A034690 -id:A034690 - cp's OEIS frontend

A106756 Primes with digit sum = 14.

59, 149, 167, 239, 257, 293, 347, 383, 419, 491, 509, 563, 617, 653, 743, 761, 941, 1049, 1193, 1229, 1283, 1319, 1373, 1409, 1427, 1481, 1553, 1571, 1607, 1733, 1823, 1913, 1931, 2039, 2129, 2237, 2273, 2309, 2381, 2417, 2543, 2633, 2741, 2903, 3083

Offset: 1

Zak Seidov, May 16 2005

Or prime numbers in A114527. - Zak Seidov, May 21 2006

Vincenzo Librandi, Table of n, a(n) for n = 1..5000

Cf. A000040 (primes), A007953 (sum of digits), A235225 (digit sum = 14).
Cf. A062339 (same for digit sum s = 4), A106755 (s = 13), A106757 (s = 16), and others listed in A244918 (s = 68).
Cf. A034690, A086793, A114527.

[p: p in PrimesUpTo(4000) | &+Intseq(p) eq 14]; // Vincenzo Librandi, Jul 08 2014

Select[Prime[Range[10000]], Total[IntegerDigits[#]]==14 &] (* Vincenzo Librandi, Jul 08 2014 *)

select( {is_A106756(n)= sumdigits(n)==14 && isprime(n)}, primes([1, 3333])) \\ M. F. Hasler, Mar 09 2022

Intersection of A000040 (primes) and A235225 (digit sum = 14); also equals { p in A000040 | A007953(p) = 14 }. - M. F. Hasler, Mar 09 2022

A134681 Number of digits of all the divisors of n.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 4, 3, 5, 3, 7, 3, 5, 5, 6, 3, 7, 3, 8, 5, 6, 3, 10, 4, 6, 5, 8, 3, 11, 3, 8, 6, 6, 5, 12, 3, 6, 6, 11, 3, 11, 3, 9, 8, 6, 3, 14, 4, 9, 6, 9, 3, 11, 6, 11, 6, 6, 3, 18, 3, 6, 8, 10, 6, 12, 3, 9, 6, 12, 3, 17, 3, 6, 9, 9, 6, 12, 3, 15, 7, 6, 3, 18, 6, 6, 6, 12, 3, 18, 6, 9, 6, 6, 6, 18

Offset: 1

Reinhard Zumkeller, Nov 06 2007

Also number of digits of the concatenation of all divisors of n (A037278). - Jaroslav Krizek, Jun 15 2011

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A000005, A037278, A055642, A190997, A034690, A134682.

A134681 := proc(n)
    add(A055642(d),d=numtheory[divisors](n)) ;
end proc:
seq(A134681(n),n=1..80) ; # R. J. Mathar, Feb 21 2025

Array[Total[IntegerLength[Divisors[#]]]&,100] (* Harvey P. Dale, Jun 08 2013 *)

a(n) = sumdiv(n, d, #digits(d)); \\ Michel Marcus, Sep 01 2023

from sympy import divisors
def a(n): return sum(len(str(d)) for d in divisors(n, generator=True))
print([a(n) for n in range(1, 97)]) # Michael S. Branicky, Nov 03 2023

a(n) = A055642(A037278(n)) = Number of digits of the concatenation of all divisors of n.
From Sida Li, Sep 01 2023: (Start)
a(n) = Sum_{d divides n} (floor(log_10(d))+1).
log_10(Product_{d divides n} d) <= a(n) <= log_10(Product_{d divides n} d) + sigma_0(n), where sigma_0(n) = A000005(n).
Equivalently, sigma_0(n)*log_10(n)/2 <= a(n) <= sigma_0(n)*log_10(n)/2 + sigma_0(n), obtained by formula in A007955.
For x >= 5, c2*log(x)^2 + c1*log(x) + c0 <= (1/x)*Sum_{n<=x} a(n) <= c2*log(x)^2 + (c1+1)*log(x) + 2*c0, where c2 = 1/(2*log(10)), c1 = (gamma-1)/log(10), c0 = 2*gamma-1, and gamma is Euler's constant. This is obtained by hyperbola trick for Sum_{n<=x} sigma_0(n), and Abel partial summation on Sum_{n<=x} sigma_0(n)*log(n). (End)

New name from Jaroslav Krizek, Jun 15 2011

A190998 Digital root of concatenation of all divisors of n (A037278).

