A324455 -id:A324455 - cp's OEIS frontend

A324456 Numbers m > 1 such that there exists a divisor g > 1 of m which satisfies s_g(m) = g.

6, 10, 12, 15, 18, 20, 21, 24, 28, 33, 34, 36, 39, 40, 45, 48, 52, 57, 63, 65, 66, 68, 72, 76, 80, 85, 87, 88, 91, 93, 96, 99, 100, 105, 111, 112, 117, 120, 126, 130, 132, 133, 135, 136, 144, 145, 148, 153, 156, 160, 165, 171, 175, 176, 185, 186, 189, 190

Offset: 1

Bernd C. Kellner, Feb 28 2019

The function s_g(m) gives the sum of the base-g digits of m.
The sequence is infinite, since it contains A324460.
The sequence also contains the 3-Carmichael numbers A087788 and the primary Carmichael numbers A324316.
A term m must have at least 2 prime factors, and the divisor g satisfies the inequalities 1 < g < m^(1/(ord_g(m)+1)) <= sqrt(m), where ord_g(m) gives the maximum exponent e such that g^e divides m.
Note that the sequence contains the 3-Carmichael numbers, but not all Carmichael numbers. This is a nontrivial fact.
The subsequence A324460 mainly gives examples in which g is composite.
See Kellner 2019.
It appears that g is usually prime: compare with A324857 (g prime) and the sparser sequence A324858 (g composite). However, g is usually composite for higher values of m. - Jonathan Sondow, Mar 17 2019

			6 is a member, since 2 divides 6 and s_2(6) = 2.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..806 from Bernd C. Kellner)
Bernd C. Kellner, On primary Carmichael numbers, Integers 22 (2022), Article #A38, 39 pp.; arXiv:1902.11283 [math.NT], 2019.
Bernd C. Kellner and Jonathan Sondow, On Carmichael and polygonal numbers, Bernoulli polynomials, and sums of base-p digits, Integers 21 (2021), Article #A52, 21 pp.; arXiv:1902.10672 [math.NT], 2019.

Subsequences are A033502, A087788, A324316, A324458, A324460.
Subsequence of A324455.
Cf. A324315, A324457, A324459.
Union of A324857 and A324858.

s[n_, g_] := If[n < 1 || g < 2, 0, Plus @@ IntegerDigits[n, g]];
f[n_] := AnyTrue[Divisors[n], s[n, #] == # &];
Select[Range[5000], f[#] &]

isok(n) = {fordiv(n, d, if ((d>1) && (sumdigits(n, d) == d), return (1)););} \\ Michel Marcus, Mar 19 2019

A324460 Numbers m > 1 that have a strict s-decomposition.

Original entry on oeis.org

45, 96, 225, 325, 405, 576, 637, 640, 891, 1225, 1377, 1408, 1536, 1701, 1729, 2025, 2541, 2821, 3321, 3751, 3825, 4225, 4608, 4961, 6400, 6517, 6525, 7381, 7840, 8125, 8281, 9216, 9537, 9801, 10625, 10935, 12025, 12288, 12825, 12936, 13125, 13312, 13357

Offset: 1

Bernd C. Kellner, Feb 28 2019

The sequence contains the primary Carmichael numbers A324316.
The sequence is infinite. If f(x) counts such numbers m below x, then f(x) > 1/11 x^(1/3) - 1/3 for x >= 1.
A number m > 1 has a strict s-decomposition if there exists a decomposition in n proper factors g_k with exponents e_k >= 1 (the factors g_k being strictly increasing but not necessarily coprime) such that
    m = g_1^e_1 * ... * g_n^e_n, where s_{g_k}(m) = g_k for all k,
  and s_g(m) gives the sum of the base-g digits of m.
A term m has the following properties:
m must have at least 2 factors g_k. If m = g_1^e_1 * g_2^e_2 with exactly two factors, then e_1 + e_2 >= 3.
Each factor g_k of m satisfies the inequalities 1 < g_k < m^(1/(ord_{g_k}(m)+1)) <= sqrt(m), where ord_g(m) gives the maximum exponent e such that g^e divides m.
See Kellner 2019.

			Since 576 = 2^4 * 6^2 with s_2(576) = 2 and s_6(576) = 6, 576 is a member.

Bernd C. Kellner, Table of n, a(n) for n = 1..254
Bernd C. Kellner, On primary Carmichael numbers, Integers 22 (2022), Article #A38, 39 pp.; arXiv:1902.11283 [math.NT], 2019.

Subsequences are A324316, A324458. Subsequence of A324459.

