A324857 -id:A324857 - 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

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[#] &]

A324858 Numbers m > 1 such that there exists a composite divisor c of m with s_c(m) = c.

Original entry on oeis.org

28, 40, 52, 66, 76, 88, 96, 100, 112, 120, 126, 136, 148, 153, 156, 160, 176, 186, 190, 196, 208, 225, 232, 246, 268, 276, 280, 288, 292, 297, 304, 306, 328, 336, 340, 344, 352, 366, 369, 370, 378, 388, 396, 400, 408, 435, 441, 448, 456, 460, 486, 496, 513, 516, 520, 532, 540, 544, 546, 550, 560, 568, 576, 580, 585, 592

Offset: 1

Jonathan Sondow, Mar 17 2019

The function s_c(m) gives the sum of the base-c digits of m.
The main entry for this sequence is A324456 = numbers m > 1 such that there exists a divisor d > 1 of m with s_d(m) = d. It appears that d is usually prime: compare the subsequence A324857 = numbers m > 1 such that there exists a prime divisor p of m with s_p(m) = p. However, d is usually composite for higher values of m.
For any composite c, 0 < b < c, and 0 < i < j, b*c^i + (c-b)*c^j is in the sequence. - Robert Israel, Mar 19 2019
The sequence does not contain the 3-Carmichael numbers A087788, but intersects the Carmichael numbers A002997 that have at least four factors. This is a nontrivial fact. Examples for such Carmichael numbers below one million: 41041 = 7*11*13*41, 172081 = 7*13*31*61, 188461 = 7*13*19*109, 278545 = 5*17*29*113, 340561 = 13*17*23*67, 825265 = 5*7*17*19*73. For further properties of the terms see A324456 and Kellner 2019. - Bernd C. Kellner, Apr 02 2019

			s_4(28) = 4 as 28 = 3 * 4 + 1 * 4^2, so 28 is a member.

Robert Israel, Table of n, a(n) for n = 1..10000
Bernd C. Kellner, On primary Carmichael numbers, Integers 22 (2022), Article #A38, 39 pp.; arXiv:1902.11283 [math.NT], 2019.

A324456 is the union of A324857 and A324858.

S:= proc(c,m) convert(convert(m,base,c),`+`) end proc:
filter:= proc(m) ormap(c -> (S(c,m)=c), remove(isprime,numtheory:-divisors(m) minus {1})) end proc:
select(filter, [$1..1000]); # Robert Israel, Mar 19 2019

s[n_, b_] := If[n < 1 || b < 2, 0, Plus @@ IntegerDigits[n, b]];
f[n_] := AnyTrue[Divisors[n], CompositeQ[#] && s[n, #] == # &];
Select[Range[600], f[#] &] (* simplified by Bernd C. Kellner, Apr 02 2019 *)

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

cp's OEIS Frontend

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

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

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A324858 Numbers m > 1 such that there exists a composite divisor c of m with s_c(m) = c.

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