Original entry on oeis.org

1, 3, 4, 7, 6, 3, 8, 6, 4, 9, 3, 1, 5, 6, 6, 4, 9, 3, 2, 6, 5, 9, 6, 6, 4, 6, 4, 2, 3, 9, 5, 9, 3, 9, 3, 1, 2, 6, 2, 9, 6, 6, 8, 3, 6, 9, 3, 7, 3, 3, 9, 8, 9, 3, 9, 3, 8, 9, 6, 6, 8, 6, 5, 1, 3, 9, 5, 9, 6, 9, 9, 6, 2, 6, 7, 5, 6, 6, 8, 6, 4, 9, 3, 8, 9, 6, 3, 9, 9, 9, 4, 6, 2, 9, 3, 9, 8, 9, 3, 1, 3, 9, 5, 3, 3

Offset: 1

Jaroslav Krizek, Jun 15 2011

Also digital root of A034690 (sum of digits of all the divisors of n).

Also digital root of A000203 (sum of divisors of n). - Michel Marcus, Sep 13 2014

			For n = 12: 1 + 2 + 3 + 4 + 6 + 1 + 2 = 19, 1 + 9 = 10, 1 + 0 = 1; a(12) = 1.

Nathaniel Johnston, Table of n, a(n) for n = 1..10000

Cf. A134681, A034690, A190997.

A190998:=proc(n) local d, i, s: d:=numtheory[divisors](n): s:=0: for i from 1 to nops(d) do s:=s+((d[i]-1) mod 9)+1: if(s>=10)then s:=((s-1) mod 9)+1: fi: od: return s: end: seq(A190998(n), n=1..105); # Nathaniel Johnston, Jun 15 2011

A190997 Product of digits of all the divisors of n.

Original entry on oeis.org

1, 2, 3, 8, 5, 36, 7, 64, 27, 0, 1, 288, 3, 56, 75, 384, 7, 2592, 9, 0, 42, 8, 6, 18432, 50, 72, 378, 3584, 18, 0, 3, 2304, 27, 168, 525, 373248, 21, 432, 243, 0, 4, 16128, 12, 512, 13500, 288, 28, 3538944, 252, 0, 105, 2880, 15, 725760, 125, 860160, 945

Offset: 1

Jaroslav Krizek, Jun 15 2011

Product of digits of concatenation of all divisors of n (A037278).

			For n = 12: a(12) = 1 * 2 * 3 * 4 * 6 * 1 * 2 = 288.

Nathaniel Johnston, Table of n, a(n) for n = 1..10000

Cf. A134681, A034690.

A190997:=proc(n) local d, i, p: d:=numtheory[divisors](n): p:=1: for i from 1 to nops(d) do p:=p*mul(d, d=convert(d[i], base, 10)): od: return p: end: seq(A190997(n),n=1..57); # Nathaniel Johnston, Jun 15 2011

Table[Times@@Flatten[IntegerDigits/@Divisors[n]],{n,100}] (* Harvey P. Dale, Nov 27 2022 *)

a007954(n) = my(d=digits(n)); prod(i=1, #d, d[i]);
a(n) = my(div=divisors(n), pdt=1); for(k=1, #div, pdt=pdt*a007954(div[k])); pdt \\ Felix Fröhlich, Sep 22 2016

a(n) = 0 for n = multiples of 10; a(A008592(n)) = 0 for n >=1.

A114527 Numbers k such that A086793(k) is 1.

Original entry on oeis.org

8, 14, 20, 26, 59, 62, 122, 123, 143, 149, 167, 206, 239, 257, 293, 302, 341, 347, 383, 419, 422, 491, 509, 563, 617, 653, 743, 761, 941, 1049, 1133, 1193, 1202, 1203, 1229, 1283, 1313, 1319, 1331, 1373, 1409, 1427, 1481, 1553, 1571, 1607

Offset: 1

Zak Seidov, May 16 2006

Prime numbers in the sequence are also primes with digit sum = 14 (A106756). - Zak Seidov, May 21 2006

Cf. A034690, A086793, A106756.

ss={8,14};Do[If[15==Total@Flatten[IntegerDigits/@Divisors[n]],AppendTo[ss,n]],{n,20,2000}];ss (* Zak Seidov, May 21 2006 *)

A119396 Numbers n such that A086793(n)=20.

