A257766 -id:A257766 - cp's OEIS frontend

A258484 Numbers m such that m equals a fixed number raised to the powers of the digits.

1, 10, 12, 100, 101, 111, 1000, 1010, 1033, 1100, 2112, 4624, 10000, 10001, 11101, 20102, 31301, 100000, 100010, 100011, 100100, 100101, 100110, 101000, 101001, 101010, 101100, 101110, 101111, 101121, 110000, 110001, 110010, 110100, 110110, 110111, 111000

Offset: 1

Pieter Post, May 31 2015

Let m = abcde... and z is a fixed radix -> m = z^a +z^b +z^c +z^d +z^e...

A number m made of k ones and h zeros is a member if m-h is divisible by k. Several other large members exist, including 12095925296900865188 (base = 113) and 115330163577499130079377256005 (base = 1500). - Giovanni Resta, Jun 01 2015

			12 = 3^1 + 3^2;
31301 = 25^3 + 25^1 + 25^3 + 25^0 + 25^1;
595968 = 4^5 + 4^9 + 4^5 + 4^9 + 4^6 + 4^8;
13177388 = 7^1 + 7^3 + 7^1 + 7^7 + 7^7 + 7^3 + 7^8 + 7^8.

Giovanni Resta, Table of n, a(n) for n = 1..956 (terms < 10^12)
Giovanni Resta, Table of a(1..956) values and corresponding bases

Cf. A005188, A023052, A257784, A257766, A257787, A257814.

okQ[v_] := Block[{b, d=IntegerDigits@ v, y, t}, t = Last@ Tally@ Sort@d; b = Floor[ (v/t[[2]]) ^ (1/t[[1]])]; While[(y = Total[b^d]) > v, b--]; v==y]; Select[Range[10^5],okQ] (* Giovanni Resta, Jun 01 2015 *)

for(n=1,10^5,d=digits(n);for(m=1,n,s=sum(i=1,#d,m^d[i]);if(s==n,print1(n,", ");break);if(s>n,break))) \\ Derek Orr, Jun 12 2015

def moda(n,a,m):
    kk = 0
    while n > 0:
        na=int(n%m)
        kk= kk+a**na
        n =int(n//m)
    return kk
for c in range (1, 10**8):
    for a in range (1,20):
        if  c==moda(c,a,10):
            print (a,c)

More terms from Giovanni Resta, Jun 01 2015

A366507 Numbers k such that the sum of the digits of k times the square of the sum of the digits cubed of k equals k.

Original entry on oeis.org

1, 4147200, 12743163, 21147075, 39143552, 52921472, 156754936, 205889445, 233935967

Offset: 1

René-Louis Clerc, Oct 11 2023

There are exactly 9 such numbers (Property 13 of Clerc).

			4147200 = (4+1+4+7+2)*(4^3+1+4^3+7^3+2^3)^2 = 18*230400.

René-Louis Clerc, Quelques nombres de Niven-Harshad particuliers, pp. 1-17, hal-04235744, 2023.
René-Louis Clerc, Nombres S+P, maxSP, minSP et |P-S|, hal-04507547 [math.nt], 2024. (In French)

Cf. A115518, A257766, A061209, A061210, A254000, A130680.

niven12()={for(a=0,9,for(b=0,9,for(c=0,9,for(d=0,9,for(e=0,9,for(f=0,9,for(g=0,9,for(h=0,9,for(i=0,9,for(j=0,9,if((a+b+c+d+e+f+g+h+i+j)*(a^3+b^3+c^3+d^3+e^3+f^3+g^3+h^3+i^3+j^3)^2==1000000000*a+100000000*b+10000000*c+1000000*d+100000*e+10000*f+1000*g+100*h+10*i+j,print1(1000000000*a+100000000*b+10000000*c+1000000*d+100000*e+10000*f+1000*g+100*h+10*i+j,";"))))))))))))}

isok(k) = my(d=digits(k)); vecsum(d)*sum(i=1, #d, d[i]^3)^2 == k; \\ Michel Marcus, Oct 12 2023

A366512 Numbers k such that the square of the sum of the digits times the sum of the cubes of the digits equals k.

Original entry on oeis.org

1, 32144, 37000, 111616, 382360

Offset: 1

René-Louis Clerc, Oct 11 2023

There are exactly 5 such numbers (Property 14 of Clerc).

			32144 = ((3+2+1+4+4)^2)*(3^3 + 2^3 + 1^3 + 4^3 + 4^3) = 196*164.

René-Louis Clerc, Quelques nombres de Niven-Harshad particuliers, pp. 1-17, hal-04235744, 2023.
René-Louis Clerc, Nombres S+P, maxSP, minSP et |P-S|, hal-04507547 [math.nt], 2024. (In French)

Cf. A115518, A257766, A061209, A061210, A254000, A130680.

