Vol. 14, No. 6 (1921), pp. 305-310; alternative link. See p. 309.

Cf. A371418, A371421, A371422.

Similar sequences: A008892, A323328, A361421.

r[n_] := n/FactorInteger[n][[1, 1]]; f[n_] := r[DivisorSigma[1, n]]; NestList[f, 222, 60]

f(n) = {my(s = sigma(n)); if(s == 1, 1, s/factor(s)[1, 1]);}
lista(nmax) = {my(m = 222); for(n = 1, nmax, print1(m, ", "); m = f(m));}

A371419 Lesser member of Carmichael's variant of amicable pair: numbers k < m such that s(k) = m and s(m) = k, where s(k) = A371418(k).

Original entry on oeis.org

12, 48, 112, 160, 192, 448, 1984, 12288, 28672, 126976, 196608, 458752, 520192, 786432, 1835008, 2031616, 8126464, 8323072, 33292288, 536805376, 2147221504, 3221225472, 7516192768, 33285996544, 34359476224, 136365211648

Offset: 1

Amiram Eldar, Mar 23 2024

Analogous to amicable numbers (A002025 and A002046) with the largest aliquot divisor of the sum of divisors (A371418) instead of the sum of aliquot divisors (A001065).
Carmichael (1921) proposed this function (A371418) for the purpose of studying periodic chains that are formed by repeatedly applying the mapping x -> A371418(x). The chains of cycle 2 are analogous to amicable numbers.
Carmichael noted that if q < p are two different Mersenne exponents (A000043), then 2^(p-1)*(2^q-1) and 2^(q-1)*(2^p-1) are an amicable pair. With the 51 Mersenne exponents that are currently known it is possible to calculate 51 * 50 / 2 = 1275 amicable pairs. (160, 189) is a pair that is not of this "Mersenne form". Are there any other pairs like it? There are no other such pairs with lesser member below a(26).
a(27) <= 8795019280384.
The greater counterparts are in A371420.

			12 is a term since A371418(12) = 14 > 12, and A371418(14) = 12.

Robert D. Carmichael, Empirical Results in the Theory of Numbers, The Mathematics Teacher, Vol. 14, No. 6 (1921), pp. 305-310; alternative link. See p. 309.

Cf. A000043, A001065, A002046, A337874, A337876, A371418, A371420.

Similar sequences: A002025, A002952, A126165, A126169, A259038, A292980, A322541, A324708, A348343.

r[n_] := n/FactorInteger[n][[1, 1]]; s[n_] := r[DivisorSigma[1, n]]; seq = {}; Do[m = s[n]; If[m > n && s[m] == n, AppendTo[seq, n]], {n, 1, 10^6}]; seq

f(n) = {my(s = sigma(n)); if(s == 1, 1, s/factor(s)[1, 1]);}
lista(nmax) = {my(m); for(n = 1, nmax, m = f(n); if(m > n && f(m) == n, print1(n, ", ")));}

A371420 Greater member of Carmichael's variant of amicable pair: numbers k < m such that s(k) = m and s(m) = k, where s(k) = A371418(k).

Original entry on oeis.org

14, 62, 124, 189, 254, 508, 2032, 16382, 32764, 131056, 262142, 524284, 524224, 1048574, 2097148, 2097136, 8388592, 8388544, 33554368, 536866816, 2147479552, 4294967294, 8589934588, 34359738352, 34359672832, 137438953408

Offset: 1

Amiram Eldar, Mar 23 2024

The terms are ordered according to their lesser counterparts (A371419).

			14 is a term since A371418(14) = 12 < 14, and A371418(12) = 14.

Robert D. Carmichael, Empirical Results in the Theory of Numbers, The Mathematics Teacher, Vol. 14, No. 6 (1921), pp. 305-310; alternative link. See p. 309.

Cf. A371418, A371419.

Similar sequences: A002046, A002953, A126166, A126170, A259039, A292981, A322542, A324709, A348344.

r[n_] := n/FactorInteger[n][[1, 1]]; s[n_] := r[DivisorSigma[1, n]]; seq = {}; Do[m = s[n]; If[m > n && s[m] == n, AppendTo[seq, m]], {n, 1, 10^6}]; seq

f(n) = {my(s = sigma(n)); if(s == 1, 1, s/factor(s)[1, 1]);}
lista(nmax) = {my(m); for(n = 1, nmax, m = f(n); if(m > n && f(m) == n, print1(m, ", ")));}

cp's OEIS Frontend

A371421 Numbers whose aliquot-like sequence based on the largest aliquot divisor of the sum of divisors of n (A371418) terminates in a fixed point.

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A371422 Numbers whose aliquot-like sequence based on the largest aliquot divisor of the sum of divisors of n (A371418) terminates in a cycle of length 2.

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A371423 Aliquot-like sequence based on the largest aliquot divisor of the sum of divisors of n (A371418) that starts with 222.

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A371419 Lesser member of Carmichael's variant of amicable pair: numbers k < m such that s(k) = m and s(m) = k, where s(k) = A371418(k).

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A371420 Greater member of Carmichael's variant of amicable pair: numbers k < m such that s(k) = m and s(m) = k, where s(k) = A371418(k).

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