A324456 -id:A324456 - cp's OEIS frontend

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

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[#] &]

A324455 Numbers m > 1 such that there exists a divisor g > 1 of m which satisfies s_g(m) >= g.

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, 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, 96, 98, 99, 100, 102, 104, 105

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 and the Carmichael numbers A002997.
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.
See Kellner 2019.

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

Bernd C. Kellner, Table of n, a(n) for n = 1..743
Bernd C. Kellner, On primary Carmichael numbers, Integers 22 (2022), Article #A38, 39 pp.; arXiv:1902.11283 [math.NT], 2019.

Subsequences are A002997, A324315, A324316, A324456, A324457, A324458, A324459, A324460.

s[n_, g_] := If[n < 1 || g < 2, 0, Plus @@ IntegerDigits[n, g]];
f[n_] := AnyTrue[Divisors[n], s[n, #] >= # &];
Select[Range[1000], f[#] &]

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[#] &]

A324857 Numbers m > 1 such that there exists a prime divisor p of m with s_p(m) = p.

Original entry on oeis.org

6, 10, 12, 15, 18, 20, 21, 24, 33, 34, 36, 39, 40, 45, 48, 57, 63, 65, 66, 68, 72, 80, 85, 87, 91, 93, 96, 99, 105, 111, 117, 130, 132, 133, 135, 136, 144, 145, 160, 165, 171, 175, 185, 189, 192, 205, 217, 225, 231, 249, 255, 258, 259, 260, 261, 264, 265, 272, 273, 279, 285, 288, 297, 301, 305, 320, 325, 327, 333, 341, 351, 384, 385

Offset: 1

Jonathan Sondow, Mar 17 2019

The function s_p(m) gives the sum of the base-p digits of m.
m must have at least 2 prime factors, since s_p(p^k) = 1 < p.
The sequence contains the primary Carmichael numbers A324316.
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 sparser sequence A324858 = numbers m > 1 such that there exists a composite divisor c of m with s_c(m) = c. However, d is usually composite for higher values of m.
The sequence contains the 3-Carmichael numbers A087788, but not all Carmichael numbers A002997. This is a nontrivial fact. The smallest Carmichael number that is not a member is 173085121 = 11*31*53*61*157. For further properties of the terms see A324456 and Kellner 2019. - Bernd C. Kellner, Apr 02 2019

			s_p(m) = 1 < p for m = 2, 3, 4, 5 with prime p dividing m, but if m = 6 and p = 2 then s_p(m) = s_2(2 + 2^2) = 1 + 1 = 2 = p, so a(1) = 6.

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.

Includes A083558.

S:= (p,m) -> convert(convert(m,base,p),`+`):
filter:= proc(m) ormap(p -> S(p,m) = p, numtheory:-factorset(m)) end proc:
select(filter, [$2..500]); # Robert Israel, Mar 20 2019

s[n_, b_] := If[n < 1 || b < 2, 0, Plus @@ IntegerDigits[n, b]];
f[n_] := AnyTrue[Divisors[n], PrimeQ[#] && s[n, #] == # &];
Select[Range[400], f[#] &]n (* simplified by Bernd C. Kellner, Apr 02 2019 *)

isok(n) = {if (n>1, my(vp=factor(n)[,1]); for (k=1, #vp, if (sumdigits(n, vp[k]) == vp[k], return (1)))); } \\ Michel Marcus, Mar 19 2019

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

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

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A324455 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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A324458 Numbers m > 1 such that every prime divisor p of m satisfies s_p(m) = p.

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

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A324857 Numbers m > 1 such that there exists a prime divisor p of m with 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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