Original entry on oeis.org

924, 1104, 1134, 1540, 1650, 1760, 1820, 1908, 1992, 2016, 2288, 2556, 2632, 2744, 2860, 2940, 2970, 3000, 3192, 3204, 3220, 3248, 3400, 3630, 3738, 3784, 3840, 3852, 3880, 3968, 3990, 4134, 4260, 4410, 4464, 4674, 4736, 4860, 4875, 4930, 4992, 5016

Offset: 1

Zak Seidov, May 17 2006

Some trajectories are: 924,168,102,36,46,18,30,27,22,9,13,5,6,12,19,11,3,4,7,8,15 1104,168,102,... 1540,162,66,36,... 1650,162,66,36,... 2016,297,66,36,... 2940,297,66,36,... 3192,312,102,36,... All trajectories eventually join one of previous trajectories.

			924 is a term because it reaches 15 in 20 steps with this trajectory 924,168,102,36,46,18,30,27,22,9,13,5,6,12,19,11,3,4,7,8,15.

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

Cf. A034690, A086793, A094501, A095347, A114527, A118358, A119397, A119398, A260059.

f:= proc(n) option remember; local t;
  if kernelopts(level) > 460 then return FAIL fi;
  t:= add(convert(convert(d,base,10),`+`),d=numtheory:-divisors(n));
  1+procname(t)
end proc:
f(15):= 0:
f(1):= FAIL:
Res:= NULL: count:= 0:
for n from 1 while count < 100 do
  if f(n) = 20 then
    count:= count+1;
    Res:= Res, n;
   fi
od:
Res; # Robert Israel, Apr 03 2018

Edited by Robert Israel, Apr 03 2018

A333618 a(n) is the total number of terms (1-digits) in the dual Zeckendorf representation of all divisors of n.

Original entry on oeis.org

1, 2, 3, 4, 3, 7, 3, 7, 6, 7, 5, 12, 4, 8, 8, 11, 5, 14, 6, 12, 9, 10, 5, 20, 7, 9, 11, 14, 6, 20, 6, 17, 11, 10, 10, 23, 6, 12, 11, 21, 5, 22, 6, 17, 17, 11, 6, 30, 8, 17, 13, 17, 8, 23, 12, 22, 13, 13, 6, 33, 7, 12, 18, 23, 12, 26, 6, 17, 13, 23, 7, 37, 7, 14

Offset: 1

Amiram Eldar, Mar 29 2020

			For n = 6, its divisors are 1, 2, 3 and 6. The dual Zeckendorf representations (A104326) of the divisors are 1, 10, 11 and 111. Their total number of 1's is 1 + 1 + 2 + 3 = 7, thus a(6) = 7.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A034690, A093653, A104326, A112310, A300837.

fibTerms[n_] := Module[{k = Ceiling[Log[GoldenRatio, n*Sqrt[5]]], t = n, fr = {}}, While[k > 1, If[t >= Fibonacci[k], AppendTo[fr, 1]; t = t - Fibonacci[k], AppendTo[fr, 0]]; k--]; fr];
dualZeckSum[n_] := Module[{v = fibTerms[n]}, nv = Length[v]; i = 1; While[i <= nv - 2, If[v[[i]] == 1 && v[[i + 1]] == 0 && v[[i + 2]] == 0, v[[i]] = 0; v[[i + 1]] = 1; v[[i + 2]] = 1; If[i > 2, i -= 3]]; i++]; i = Position[v, _?(# > 0 &)]; If[i == {}, 0, Total[v[[i[[1, 1]] ;; -1]]]]];
a[n_] := DivisorSum[n, dualZeckSum[#] &]; Array[a, 100]

a(n) = Sum_{d|n} A112310(d).

A094450 Number of iterations of the sum of digits of the divisors of 10^n needed to reach 15.

