A069709 -id:A069709 - cp's OEIS frontend

A285095 Corresponding values of arithmetic means of digits of primes from A069709.

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

Offset: 1

Jaroslav Krizek, Apr 16 2017

Jaroslav Krizek, Table of n, a(n) for n = 1..1000

Cf. A069709 (primes with integer arithmetic mean of digits in base 10).

Sequences of primes such that a(n) = k for k = 1, 2, 4, 5, 7 and 8: A069710 (k = 1), A285096 (k = 2), A285225 (k = 4), A285226 (k = 5), A285227 (k = 7), A285228 (k = 8).

[&+Intseq(n) / #Intseq(n): n in [1..100000] | IsPrime(n) and &+Intseq(n) mod #Intseq(n) eq 0];

A069710 Primes with arithmetic mean of digits = 1 (sum of digits = number of digits).

Original entry on oeis.org

11, 1021, 1201, 2011, 3001, 10103, 10211, 10301, 11003, 12011, 12101, 13001, 20021, 20201, 21011, 21101, 30011, 1000033, 1000213, 1000231, 1000303, 1001023, 1001041, 1001311, 1001401, 1002121, 1003003, 1003111, 1003201, 1010131

Offset: 1

Amarnath Murthy, Apr 08 2002

The sum of the digits of a prime > 3 cannot be a multiple of 3, hence no prime with 3*k digits can be here. - David Radcliffe, May 05 2015

Subsequence of primes of A061384. - Michel Marcus, May 05 2015

David Radcliffe, Table of n, a(n) for n = 1..13376

Cf. A069709, A069711, A069712.

F:= proc(d,s) option remember;
  local t,r;
  if d = 1 then
    if s >= 1 and s <= 9 then {s}
    else {}
    fi
  else
    `union`(seq(map(t -> 10*t+r, procname(d-1,s-r)), r=0..min(s,9)))
  fi
end proc:
`union`(seq(select(isprime,F(i,i)), i = remove(d -> d mod 3 = 0, [$1..8]));
# if using Maple 11 or earlier, uncomment the next line
# sort(convert(%,list)); # Robert Israel, May 05 2015

Do[p = Prime[n]; If[ Apply[ Plus, IntegerDigits[p]] == Floor[ Log[10, p] + 1], Print[p]], {n, 1, 10^5}]

from itertools import count, islice
from collections import Counter
from sympy.utilities.iterables import partitions, multiset_permutations
from sympy import isprime
def A069710_gen(): # generator of terms
    for l in count(1):
        for i in range(1,min(9,l)+1):
            yield from sorted(q for q in (int(str(i)+''.join(map(str,j))) for s,p in partitions(l-i,k=9,size=True) for j in multiset_permutations([0]*(l-1-s)+list(Counter(p).elements()))) if isprime(q))
A069710_list = list(islice(A069710_gen(),30)) # Chai Wah Wu, Nov 28 2023

Edited and extended by Robert G. Wilson v, Apr 12 2002

A069711 Squares whose arithmetic mean of digits is an integer (i.e., the sum of digits is a multiple of the number of digits).

Original entry on oeis.org

0, 1, 4, 9, 64, 144, 225, 324, 441, 576, 729, 900, 1681, 3364, 3481, 4624, 7225, 9025, 12769, 14884, 21025, 23104, 24649, 24964, 27556, 30976, 32041, 33856, 36100, 37249, 37636, 44944, 48841, 56644, 63001, 65536, 66049, 70756, 75076, 75625, 80089, 80656, 85264

Offset: 1

Amarnath Murthy, Apr 08 2002

Jon E. Schoenfield, Table of n, a(n) for n = 1..10000

Cf. A069709, A069710, A069712.

Do[ If[ IntegerQ[ Apply[ Plus, IntegerDigits[n^2]] / Floor[ Log[10, n^2] + 1]], Print[n^2]], {n, 1, 10^3}]

Edited and extended by Robert G. Wilson v, Apr 12 2002

0 prepended as a(1) and a(27)-a(43) from Jon E. Schoenfield, Jun 28 2018

A285096 Primes with integer arithmetic mean of digits = 2 in base 10.

Original entry on oeis.org

2, 13, 31, 1061, 1151, 1223, 1511, 1601, 2141, 2213, 2411, 3023, 3041, 3203, 3221, 4013, 4211, 5003, 5021, 6011, 6101, 7001, 10009, 10243, 10333, 10513, 10531, 10711, 11071, 11161, 11251, 11503, 11701, 12007, 12043, 12241, 12421, 12511, 12601, 13033, 13411

Offset: 1

Jaroslav Krizek, Apr 16 2017

Jaroslav Krizek, Table of n, a(n) for n = 1..1000

Primes in A061385. Subsequence of A069709.

Sequences of primes such that a(n) = k for k = 1, 2, 4, 5, 7 and 8: A069710 (k = 1), this sequence (k = 2), A285225 (k = 4), A285226 (k = 5), A285227 (k = 7), A285228 (k = 8).

