A061383 -id:A061383 - cp's OEIS frontend

A268620 Numbers whose digital sum is a multiple of 4.

0, 4, 8, 13, 17, 22, 26, 31, 35, 39, 40, 44, 48, 53, 57, 62, 66, 71, 75, 79, 80, 84, 88, 93, 97, 103, 107, 112, 116, 121, 125, 129, 130, 134, 138, 143, 147, 152, 156, 161, 165, 169, 170, 174, 178, 183, 187, 192, 196, 202, 206, 211, 215, 219, 220, 224, 228, 233, 237, 242, 246

Offset: 1

Bruno Berselli, Feb 09 2016

a(1498) = 5999 is the smallest term that is congruent to 5 modulo 9.

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

Cf. A007953, A061383 (supersequence).
Cf. numbers whose digital sum is a multiple of k: A054683 (k=2), A008585 (k=3), this sequence (k=4), A227793 (k=5).
Subsequences: A002278, A038446, A052218, A052222, A061387, A169967, A235151, A235227, A235229.

[n: n in [0..250] | IsIntegral(&+Intseq(n)/4)];

select(t -> convert(convert(t,base,10),`+`) mod 4 = 0, [$1..1000]); # Robert Israel, Feb 09 2016

Select[Range[0, 250], IntegerQ[Total[IntegerDigits[#]]/4] &]

A062182 Harmonic mean of digits is 4.

Original entry on oeis.org

4, 36, 44, 63, 288, 346, 364, 436, 444, 463, 634, 643, 828, 882, 2488, 2666, 2848, 2884, 3366, 3446, 3464, 3636, 3644, 3663, 4288, 4346, 4364, 4436, 4444, 4463, 4634, 4643, 4828, 4882, 6266, 6336, 6344, 6363, 6434, 6443, 6626, 6633, 6662, 8248, 8284

Offset: 1

Vladeta Jovovic, Jun 12 2001

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

Cf. A062179-A062185, A061383-A061388, A061423-A061425.

filter:= proc(n) local L;
  L:= convert(n,base,10);
  if has(L,0) then return false fi;
  nops(L)/add(1/i,i=L)=4
end proc:
select(filter, [$1..10^4]); # Robert Israel, Aug 20 2018

Do[ h = IntegerDigits[n]; If[ Sort[h][[1]] != 0 && Length[h]/Apply[Plus, 1/h] == 4, Print[n]], {n, 1, 10^5}]
hm4Q[n_]:=DigitCount[n,10,0]==0&&HarmonicMean[IntegerDigits[n]]==4; Select[Range[9000],hm4Q]  (* Harvey P. Dale, Mar 23 2011 *)

More terms from Henry Bottomley, Jul 25 2001

A062183 Numbers such that harmonic mean of digits is 5.

Original entry on oeis.org

5, 55, 555, 5555, 26999, 28888, 29699, 29969, 29996, 33999, 34688, 34868, 34886, 36488, 36666, 36848, 36884, 38468, 38486, 38648, 38684, 38846, 38864, 39399, 39939, 39993, 43688, 43868, 43886, 44488, 44666, 44848, 44884, 46388, 46466

Offset: 1

Vladeta Jovovic, Jun 12 2001

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A062179-A062185, A061383-A061388, A061423-A061425.

Do[ h = IntegerDigits[n]; If[ Sort[h][[1]] != 0 && Length[h]/Apply[Plus, 1/h] == 5, Print[n]], {n, 1, 10^6}]
Select[Range[50000],HarmonicMean[IntegerDigits[#]]==5&] (* Harvey P. Dale, Sep 27 2018 *)

More terms from Robert G. Wilson v, Aug 08 2001

A175762 Primes with an arithmetic mean of digits which is also prime.

Original entry on oeis.org

2, 3, 5, 7, 13, 19, 31, 37, 59, 73, 1061, 1151, 1223, 1289, 1487, 1511, 1559, 1601, 1667, 1847, 1973, 1999, 2099, 2141, 2213, 2297, 2411, 2459, 2477, 2549, 2657, 2693, 2729, 2819, 2837, 2909, 2927, 2963, 3023, 3041, 3089, 3203, 3221, 3359, 3449, 3467, 3539, 3557

Offset: 1

Juri-Stepan Gerasimov, Aug 29 2010

			a(10)=73 because (7+3)/2=5=prime. a(12)=1151 because (1+1+5+1)/4=2=prime.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A000040, A007605, A061383.

Select[Prime[Range[500]],PrimeQ[Mean[IntegerDigits[#]]]&] (* Harvey P. Dale, Jan 16 2013 *)

{A000040(i): A007605(i)/A097944(i) in A000040}.

