A306509 -id:A306509 - cp's OEIS frontend

A306510 Numbers k such that twice the number of divisors of k is equal to the number of divisors of the sum of digits of k.

17, 19, 37, 53, 59, 71, 73, 107, 109, 127, 149, 163, 167, 181, 233, 239, 251, 257, 271, 293, 307, 347, 383, 419, 431, 433, 491, 499, 503, 509, 521, 523, 541, 563, 613, 617, 631, 653, 699, 701, 743, 761, 769, 787, 789, 811, 859, 877, 879, 941, 967

Offset: 1

Ctibor O. Zizka, Feb 20 2019

From Robert Israel, Jul 28 2020: (Start)
The first even term is a(2747)=68998.
Includes primes p such that A007953(p) is in A030513. (End)

			For k = 19, 2*A000005(19) = A000005(A007953(19)), 2*A000005(19) = A000005(10), thus k = 19 is a member of the sequence.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000005, A007953, A030513, A306509.

filter:= proc(n) 2*numtheory:-tau(n) = numtheory:-tau(convert(convert(n,base,10),`+`)) end proc:
select(filter, [$1..1000]); # Robert Israel, Jul 28 2020

isok(k) = (k >= 1) && (2*numdiv(k) == numdiv(sumdigits(k, 10))); \\ Daniel Suteu, Feb 20 2019

2*A000005(k) = A000005(A007953(k)).

A356061 Numbers whose sum of digits is a refactorable number.

Original entry on oeis.org

1, 2, 8, 9, 10, 11, 17, 18, 20, 26, 27, 35, 36, 39, 44, 45, 48, 53, 54, 57, 62, 63, 66, 71, 72, 75, 80, 81, 84, 90, 93, 99, 100, 101, 107, 108, 110, 116, 117, 125, 126, 129, 134, 135, 138, 143, 144, 147, 152, 153, 156, 161, 162, 165, 170, 171, 174, 180, 183, 189, 192, 198, 200

Offset: 1

Ctibor O. Zizka, Aug 10 2022

Also numbers k such that A007953(k) = c * A000005(A007953(k)); c >= 1 is a positive integer. For c = 1 see A356520.

			k = 17; A007953(17) = 2 * A000005(A007953(17)), thus k = 17 is in the sequence.

Cf. A000005, A033950, A007953, A306509, A356520.

filter:= proc(n) local s; s:= convert(convert(n,base,10),`+`); s mod numtheory:-tau(s) = 0 end proc:
select(filter, [$1..200]); # Robert Israel, Aug 10 2022

refQ[n_] := Divisible[n, DivisorSigma[0,n]]; Select[Range[2000], refQ[Plus @@ IntegerDigits[#]] &] (* Amiram Eldar, Aug 10 2022 *)

isok(k) = my(s=sumdigits(k)); denominator(s/numdiv(s)) == 1; \\ Michel Marcus, Aug 10 2022

from sympy import divisor_count
def ok(n): sd = sum(map(int, str(n))); return sd%divisor_count(sd) == 0
print([k for k in range(1, 200) if ok(k)]) # Michael S. Branicky, Aug 10 2022

A356520 Numbers k such that A000005(A007953(k)) = A007953(k).

Original entry on oeis.org

1, 2, 10, 11, 20, 100, 101, 110, 200, 1000, 1001, 1010, 1100, 2000, 10000, 10001, 10010, 10100, 11000, 20000, 100000, 100001, 100010, 100100, 101000, 110000, 200000, 1000000, 1000001, 1000010, 1000100, 1001000, 1010000, 1100000, 2000000, 10000000, 10000001

Offset: 1

Ctibor O. Zizka, Aug 10 2022

Union of A011557 and A052216. I.e., numbers with digital sum 1 or 2. - David A. Corneth, Aug 10 2022

			k = 101; A000005(A007953(101)) = A007953(101) = 2, thus k = 101 is in the sequence.

Cf. A000005, A007953, A011557, A052216, A101318, A306509, A356061.

Select[Range[1,10000001], Plus @@ IntegerDigits[#] < 3 &] (* Amiram Eldar, Aug 10 2022 *)

isok(k) = my(s=sumdigits(k)); numdiv(s) == s; \\ Michel Marcus, Aug 10 2022

is(n) = my(s = sumdigits(n)); s == 1 || s == 2 \\ David A. Corneth, Aug 10 2022

from itertools import count, islice
def agen():
    for i in count(0):
        yield from [10**i] + [10**i + 10**j for j in range(i+1)]
print(list(islice(agen(), 37))) # Michael S. Branicky, Aug 10 2022

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A306510 Numbers k such that twice the number of divisors of k is equal to the number of divisors of the sum of digits of k.

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A356061 Numbers whose sum of digits is a refactorable number.

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A356520 Numbers k such that A000005(A007953(k)) = A007953(k).

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