[n: n in [1..100000] | IsPrime(n) and &+Intseq(n) mod #Intseq(n) eq 0 and &+Intseq(n) / #Intseq(n) eq 2];

S:= proc(d,k,flag) option remember;
  if d = 1 then
    if k >= 0 and k <= 9 then return [k]
    else return []
    fi
  fi;
  [seq(op(map(`+`, procname(d-1,k-i,0), i*10^(d-1))),i=flag..min(k,9))]
end proc:
seq(op(select(isprime,S(d,2*d,1))),d=1..5);# Robert Israel, Apr 23 2017

Select[Prime[Range[1600]],Mean[IntegerDigits[#]]==2&] (* Harvey P. Dale, Aug 07 2021 *)

A285225 Primes with integer arithmetic mean of digits = 4 in base 10.

Original entry on oeis.org

17, 53, 71, 1069, 1087, 1249, 1429, 1447, 1483, 1609, 1627, 1663, 1753, 1861, 1933, 1951, 2239, 2293, 2347, 2383, 2437, 2473, 2617, 2671, 2707, 2833, 2851, 3049, 3067, 3229, 3319, 3373, 3391, 3463, 3517, 3571, 3607, 3643, 3733, 3823, 3931, 4057, 4093, 4129

Offset: 1

Jaroslav Krizek, Apr 16 2017

Jaroslav Krizek, Table of n, a(n) for n = 1..1000

Primes in A061387. Subsequence of A069709.

Sequences of primes such that a(n) = k for k = 1, 2, 4, 5, 7 and 8: A069710 (k = 1), A285096 (k = 2), this sequence (k = 4), A285226 (k = 5), A285227 (k = 7), A285228 (k = 8).

[n: n in [1..100000] | IsPrime(n) and &+Intseq(n) mod #Intseq(n) eq 0 and &+Intseq(n) / #Intseq(n) eq 4];

Select[Prime[Range[600]],Mean[IntegerDigits[#]]==4&] (* Harvey P. Dale, Jun 11 2024 *)

A285226 Primes with integer arithmetic mean of digits = 5 in base 10.

Original entry on oeis.org

5, 19, 37, 73, 1289, 1487, 1559, 1667, 1847, 1973, 2099, 2297, 2459, 2477, 2549, 2657, 2693, 2729, 2819, 2837, 2909, 2927, 2963, 3089, 3359, 3449, 3467, 3539, 3557, 3593, 3719, 3863, 3881, 3917, 4079, 4259, 4349, 4457, 4493, 4547, 4583, 4637, 4673, 4691, 4817

Offset: 1

Jaroslav Krizek, Apr 19 2017

Jaroslav Krizek, Table of n, a(n) for n = 1..1000

Primes from A061388. Subsequence of A069709.

Sequences of primes such that a(n) = k for k = 1, 2, 4, 5, 7 and 8: A069710 (k = 1), A285096 (k = 2), A285225 (k = 4), this sequence (k = 5), A285227 (k = 7), A285228 (k = 8).

[n: n in [1..100000] | IsPrime(n) and &+Intseq(n) mod #Intseq(n) eq 0 and &+Intseq(n) / #Intseq(n) eq 5];

Select[Prime@ Range@ PrimePi@ 5000, Mean@ IntegerDigits@ # == 5 &] (* Michael De Vlieger, Apr 22 2017 *)

A285227 Primes with integer arithmetic mean of digits = 7 in base 10.

Original entry on oeis.org

7, 59, 1999, 3889, 4789, 4969, 4987, 5689, 5779, 5869, 6679, 6949, 6967, 7489, 7669, 7687, 7759, 7867, 7993, 8389, 8677, 8839, 8893, 8929, 9199, 9397, 9649, 9739, 9829, 9883, 9973, 18899, 19889, 19979, 19997, 28979, 29789, 29879, 35999, 36899, 37799, 37889

Offset: 1

Jaroslav Krizek, Apr 19 2017

Jaroslav Krizek, Table of n, a(n) for n = 1..1000

Primes from A061424. Subsequence of A069709.

Sequences of primes such that a(n) = k for k = 1, 2, 4, 5, 7 and 8: A069710 (k = 1), A285096 (k = 2), A285225 (k = 4), A285226 (k = 5), this sequence (k = 7), A285228 (k = 8).

[n: n in [1..100000] | IsPrime(n) and &+Intseq(n) mod #Intseq(n) eq 0 and &+Intseq(n) / #Intseq(n) eq 7]

Select[Prime@ Range@ PrimePi@ 40000, Mean@ IntegerDigits@ # == 7 &] (* Michael De Vlieger, Apr 22 2017 *)

from itertools import count, islice
from collections import Counter
from sympy.utilities.iterables import partitions, multiset_permutations
from sympy import isprime
def A285227_gen(): # generator of terms
    yield 7
    for l in count(2):
        for i in range(1,10):
            yield from sorted(q for q in (int(str(i)+''.join(map(str,j))) for s,p in partitions(7*l-i,m=l-1,k=9,size=True) for j in multiset_permutations([0]*(l-1-s)+list(Counter(p).elements()))) if isprime(q))
A285227_list = list(islice(A285227_gen(),30)) # Chai Wah Wu, Nov 29 2023

A285228 Primes with integer arithmetic mean of digits = 8 in base 10.