Cf. A324455, A324456, A324457.

s[n_, g_] := If[n < 1 || g < 2, 0, Plus @@ IntegerDigits[n, g]];
HasDecompS[m_] := Module[{E0, EV, G, R, k, n, v},
If[m < 1 || !CompositeQ[m], Return[False]];
G = Select[Divisors[m], s[m, #] == # &];
n = Length[G]; If[n < 2, Return[False]];
E0 = Array[0 &, n]; EV = Array[v, n];
R = Solve[Product[G[[k]]^EV[[k]], {k, 1, n}] == m && EV >= E0, EV, Integers]; Return[R != {}]];
Select[Range[10^4], HasDecompS[#] &]

A324457 Numbers m > 1 such that every prime divisor p of m satisfies s_p(m) >= p.

Original entry on oeis.org

24, 45, 48, 72, 96, 120, 144, 189, 192, 216, 224, 225, 231, 240, 288, 315, 320, 325, 336, 352, 360, 384, 405, 432, 450, 480, 525, 540, 560, 561, 567, 576, 594, 600, 637, 648, 672, 704, 720, 768, 792, 819, 825, 832, 850, 864, 891, 896, 924, 945, 960, 975, 980

Offset: 1

Bernd C. Kellner, Feb 28 2019

The function s_p(m) gives the sum of the base-p digits of m.
The sequence is infinite, since it contains A324315, and thus the Carmichael numbers A002997.
Being a subsequence of A324459, a term m has the following properties:
m must have at least 2 prime factors. If m = p1^e1 * p2^e2 with two primes p1 and p2, then e1 + e2 >= 3.
Each prime factor p of m satisfies the inequalities p < m^(1/(ord_p(m)+1)) <= sqrt(m), where ord_p(m) gives the maximum exponent e such that p^e divides m.
In the terminology of A324459, the prime factorization of m equals an s-decomposition of m.
See Kellner 2019.
a(n) is a Carmichael number A002997 iff a(n) is squarefree and s_p(a(n)) == 1 (mod p-1) for every prime factor p of a(n). See Kellner and Sondow 2019. - Jonathan Sondow, Mar 16 2019

			The number 45 has the prime factors 3 and 5. Since s_3(45) = 3 and s_5(45) = 5, 45 is a member.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..477 from Bernd C. Kellner)
Bernd C. Kellner, On primary Carmichael numbers, Integers 22 (2022), Article #A38, 39 pp.; arXiv:1902.11283 [math.NT], 2019.
Bernd C. Kellner and Jonathan Sondow, On Carmichael and polygonal numbers, Bernoulli polynomials, and sums of base-p digits, Integers 21 (2021), Article #A52, 21 pp.; arXiv:1902.10672 [math.NT], 2019.

Subsequences are A002997, A324315, and A324458.
Subsequence of A324459 and A324857.
Cf. A324316, A324455, A324456, A324460.

s[n_, p_] := If[n < 1 || p < 2, 0, Plus @@ IntegerDigits[n, p]];
f[n_] := AllTrue[Transpose[FactorInteger[n]][[1]], s[n, #] >= # &];
Select[Range[10^4], f[#] &]

A324458 Numbers m > 1 such that every prime divisor p of m satisfies s_p(m) = p.

Original entry on oeis.org

45, 325, 405, 637, 891, 1729, 2821, 3751, 4961, 6517, 7381, 8125, 8281, 10625, 13357, 21141, 26353, 28033, 29341, 31213, 33125, 35443, 46657, 47081, 58621, 65341, 74431, 78625, 81289, 94501, 98125, 99937, 123823, 146461, 231601, 236321, 252601, 254221, 294409

Offset: 1

Bernd C. Kellner, Feb 28 2019

The function s_p(m) gives the sum of the base-p digits of m.
The sequence contains the primary Carmichael numbers A324316.
Being a subsequence of A324460, a term m has the following properties:
m must have at least 2 prime factors. If m = p1^e1 * p2^e2 with two primes p1 and p2, then e1 + e2 >= 3.
Each prime factor p of m satisfies the inequalities p < m^(1/(ord_p(m)+1)) <= sqrt(m), where ord_p(m) gives the maximum exponent e such that p^e divides m.
In the terminology of A324460, the prime factorization of m equals a strict s-decomposition of m.
See Kellner 2019.
a(n) is squarefree iff a(n) is a primary Carmichael number A324316. - Jonathan Sondow, Mar 16 2019

			The number 45 has the prime factors 3 and 5. Since s_3(45) = 3 and s_5(45) = 5, 45 is a member.

Amiram Eldar, Table of n, a(n) for n = 1..1000 (terms 1..117 from Bernd C. Kellner)
Bernd C. Kellner, On primary Carmichael numbers, Integers 22 (2022), Article #A38, 39 pp.; arXiv:1902.11283 [math.NT], 2019.
Bernd C. Kellner and Jonathan Sondow, On Carmichael and polygonal numbers, Bernoulli polynomials, and sums of base-p digits, Integers 21 (2021), Article #A52, 21 pp.; arXiv:1902.10672 [math.NT], 2019.

Subsequence is A324316. Subsequence of A324457, A324459, and A324460.