Corrected (59 inserted, 1061 inserted etc.) by R. J. Mathar, Sep 24 2010

A190760 Product of digits is divisible by number of digits.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 16, 18, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 32, 34, 36, 38, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 52, 54, 56, 58, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 72, 74, 76, 78, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 92, 94, 96, 98, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 113, 116, 119

Offset: 1

Kyle Stern, May 18 2011

Almost all numbers are in this sequence: there are at least n - 1.125 n^0.95... elements up to n, where the exponent is log(9)/log(10). - Charles R Greathouse IV, May 20 2011

			3*8*2 = 48 and 48 is divisible by the number of digits, 3, so 382 is included.

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

Cf. A007954, A055642, A061383.

A190760 := proc(n) option remember: local k: if(n=1)then return 0: fi: for k from procname(n-1)+1 do if(mul(d,d=convert(k,base,10)) mod length(k) = 0)then return k: fi: od: end: seq(A190760(n),n=1..100); # Nathaniel Johnston, May 19 2011

A306520 Numbers k with property that the arithmetic mean of any subset of its digits is an integer.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 13, 15, 17, 19, 20, 22, 24, 26, 28, 31, 33, 35, 37, 39, 40, 42, 44, 46, 48, 51, 53, 55, 57, 59, 60, 62, 64, 66, 68, 71, 73, 75, 77, 79, 80, 82, 84, 86, 88, 91, 93, 95, 97, 99, 111, 117, 135, 153, 159, 171, 177, 195

Offset: 1

R. J. Cano, Feb 21 2019

This sequence is different from A061383. Here digits in k must have all the same parity, otherwise the average of at least a pair of digits wouldn't be an integer. Note that for every 2-digit term in A061383 both digits have the same parity. But not every number whose digits have all the same parity (sequence A059708) belongs here.

			17 is in this sequence because the set of digits (1,7) has an integer average: 4.
159 and 195 are in this sequence because the sets of digits (1,5), (1,9), (5,9), and (1,5,9) all have integer averages, respectively: 3, 5, 7, and 5.

Harvey P. Dale, Table of n, a(n) for n = 1..177 (all terms up to 1 million)

Cf. A061383, A059708, A010785, A165165.

Select[Range[0,200],AllTrue[Mean/@Subsets[IntegerDigits[#],{2, IntegerLength[ #]}],IntegerQ]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Aug 09 2020 *)

firstTerms_vec(n)={my(v=vector(n),c,t,w:list,h);for(i=1,+oo,w=List();forsubset(i,k,listput(w,k));listpop(w,1);forvec(j=vector(i,z,[(z==1)&&(i>1),9]),h=j[1]%2;for(l=2,#j,if((j[l]%2)!=h,next(2)));for(k=1,#w,t=vecextract(j,w[k]);if(vecsum(t)%(#w[k]),next(2)));v[c++]=fromdigits(j);if(c==n,return(v))))}

isok(m,{B=10})={my(w=digits(m,B));forsubset(#w,y,if(y!=Vecsmall([]),if(vecsum(vecextract(w,y))%(#y),return(0)),next));1}

Apparently a(158+n) = A010785(35+n).

A335743 Keep the first two digits of a(n) and insert a dot between them; this is now the arithmetic mean (truncated after the first decimal) of the digits used so far in the sequence. Lexicographically earliest sequence of distinct positive terms with this property.

Original entry on oeis.org

45, 30, 38, 301, 306, 307, 305, 308, 304, 318, 303, 309, 316, 302, 2900, 2901, 2910, 3019, 3009, 3018, 3027, 3028, 3008, 3029, 3007, 3036, 3037, 3017, 3038, 3016, 3039, 3006, 3045, 3046, 3026, 3047, 3025, 3048, 3015, 3049, 3005, 3054, 3055, 3035, 3056, 3034, 3057, 3024, 3058, 3014, 3059, 3004, 3063

Offset: 1

Eric Angelini and Carole Dubois, Jul 02 2020

The sequence starts with a(1) = 45 as any a(1) < 45 would not produce an infinite sequence.

			a(1) = 45; inserting a dot between the first two digits produces 4.5; this is now the arithmetic mean (AM) of the digits used so far in the sequence as (4 + 5)/2 = 9/2 = 4.5 (and 4.5 is 45 with a dot);
a(2) = 30; inserting a dot between the first two digits produces 3.0; this is the AM of the digits used so far in the sequence as (4 + 5 + 3 + 0)/4 = 12/4 = 3 (and 3 is 30 with a dot);
a(3) = 38; inserting a dot between the first two digits produces 3.8; this is the AM of the digits used so far (when truncated after the first decimal) as (4 + 5 + 3 + 0 + 3 + 8)/6 = 23/6 = 3.83333... which produces 38, and 3.8 is 38 with a dot);
a(4) = 301; inserting a dot between the first two digits produces 3.0; this is the AM of the digits used so far as (4 + 5 + 3 + 0 + 3 + 8 + 3 + 0 + 1)/9 = 27/9 = 3 [and 3 is 30 with a dot, this 30 being formed by the first two digits of a(4)];
a(5) = 306; inserting a dot between the first two digits produces 3.0; this is the AM of the digits used so far as (4 + 5 + 3 + 0 + 3 + 8 + 3 + 0 + 1 + 3 + 0 + 6)/12 = 36/12 = 3 (and 3 is 30 with a dot, this 30 being formed by the first two digits of a(5)]);
a(6) = 307; inserting a dot between the first two digits produces 3.0; this is the AM of the digits used so far (truncated after the first decimal) as (4 + 5 + 3 + 0 + 3 + 8 + 3 + 0 + 1 + 3 + 0 + 6 + 3 + 0 + 7)/15 = 46/15 = 3.0666... which produces 30, this 30 being formed by the first two digits of a(6)]; etc.