Select[Range[10^6], #1 == Total[#2]^2*Total[#2^3] & @@ {#, IntegerDigits[#]} &] (* Michael De Vlieger, Mar 25 2024 *)

niven23()={for(a=0,9,for(b=0,9,for(c=0,9,for(d=0,9,for(e=0,9,for(f=0,9,for(g=0,9,for(h=0,9,if((a+b+c+d+e+f+g+h)^2*(a^3+b^3+c^3+d^3+e^3+f^3+g^3+h^3)==10000000*a+1000000*b+100000*c+10000*d+1000*e+100*f+10*g+h,print1(10000000*a+1000000*b+100000*c+10000*d+1000*e+100*f+10*g+h,", "))))))))))}

isok(k) = my(d=digits(k)); vecsum(d)^2*sum(i=1, #d, d[i]^3) == k; \\ Michel Marcus, Oct 12 2023

A257860 Numbers n such that a digit of n to the power k plus the sum of the other digits of n equals n, where k is a positive integer.

Original entry on oeis.org

1, 89, 132, 264, 518, 739, 2407, 6579, 8200, 8201, 8202, 8203, 8204, 8205, 8206, 8207, 8208, 8209, 32780, 32781, 32782, 32783, 32784, 32785, 32786, 32787, 32788, 32789, 59060, 59061, 59062, 59063, 59064, 59065, 59066, 59067, 59068, 59069, 78145, 524300, 524301, 524302, 524303, 524304, 524305, 524306, 524307, 524308, 524309, 531459, 823567, 2097178

Offset: 1

Pieter Post, May 11 2015

There are numbers that come in groups of 10, like 8200, 32780 and 524300. But there are also a few stand-alone numbers. Like 531459 (=5+3+1+4+5+9^6).

It is easy to generate large terms in the sequence, for example, 9^104+409 and 9^1047+4561 are the smallest terms with 100 and 1000 digits, respectively. - Giovanni Resta, May 12 2015

			89 is in the sequence because 89 = 8+9^2.
2407 is in the sequence because 2407 = 2+4+0+7^4.
8202 is in the sequence because 8202 = 8+ 2^13 +0+2, also 8202 = 8+2+0+2^13.

Cf. A061209, A130680, A257766, A257768, A257784,

Cf. A007953.

import Data.List (nub); import Data.List.Ordered (member)
a257860 n = a257860_list !! (n-1)
a257860_list = 1 : filter f [1..] where
   f x = any (\d -> member (x - q + d) $ ps d) $ filter (> 1) $ nub ds
         where q = sum ds; ds = (map (read . return) . show) x
   ps x = iterate (* x) (x ^ 2)
-- Reinhard Zumkeller, May 12 2015

def sod(n):
    kk = 0
    while n > 0:
        kk= kk+(n%10)
        n =int(n//10)
    return kk
for i in range (1,10**7):
    for j in range(1,len(str(i))+1):
        k=(i//(10**(j-1)))%10
        for m in range (2,30):
            if i==sod(i)+k**m-k:
                print (i)

One more term and some missing data added by Reinhard Zumkeller, May 12 2015

A368939 Numbers k such that the sum of the digits times the sum of the fourth powers of the digits equals k.

Original entry on oeis.org

0, 1, 182380, 444992

Offset: 1

René-Louis Clerc, Jan 10 2024

There are exactly 4 such numbers (Property 16 of Clerc).

			182380 = (1+8+2+3+8)*(1^4 + 8^4 + 2^4 + 3^4 + 8^4) = 22*8290.

René-Louis Clerc, Quelques nombres de Niven-Harshad particuliers, 2023.

Cf. A115518, A257766, A061209, A061210, A254000, A130680, A366507, A366512.

Select[Range[0,10^7],#==Total[IntegerDigits[#]]*Total[IntegerDigits[#]^4]&] (* James C. McMahon, Jan 11 2024 *)

niven14(k) = my(d=digits(k)); vecsum(d)*sum(i=1, #d, d[i]^4) == k;
for(k=1,10^7,if(niven14(k)==1,print1(k,", ")))

A370250 Numbers k such that the sum of the digits times the square of the sum of the fourth powers of the digits equals k.

Original entry on oeis.org

0, 1, 5873656512, 7253758561, 29961747275

Offset: 1

René-Louis Clerc, Feb 13 2024

There are exactly 5 such numbers (Property 17 of Clerc).

			7253758561 = (7+2+5+3+7+5+8+5+6+1)*(7^4 + 2^4 + 5^4 + 3^4 + 7^4 + 5^4 + 8^4 + 5^4 + 6^4 + 1^4)^2 = 49*148035889 = 7253758561.

René-Louis Clerc, Quelques nombres de Niven-Harshad particuliers, 2023.

Cf. A368939, A115518, A257766, A061209, A061210, A254000, A130680, A366507, A366512.

niven142(k) = my(d=digits(k)); vecsum(d)*sum(i=1, #d, d[i]^4)^2 == k;
for(k=0,10^12,if(niven142(k)==1,print1(k, ", ")))

cp's OEIS Frontend

A258484 Numbers m such that m equals a fixed number raised to the powers of the digits.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Python

Extensions

A366507 Numbers k such that the sum of the digits of k times the square of the sum of the digits cubed of k equals k.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

PARI

A366512 Numbers k such that the square of the sum of the digits times the sum of the cubes of the digits equals k.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

A257860 Numbers n such that a digit of n to the power k plus the sum of the other digits of n equals n, where k is a positive integer.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Haskell

Python

Extensions

A368939 Numbers k such that the sum of the digits times the sum of the fourth powers of the digits equals k.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A370250 Numbers k such that the sum of the digits times the square of the sum of the fourth powers of the digits equals k.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI