A306509 -id:A306509

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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.

+10
1

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

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)

LINKS

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

FORMULA

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

EXAMPLE

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

MAPLE

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

PROG

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

CROSSREFS

Cf. A000005, A007953, A030513, A306509.

KEYWORD

nonn,base

AUTHOR

Ctibor O. Zizka, Feb 20 2019

STATUS

approved

A356061

Numbers whose sum of digits is a refactorable number.

+10
1

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

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

LINKS

Table of n, a(n) for n=1..63.

EXAMPLE

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

MAPLE

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

MATHEMATICA

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

PROG

(Python)

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

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

CROSSREFS

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

KEYWORD

nonn,base,easy

AUTHOR

Ctibor O. Zizka, Aug 10 2022

STATUS

approved

A356520

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

+10
1

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

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

LINKS

Table of n, a(n) for n=1..37.

EXAMPLE

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

MATHEMATICA

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

PROG

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

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

(Python)

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

CROSSREFS

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

KEYWORD

nonn,easy,base

AUTHOR

Ctibor O. Zizka, Aug 10 2022

STATUS

approved

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