Cf. A061383 (arithmetic mean of digits is an integer).

A364606 Numbers k such that the average digit of 2^k is an integer.

Original entry on oeis.org

0, 1, 2, 3, 6, 13, 16, 26, 46, 51, 56, 73, 122, 141, 166, 313, 383

Offset: 1

Jon E. Schoenfield, Jul 29 2023

			2^26 = 67108864 is an 8-digit number; its average digit is (6+7+1+0+8+8+6+4)/8 = 40/8 = 5, an integer, so 26 is a term.

Cf. A000079, A001370, A034887, A061383.

q:= n-> (l-> irem(add(i, i=l), nops(l))=0)(convert(2^n, base, 10)):
select(q, [$0..400])[];  # Alois P. Heinz, Jul 29 2023

Select[Range[0, 2^12], IntegerQ@ Mean@ IntegerDigits[2^#] &] (* Michael De Vlieger, Jul 29 2023 *)

isok(k) = my(d=digits(2^k)); !(vecsum(d) % #d); \\ Michel Marcus, Jul 29 2023

from itertools import count, islice
from gmpy2 import mpz, digits
def A364606_gen(startvalue=0): # generator of terms >= startvalue
    m = mpz(1)<A364606_list = list(islice(A364606_gen(),10)) # Chai Wah Wu, Jul 31 2023

{ k : A001370(k) mod A034887(k) = 0 }.

A383304 Nonnegative integers whose difference between the largest and smallest digits is equal to the arithmetic mean of its digits.

Original entry on oeis.org

0, 13, 26, 31, 39, 62, 93, 123, 132, 144, 213, 225, 231, 246, 252, 264, 267, 276, 288, 312, 321, 348, 369, 384, 396, 414, 426, 438, 441, 462, 483, 522, 624, 627, 639, 642, 672, 693, 726, 762, 828, 834, 843, 882, 936, 963, 1133, 1223, 1232, 1313, 1322, 1331, 1344, 1434, 1443

Offset: 1

Stefano Spezia, Apr 22 2025

			144 is a term since 4 - 1 = 3 = (1 + 4 + 4)/3.

Cf. A037904, A054054, A054055, A179239, A371383, A371384, A383305.

Subsequence of A061383.

Select[Range[0,1500], Max[d=IntegerDigits[#]]-Min[d]==Mean[d] &]

def ok(n): return sum(d:=list(map(int, str(n)))) == (max(d) - min(d))*len(d)
print([k for k in range(1500) if ok(k)]) # Michael S. Branicky, Apr 23 2025

A061546 Harmonic mean of digits is 7.

Original entry on oeis.org

7, 77, 777, 7777, 77777, 777777, 3999999, 4688999, 4689899, 4689989, 4689998, 4698899, 4698989, 4698998, 4699889, 4699898, 4699988, 4868999, 4869899, 4869989, 4869998, 4886999, 4888888, 4889699, 4889969, 4889996, 4896899

Offset: 1

Vladeta Jovovic, Jun 13 2001

			6666999 is a term since 7/(1/6+1/6+1/6+1/6+1/9+1/9+1/9)=7.

Cf. A062179-A062185, A061383-A061388, A061423-A061425.

Do[ h = IntegerDigits[n]; If[ Sort[h][[1]] != 0 && Length[h]/Apply[Plus, 1/h] == 7, Print[n]], {n, 1, 10^6}]

More terms from Robert G. Wilson v, Aug 08 2001

cp's OEIS Frontend

A268620 Numbers whose digital sum is a multiple of 4.

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A062182 Harmonic mean of digits is 4.

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A062183 Numbers such that harmonic mean of digits is 5.

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A175762 Primes with an arithmetic mean of digits which is also prime.

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Formula

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A190760 Product of digits is divisible by number of digits.

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Maple

A306520 Numbers k with property that the arithmetic mean of any subset of its digits is an integer.

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Mathematica

PARI

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A335743 Keep the first two digits of a(n) and insert a dot between them; this is now the arithmetic mean (truncated after the first decimal) of the digits used so far in the sequence. Lexicographically earliest sequence of distinct positive terms with this property.

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A364606 Numbers k such that the average digit of 2^k is an integer.

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Maple