Vighnesh Patil - cp's OEIS frontend

A385660 Numbers k such that prime(k+1)-prime(k) divides k.

1, 2, 4, 8, 10, 12, 18, 20, 24, 26, 28, 36, 44, 48, 52, 54, 60, 64, 72, 80, 84, 88, 96, 98, 102, 104, 108, 112, 116, 120, 128, 136, 140, 142, 144, 148, 152, 168, 174, 176, 178, 180, 182, 190, 192, 206, 210, 212, 216, 224, 230, 234, 236, 240, 244, 248, 252, 256, 262, 264, 268, 276, 286, 288, 294

Offset: 1

Vighnesh Patil, Jul 06 2025

			k = 18 is a term since prime(19) - prime(18) = 6 divides 18.

Vighnesh Patil, Table of n, a(n) for n = 1..1000

Cf. A000040, A001223, A029707.

q[k_] := Divisible[k, Prime[k + 1] - Prime[k]]; Select[Range[300], q] (* Amiram Eldar, Jul 09 2025 *)

from sympy import prime
def ok(k): return k % (prime(k+1) - prime(k)) == 0
print([k for k in range(1, 300) if ok(k)])

A385656 Numbers k such that the sum of the decimal digits of k^2 divides k^2.

Original entry on oeis.org

1, 2, 3, 6, 9, 10, 12, 15, 18, 20, 21, 24, 30, 36, 39, 42, 45, 48, 49, 51, 52, 54, 60, 63, 65, 66, 68, 72, 78, 80, 84, 88, 90, 96, 100, 102, 104, 105, 108, 110, 111, 112, 117, 120, 126, 132, 138, 140, 144, 148, 150, 156, 162, 168, 174, 180, 182, 190, 198, 200, 201, 204, 207

Offset: 1

Vighnesh Patil, Jul 06 2025

			15 is a term 15^2 = 225; digit sum of 225 = 2 + 2 + 5 = 9; 225 mod 9 = 0, so 15 is included.
18 is a term 18^2 = 324; digit sum of 324 = 3 + 2 + 4 = 9; 324 mod 9 = 0, so 16 is included.

Vighnesh Patil, Table of n, a(n) for n = 1..1000

Cf. A005349, A007953, A034706, A118547, A385640.

digitSum := n -> add(i,i=convert(n, base, 10)):
isok := n -> modp(n^2, digitSum(n^2)) = 0:
select(isok, [$1..400])[];

DigitSum[n_] := Total[IntegerDigits[n]];
Select[Range[400], Mod[#^2, DigitSum[#^2]] == 0 &]

isok(k) = (k^2 % sumdigits(k^2)) == 0; \\ Michel Marcus, Jul 06 2025

def digit_sum(n): return sum(int(d) for d in str(n))
def ok(n): return (n**2) % digit_sum(n**2) == 0
print([n for n in range(1, 1000) if ok(n)])

a(n) = sqrt(A118547(n)). - Michel Marcus, Jul 07 2025

A385640 Numbers k such that the sum of the digits of k divides k and the sum of the digits of k^2 divides k^2.

Original entry on oeis.org

1, 2, 3, 6, 9, 10, 12, 18, 20, 21, 24, 30, 36, 42, 45, 48, 54, 60, 63, 72, 80, 84, 90, 100, 102, 108, 110, 111, 112, 117, 120, 126, 132, 140, 144, 150, 156, 162, 180, 190, 198, 200, 201, 204, 207, 210, 216, 220, 234, 240, 243, 252, 264, 270, 288, 300, 306, 315

Offset: 1

Vighnesh Patil, Jul 05 2025

If k is in the sequence then so is 10*k. - David A. Corneth, Jul 06 2025

			18 is a term since 1+8 = 9 and 18 mod 9 = 0; also, 18^2 = 324, and 3+2+4 = 9 and 324 mod 9 = 0.

Vighnesh Patil, Table of n, a(n) for n = 1..1000

Intersection of A005349 and A385656.

Cf. A007953.

A385640Q[k_] :=  Divisible[k, DigitSum[k]] && Divisible[k^2, DigitSum[k^2]];
Select[Range[500], A385640Q] (* Paolo Xausa, Jul 06 2025 *)

def digit_sum(n): return sum(int(d) for d in str(n))
def ok(n):
    return n % digit_sum(n) == 0 and (n**2) % digit_sum(n**2) == 0
print([n for n in range(1, 400) if ok(n)])

A005349(a(n)) | a(n) and A005349(a(n)^2) | a(n)^2.

{k | k in A005349 and k^2 in A005349}. - Michael S. Branicky, Jul 05 2025

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User: Vighnesh Patil

A385660 Numbers k such that prime(k+1)-prime(k) divides k.

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A385656 Numbers k such that the sum of the decimal digits of k^2 divides k^2.

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A385640 Numbers k such that the sum of the digits of k divides k and the sum of the digits of k^2 divides k^2.

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