Cf. A324455, A324456.

s[n_, p_] := If[n < 1 || p < 2, 0, Plus @@ IntegerDigits[n, p]];
f[n_] := AllTrue[Transpose[FactorInteger[n]][[1]], s[n, #] == # &];
Select[Range[10^7], f[#] &]

A324459 Numbers m > 1 that have an s-decomposition.

Original entry on oeis.org

24, 45, 48, 72, 96, 120, 144, 189, 192, 216, 224, 225, 231, 240, 280, 288, 315, 320, 325, 336, 352, 360, 378, 384, 405, 432, 450, 480, 525, 540, 560, 561, 567, 576, 594, 600, 637, 640, 648, 672, 704, 720, 768, 792, 819, 825, 832, 850, 864, 891, 896, 924, 945

Offset: 1

Bernd C. Kellner, Feb 28 2019

The sequence is infinite, since it contains A324460 and the Carmichael numbers A002997.
A number m > 1 has an s-decomposition if there exists a decomposition in n proper factors g_k with exponents e_k >= 1 (the factors g_k being strictly increasing but not necessarily coprime) such that
    m = g_1^e_1 * ... * g_n^e_n, where s_{g_k}(m) >= g_k for all k,
  and s_g(m) gives the sum of the base-g digits of m.
A term m has the following properties:
m must have at least 2 factors g_k. If m = g_1^e_1 * g_2^e_2 with exactly two factors, then e_1 + e_2 >= 3.
Each factor g_k of m satisfies the inequalities 1 < g_k < m^(1/(ord_{g_k}(m)+1)) <= sqrt(m), where ord_g(m) gives the maximum exponent e such that g^e divides m.
See Kellner 2019.

			Since 225 = 5^2 * 9 with s_5(225) = 5 and s_9(225) = 9, 225 is a member.

Bernd C. Kellner, Table of n, a(n) for n = 1..532
Bernd C. Kellner, On primary Carmichael numbers, Integers 22 (2022), Article #A38, 39 pp.; arXiv:1902.11283 [math.NT], 2019.

Subsequences are A002997, A324457, A324458, A324460.

Cf. A324455, A324456.

s[n_, g_] := If[n < 1 || g < 2, 0, Plus @@ IntegerDigits[n, g]];
HasDecomp[m_] := Module[{E0, EV, G, R, k, n, v},
If[m < 1 || !CompositeQ[m], Return[False]];
G = Select[Divisors[m], s[m, #] >= # &];
n = Length[G]; If[n < 2, Return[False]];
E0 = Array[0 &, n]; EV = Array[v, n];
R = Solve[Product[G[[k]]^EV[[k]], {k, 1, n}] == m && EV >= E0, EV, Integers]; Return[R != {}]];
Select[Range[10^3], HasDecomp[#] &]

A386207 Numbers m > 1 such that there exists k such that k | m, k^k = k mod m and 1 < k < m.

Original entry on oeis.org

6, 10, 12, 14, 15, 18, 20, 21, 22, 24, 26, 28, 30, 33, 34, 36, 38, 39, 40, 42, 44, 45, 46, 48, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 65, 66, 68, 69, 70, 72, 74, 75, 76, 78, 80, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 98, 100, 102, 104, 105, 106, 108, 110, 111, 112, 114, 116

Offset: 1

Arjun Maneesh Agarwal, Jul 15 2025

nonn

The sequence seems to be very similar to A324455 but there are terms present here, which are absent there and vice versa. Also, the k satisfying the property for a given n, even when it exists, is different for both.
The sequence is infinite as A016825 (4m+2, m > 0) is a subsequence with the corresponding k = 2m + 1.
Another subsequence is A002997 (Carmichael numbers).

			6 is a term as 3^3 = 27 = 3 mod 6.
10 is a term as 5^5 = 3125 = 5 mod 10.

Arjun Maneesh Agarwal, Table of n, a(n) for n = 1..10000

Cf. A324455, A016825, A002997.

divisors n = [x | x <- [2..n], n `mod` x == 0]
property n = any (\x -> x^x `mod` n == x) $ divisors n
inRange t = [x | x <- [2..t], property x]

isok(m) = fordiv(m, k, if((k>1) && (kMichel Marcus, Jul 16 2025

cp's OEIS Frontend

A324456 Numbers m > 1 such that there exists a divisor g > 1 of m which satisfies s_g(m) = g.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A324460 Numbers m > 1 that have a strict s-decomposition.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A324457 Numbers m > 1 such that every prime divisor p of m satisfies s_p(m) >= p.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A324458 Numbers m > 1 such that every prime divisor p of m satisfies s_p(m) = p.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A324459 Numbers m > 1 that have an s-decomposition.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A386207 Numbers m > 1 such that there exists k such that k | m, k^k = k mod m and 1 < k < m.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

PARI