Original entry on oeis.org

12, 10, 16, 15, 3, 11, 4, 13, 6, 6, 5, 19, 16, 3, 11, 13, 19, 7, 5, 9, 6, 16, 16, 19, 5, 3, 12, 3, 18, 16, 4, 10, 6, 16, 18, 12, 4, 16, 12, 13, 12, 5, 12, 5, 20, 15, 16, 12, 4, 16, 4, 20, 5, 19, 4, 6, 21, 5, 6, 5, 21, 12, 5, 16, 13, 17, 6, 5, 7, 21, 20, 18, 12, 10, 6, 18, 13, 13, 6, 13, 15

Offset: 1

Jason Earls, Jun 04 2004

Michael S. Branicky, Table of n, a(n) for n = 1..1400

Cf. A034690, A086793.

Table[Length[NestWhileList[Total[Flatten[IntegerDigits/@Divisors[#]]]&,10^n, #!= 15&]]-1,{n,90}] (* Harvey P. Dale, Mar 05 2019 *)

from sympy import divisors
def sd(n): return sum(map(int, str(n)))
def f(n): return sum(sd(d) for d in divisors(n, generator=True))
def a(n):
    i, c = 10**n, 0
    while i != 15: i = f(i); c += 1
    return c
print([a(n) for n in range(1, 82)]) # Michael S. Branicky, Dec 10 2021

A246841 Sum of digits of all the anti-divisors of n.

Original entry on oeis.org

0, 0, 2, 3, 5, 4, 10, 8, 8, 14, 12, 13, 19, 16, 9, 5, 19, 19, 9, 15, 13, 27, 25, 14, 21, 15, 24, 28, 15, 9, 24, 31, 21, 12, 16, 14, 23, 34, 25, 28, 23, 30, 29, 22, 32, 22, 24, 20, 27, 26, 15, 40, 34, 16, 20, 20, 29, 42, 45, 35, 12, 24, 40, 10, 21, 32, 60, 49

Offset: 1

Paolo P. Lava, Sep 05 2014

Sum of the digits of the terms in row n of A130799.

First occurrence of k, or 0 if k never appears: 0, 3, 4, 6, 5, 0, 0, 8, 15, 7, 0, 11, 12, 10, 20, 14, 75, 69, 13, 48, 25, 44, 37, 27, 23, 50, 22, 28, 43, 42, 32, 45, 92, 38, 60, 82, 208, 81, 110, 52, 72, 58, 97, 73, 59, 77, 255, 85, 68, 127, ...

			Anti-divisors of 20 are 3, 8, 13 and the sum of their digits is 3 + 8 + 1 + 3 = 15.

Paolo P. Lava, Table of n, a(n) for n = 1..1000

Cf. A034690, A066272.

# function antidivisors defined in A066272. transforms is https://oeis.org/transforms.txt
read("transforms");
A246841 := proc(n)
    a :=0 ;
    for adiv in antidivisors(n) do
        a := a+digsum(adiv) ;
    end do:
    a ;
end proc:
seq(A246841(n),n=1..30) ; # R. J. Mathar, Sep 07 2014

A094150 Number of iterations of the sum of digits of the divisors of R(n) needed to reach 15, where R(n) = A002275.

Original entry on oeis.org

5, 13, 12, 3, 11, 10, 15, 3, 14, 3, 5, 18, 13, 4, 18, 18, 5, 2, 4, 3, 12, 10, 21, 12, 18, 16, 11, 19, 20, 4, 6, 12, 13, 6, 6, 5, 3, 16, 5, 12, 12, 11, 18, 16, 18, 17, 12, 5, 12, 12, 18, 19, 14, 4, 5, 12, 18, 10, 16, 6, 13, 17, 21, 19, 6, 13, 19, 4, 3, 7, 15, 10, 21, 4, 18, 3, 5, 20, 11, 18

Offset: 2

Jason Earls, Jun 01 2004

			a(9)=3 because 111111111 -> 151 -> 8 -> 15.
a(1) does not exist because 1 -> 1 and never reaches 15.

Cf. A002275, A034690, A086793.

More terms from David Wasserman, May 22 2007

cp's OEIS Frontend

A106756 Primes with digit sum = 14.

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A134681 Number of digits of all the divisors of n.

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A190998 Digital root of concatenation of all divisors of n (A037278).

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A190997 Product of digits of all the divisors of n.

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A114527 Numbers k such that A086793(k) is 1.

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A119396 Numbers n such that A086793(n)=20.

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A333618 a(n) is the total number of terms (1-digits) in the dual Zeckendorf representation of all divisors of n.

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A094450 Number of iterations of the sum of digits of the divisors of 10^n needed to reach 15.