A302649
Numbers that are the sum of some fixed power of the digits of their ten's complement.
Original entry on oeis.org
5, 8, 14, 3953, 33626, 89843301, 71341793655800, 245916794707565, 19429639306542698, 36106092555634673, 1818632037625982420, 4099389352522800257, 51096092690519702666, 1361788669288181208317, 80939622935362328928524, 3061856409269150191916609
Offset: 1
(10 - 5) = 5 and 5^1 = 5;
(10 - 8) = 2 and 2^3 = 8;
(100 - 14) = 86 and 8^1 + 6^1 = 14;
(10000 - 3953) = 6047 and 6^4 + 0^4 + 4^4 + 7^4 = 3953;
(100000 - 33626) = 66374 and 6^5 + 6^5 + 3^5 + 7^5 + 4^5 = 33626;
(100000000 - 89843301) = 10156699 and 1^8 + 0^8 + 1^8 + 5^8 + 6^8 + 6^8 + 9^8 + 9^8 = 89843301.
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with(numtheory): P:=proc(q) local a,b,i,j,k,n;
for n from 1 to q do a:=convert(10^(ilog10(n)+1)-n,base,10);
b:=convert(a,`+`); j:=1; i:=0; while n>b do
if i=b then break; else i:=b; j:=j+1; b:=add(a[k]^j,k=1..nops(a)); fi; od;
if n=b then print(n); fi; od; end: P(10^9);
A302651
Numbers that are the product of some fixed power of the digits of their ten's complement.
Original entry on oeis.org
5, 8, 81, 2016, 2205
Offset: 1
(10 - 5) = 5 and 5^1 = 5;
(10 - 8) = 2 and 2^3 = 8;
(100 - 81) = 19 and 1^2 * 9^2 = 81;
(10000 - 2016) = 7984 and 7^1 * 9^1 * 8^1 * 4^1 = 2016;
(10000 - 2205) = 7795 and 7^1 * 7^1 * 9^1 * 5^1 = 2205;
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with(numtheory): P:=proc(q) local a,b,i,j,k,n;
for n from 1 to q do a:=convert(10^(ilog10(n)+1)-n,base,10);
b:=convert(a,`*`); j:=1; i:=0; while n>b do
if i=b then break; else i:=b; j:=j+1; b:=add(a[k]^j,k=1..nops(a)); fi; od;
if n=b then print(n); fi; od; end: P(10^9);
A380810
Integers m for which m = Sum (d_i - 1)^k, where m is k decimal digits long and d_i are the digits of m.
Original entry on oeis.org
26, 126, 217, 729, 4193, 134068, 10875747, 24228197, 2491591748, 106557756999043
Offset: 1
134068 is a term since it is k=6 digits long and its digit powers are (1-1)^6 + (3-1)^6 + (4-1)^6 + (0-1)^6 + (6-1)^6 + (8-1)^6 = 134068.
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with(numtheory): P:=proc(q) local a, b, d, k, n;
for n from 1 to q do a:=convert(n,base,10); d:=length(n)
if add((a[k]-1)^d,k=1..d)=n then print(n); fi; od; end: P(3*10^7);
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Select[Range[150000], # == Sum[(Part[IntegerDigits[#], l] - 1)^IntegerLength[#],{l, IntegerLength[#]}] &] (* Stefano Spezia, Feb 04 2025 *)
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isok(m) = my(d=digits(m), k=#d); m == sum(i=1, k, (d[i]-1)^k); \\ Michel Marcus, Feb 04 2025
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from itertools import chain, combinations_with_replacement, islice
def A380810_gen(): # generator of terms
yield from chain.from_iterable(sorted(map(lambda s:sum((int(d)-1)**l for d in s),sorted(filter(lambda s:sorted(str(m:=sum((int(d)-1)**l for d in s)))==list(s) and 10**l>m>=10**(l-1),combinations_with_replacement('0123456789',l))))) for l in range(1,25))
A380810_list = list(islice(A380810_gen(),10)) # Chai Wah Wu, Feb 05 2025
A380854
Integers m for which m = Sum (d_i + 1)^k, where m is k decimal digits long and d_i are the digits of m.
Original entry on oeis.org
141, 251, 560, 664807556, 424710875510, 863812804425, 137134427278403350052, 366828486147473227474, 186740753582576522645847734
Offset: 1
664807556 is a term since it has 9 digits and (6+1)^9+(6+1)^9+(4+1)^9+(8+1)^9+(0+1)^9+(7+1)^9+(5+1)^9+(5+1)^9+(6+1)^9 = 664807556.
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from itertools import chain, combinations_with_replacement, count, islice
def A380854_gen(): # generator of terms
yield from chain.from_iterable(sorted(map(lambda s:sum((d+1)**l for d in s),sorted(filter(lambda s:tuple(sorted(map(int,str(m:=sum((d+1)**l for d in s)))))==s and 10**l>m>=10**(l-1),combinations_with_replacement(range(10),l))))) for l in count(1))
A380854_list = print(list(islice(A380854_gen(),8)))
Showing 1-4 of 4 results.
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