Original entry on oeis.org

79, 97, 6899, 8699, 8969, 9689, 9887, 49999, 68899, 69997, 77899, 78889, 78979, 79699, 79987, 85999, 88789, 88897, 88969, 89599, 89689, 89779, 89797, 89959, 89977, 94999, 95989, 96799, 96979, 96997, 97789, 97879, 97987, 98689, 98779, 98869, 98887, 99679, 99787

Offset: 1

Jaroslav Krizek, Apr 19 2017

Jaroslav Krizek, Table of n, a(n) for n = 1..1000

Primes from A061425. Subsequence of A069709.

Sequences of primes such that a(n) = k for k = 1, 2, 4, 5, 7 and 8: A069710 (k = 1), A285096 (k = 2), A285225 (k = 4), A285226 (k = 5), A285227 (k = 7), this sequence (k = 8).

[n: n in [1..100000] | IsPrime(n) and &+Intseq(n) mod #Intseq(n) eq 0 and &+Intseq(n) / #Intseq(n) eq 8];

Select[Prime@ Range@ PrimePi[10^5], Mean@ IntegerDigits@ # == 8 &] (* Michael De Vlieger, Apr 22 2017 *)

from itertools import count, islice
from collections import Counter
from sympy.utilities.iterables import partitions, multiset_permutations
def A285228_gen(): # generator of terms
    for l in count(2):
        for i in range(1,10):
            yield from sorted(q for q in (int(str(i)+''.join(map(str,j))) for s,p in partitions((l<<3)-i,m=l-1,k=9,size=True) for j in multiset_permutations([0]*(l-1-s)+list(Counter(p).elements()))) if isprime(q))
A285228_list = list(islice(A285228_gen(),30)) # Chai Wah Wu, Nov 28 2023

A069712 Triangular numbers with arithmetic mean of digits = integer (sum of digits = A multiple of the number of digits).

Original entry on oeis.org

1, 3, 6, 15, 28, 55, 66, 91, 105, 120, 153, 171, 210, 231, 276, 300, 351, 378, 435, 465, 528, 561, 630, 666, 741, 780, 861, 903, 990, 1128, 1326, 1830, 2145, 2415, 3081, 5151, 5995, 6105, 7140, 8385, 8646, 9591, 9870, 10153, 11026, 11175, 12246, 12403

Offset: 0

Amarnath Murthy, Apr 08 2002

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Cf. A069709, A069710, A069711.

Do[ If[ IntegerQ[ Apply[ Plus, IntegerDigits[n(n + 1)/2]] / Floor[ Log[10, n(n + 1)/2] + 1]], Print[n(n + 1)/2]], {n, 1, 10^3}]
Select[Accumulate[Range[200]],IntegerQ[Mean[IntegerDigits[#]]]&] (* Harvey P. Dale, Jun 17 2017 *)

Edited and extended by Robert G. Wilson v, Apr 12 2002

A329192 Fibonacci numbers with arithmetic mean of digits an integer (sum of digits = a multiple of number of digits).

Original entry on oeis.org

1, 2, 3, 5, 8, 13, 55, 144, 987, 6765, 10946, 9227465, 225851433717, 8944394323791464, 160500643816367088, 83621143489848422977, 59425114757512643212875125, 30960598847965113057878492344, 127127879743834334146972278486287885163

Offset: 1

José de Jesús Camacho Medina, Nov 07 2019

			55 is a term as the arithmetic mean of digits is an integer: (5+5)/2 = 5.
144 is a term as the arithmetic mean of digits is an integer: (1+4+4)/3 = 3.
6765 is a term as the arithmetic mean of digits is an integer: (6+7+6+5)/4 = 6.

Cf. A000045, A004090, A060384, A069709, A164947.

Mathematica
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a(n) = A000045(A164947(n+1)).

cp's OEIS Frontend

A285095 Corresponding values of arithmetic means of digits of primes from A069709.

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A069710 Primes with arithmetic mean of digits = 1 (sum of digits = number of digits).

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A069711 Squares whose arithmetic mean of digits is an integer (i.e., the sum of digits is a multiple of the number of digits).

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A285096 Primes with integer arithmetic mean of digits = 2 in base 10.

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A285225 Primes with integer arithmetic mean of digits = 4 in base 10.

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A285226 Primes with integer arithmetic mean of digits = 5 in base 10.

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A285227 Primes with integer arithmetic mean of digits = 7 in base 10.

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A285228 Primes with integer arithmetic mean of digits = 8 in base 10.

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A069712 Triangular numbers with arithmetic mean of digits = integer (sum of digits = A multiple of the number of digits).

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