A324404 Squarefree integers m > 1 such that if prime p divides m, then s_p(m) >= p and s_p(m) == 2 (mod p-1), where s_p(m) is the sum of the base p digits of m.

1122, 3458, 5642, 6734, 11102, 13202, 17390, 17822, 21170, 22610, 27962, 31682, 46002, 58682, 61778, 79730, 82082, 93314, 105266, 106262, 125490, 127946, 136202, 150722, 153254, 177122, 182002, 202202, 203870, 214370, 231842, 252434, 274298, 278462, 305102, 315282

Offset: 1

Author

Bernd C. Kellner and Jonathan Sondow, Feb 26 2019

Keywords

• nonn
• base

Comments

For d >= 1 define S_d = (terms m in A324315 such that s_p(m) == d (mod p-1) if prime p divides m). Then S_1 is precisely the Carmichael numbers (A002997), S_2 is A324404, S_3 is A324405, and the union of all S_d for d >= 1 is A324315.

Subsequence of the 2-Knödel numbers (A050990). Generally, for d > 1 the terms of S_d that are greater than d form a subsequence of the d-Knödel numbers.

See Kellner and Sondow 2019.

Examples

			1122 = 2*3*11*17 is squarefree and equals 10001100010_2, 1112120_3, 930_11, and 3f0_17 in base p = 2, 3, 11, and 17. Then s_2(1122) = 1+1+1+1 = 4 >= 2, s_3(1122) = 1+1+1+2+1+2 = 8 >= 3, s_11(1122) = 9+3 = 12 >= 11, and s_17(1122) = 3+f = 3+15 = 18 >= 17. Also, s_2(1122) = 4 == 2 (mod 1), s_3(1122) = 8 == 2 (mod 2), s_11(1122) = 12 == 2 (mod 10), and s_17(1122) = 18 == 2 (mod 16), so 1122 is a member.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..2200
• Bernd C. Kellner, On a product of certain primes, J. Number Theory, 179 (2017), 126-141; arXiv:1705.04303 [math.NT], 2017.
• Bernd C. Kellner and Jonathan Sondow, Power-Sum Denominators, Amer. Math. Monthly, 124 (2017), 695-709; arXiv:1705.03857 [math.NT], 2017.
• 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], 2019-2021.

Crossrefs

Cf. A002997, A050990, A324315, A324316, A324317, A324318, A324319, A324320, A324369, A324370, A324371, A324405.

Programs

Mathematica

SD[n_, p_] := If[n < 1 || p < 2, 0, Plus @@ IntegerDigits[n, p]];
LP[n_] := Transpose[FactorInteger[n]][[1]];
TestSd[n_, d_] := (n > 1) && (d > 0) && SquareFreeQ[n] && VectorQ[LP[n], SD[n, #] >= # && Mod[SD[n, #] - d, # - 1] == 0 &];
Select[Range[200000], TestSd[#, 2] &]

Extensions

More terms from Amiram Eldar, Dec 05 2020