Jonathan Frech - cp's OEIS frontend

A325902 Numbers whose neighbor's prime factors with multiplicity can be partitioned into two multisets of equal sum.

11, 17, 21, 23, 27, 31, 50, 55, 56, 65, 71, 89, 129, 131, 144, 155, 169, 204, 209, 216, 229, 239, 241, 244, 251, 265, 287, 288, 300, 305, 337, 344, 351, 371, 373, 379, 407, 415, 493, 494, 517, 526, 545, 577, 645, 647, 664, 681, 685, 737, 749, 755, 769, 776, 780, 783, 815

Offset: 1

Jonathan Frech, Sep 07 2019

nonn

The neighbors of n are the two numbers n-1 and n+1.

			71 is in the sequence since 70 = 2*5*7 < 71 < 2*2*2*3*3 = 72 with 2 + 5 + 3 + 3 = 7 + 2 + 2 + 2.

Jonathan Frech, Table of n, a(n) for n = 1..10000

Cf. A063968.

import Data.List (subsequences, (\\))
factors 1 = []
factors n | p <- head $ filter ((== 0) . mod n) [2..]
          = p : factors (n `div` p)
sumPartitionable ns | p <- \ms -> sum ms == sum (ns \\ ms)
                    = any p $ subsequences ns
a325902 = filter (\n -> sumPartitionable $ factors (n-1) ++ factors (n+1)) [2..]

ok[n_] := Block[{t, p, m, z}, {p, m} = Transpose@ Tally@ Sort[ Join@ Flatten[ ConstantArray @@@ FactorInteger[#] & /@ {n-1, n+1}]]; t = Total[p m]; If[ OddQ@ t, False, z = Quiet@ LinearProgramming[1 + 0 p, {p}, {{t/2, 0}}, Prepend[#, 0] & /@ Transpose@{m}, Integers]; ListQ@z && Total[z p]==t/2]]; Select[ Range[3, 815], ok] (* Giovanni Resta, Sep 10 2019 *)

A288040 Integers whose number of distinct decimal digits is prime.

Original entry on oeis.org

10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 100, 101

Offset: 1

Jonathan Frech, Jun 04 2017

Differs from A139819 (which contains, for example, 1234, a number with 4 distinct decimal digits). - R. J. Mathar, Jun 14 2017

Union of A031955 and A031962 and ....

Select[Range@ 101, PrimeQ@ Count[DigitCount[#], ?(# != 0 &)] &] (* _Michael De Vlieger, Jun 06 2017 *)

isok(m) = isprime(#Set(digits(m))); \\ Michel Marcus, May 10 2020

from sympy import isprime
print([n for n in range(1, 100) if isprime(len(set(str(n))))])

A286193 Decimal expansion of Sum_{n>=1} 1/(n^n+n).

Original entry on oeis.org

7, 0, 4, 1, 8, 8, 3, 4, 9, 9, 2, 3, 3, 1, 4, 7, 1, 8, 1, 7, 1, 2, 6, 2, 0, 0, 0, 4, 4, 2, 2, 1, 8, 5, 7, 8, 0, 3, 0, 4, 1, 5, 8, 5, 3, 7, 7, 2, 9, 0, 9, 4, 4, 5, 0, 7, 4, 2, 2, 2, 3, 4, 0, 3, 8, 6, 1, 3, 9, 0, 6, 9, 3, 8, 2, 4, 0, 7, 7, 6, 1, 5, 5, 9, 0, 4, 5, 6, 8, 5, 6, 2, 3, 5, 3, 0, 5, 6

Offset: 0

Jonathan Frech, May 04 2017

			0.70418834992331471817126200044221857803...

Cf. A073009, A134883.

RealDigits[N[Sum[1/(n^n+n),{n,1,Infinity}],20]][[1]]

suminf(n=1, 1/(n^n+n)) \\ Michel Marcus, May 04 2017

from decimal import *;getcontext().prec=300;print(sum([Decimal(1)/Decimal(n**n+n)for n in range(1,200)]))

A285494 Numbers k such that digit sum of k = number of distinct prime factors of k.

Original entry on oeis.org

20, 30, 102, 120, 200, 300, 1002, 1200, 2000, 2001, 2002, 3000, 3010, 10001, 10002, 10011, 10030, 10120, 11001, 11020, 11110, 12000, 20000, 20001, 21010, 30000, 30030, 30100, 100001, 100030, 100101, 100130, 100210, 100300, 101001, 101101, 101200, 102102, 110001, 110200

Offset: 1

Jonathan Frech, Apr 19 2017

Giovanni Resta, Table of n, a(n) for n = 1..10000

Cf. A001221, A007953, A057531, A057532, A050689, A070274, A070275, A063737, A067077.

Select[Range[2,10000],Total[IntegerDigits[#]]==Length[FactorInteger[#]]&] (* or *)
nd = 10; s = 1; Sort@ Flatten@ Reap[ While[ Times @@ Prime[ Range@ s] < 10^nd, pa = IntegerPartitions[s, {nd}, Range[0, 9]]; Do[Sow@ Select[ FromDigits /@ Permutations[p], PrimeNu[#] == s &], {p, pa}]; s++]][[2, 1]] (* terms < 10^10, Giovanni Resta, Apr 21 2017 *)

isok(n) = sumdigits(n) == omega(n); \\ Michel Marcus, Apr 20 2017

More terms from Michel Marcus, Apr 20 2017

A278328 Numbers n such that abs(n - rev(n)) is a square, where rev(n) is the decimal reverse of n (A004086).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 15, 21, 22, 23, 26, 32, 33, 34, 37, 40, 43, 44, 45, 48, 51, 54, 55, 56, 59, 62, 65, 66, 67, 73, 76, 77, 78, 84, 87, 88, 89, 90, 95, 98, 99, 101, 111, 121, 131, 141, 151, 161, 171, 181, 191, 202, 212, 222, 232, 242, 252, 262

Offset: 1

Jonathan Frech, Nov 18 2016

All palindromes are in this sequence, hence it is infinite.

Alois P. Heinz, Table of n, a(n) for n = 1..20000

A002113 is a subsequence.

Cf. A000290, A004086, A016766, A056965.

a:= proc(n) option remember; local k; for k from 1+
      `if`(n=1, -1, a(n-1)) while not issqr(abs(k-(s->
       parse(cat(s[-i]$i=1..length(s))))(""||k))) do od: k
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Nov 18 2016

Select[Range@ 262, IntegerQ@ Sqrt@ Abs[# - FromDigits@ Reverse@ IntegerDigits@ #] &] (* Michael De Vlieger, Nov 18 2016 *)

is(n) = issquare(abs(n - eval(concat(Vecrev(Str(n)))))) \\ Felix Fröhlich, Nov 18 2016

is(n, {b=10}) = issquare(abs(n - subst(Polrev(digits(n, b),'x),'x,b))); \\ Gheorghe Coserea, Nov 27 2016

import math
n = 0
while True:
    if math.sqrt(abs(n-int(str(n)[::-1])))%1 == 0:
        print(n)
    n += 1 # Jonathan Frech, Nov 18 2016

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User: Jonathan Frech

A325902 Numbers whose neighbor's prime factors with multiplicity can be partitioned into two multisets of equal sum.

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A288040 Integers whose number of distinct decimal digits is prime.

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A286193 Decimal expansion of Sum_{n>=1} 1/(n^n+n).

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A285494 Numbers k such that digit sum of k = number of distinct prime factors of k.

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A278328 Numbers n such that abs(n - rev(n)) is a square, where rev(n) is the decimal reverse of n (A004086).

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Maple

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