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6-white numbers: partition digits of n^6 into blocks of 6 starting at right; sum of these 6-digit numbers equals n.
+10
7
0, 1, 1208494, 1358344, 1415583, 1538460, 1734265, 1773226, 1818180, 1994707, 2155140, 2187108, 2208493, 2215486, 2272725, 2272726, 2311687, 2318680, 2351350, 2356641, 2358343, 2363634, 2390311, 2402596, 2420874, 2449252, 2454544, 2459835, 2481220, 2500498, 2533168
COMMENTS
Three pairs of consecutive terms: 2272725 and 2272726; 2999997 and 2999998; 3272724 and 3272725.
EXAMPLE
1208494^6 = 3115064124992224583219040254156270656 and 3 + 115064 + 124992 + 224583 + 219040 + 254156 + 270656 = 1208494.
MAPLE
P:=proc(q, h) local a, b, n;
for n from 0 to q do a:=n^h; b:=0; while a>0 do b:=b+(a mod 10^h); a:=trunc(a/10^h); od;
if n=b then print(n); fi; od; end: P(10^6, 6);
MATHEMATICA
k = 6; Select[Range[0, 10^7], Function[n, Total[FromDigits /@ Partition[PadLeft[#, Length@ # + k - Mod[Length@ #, k]], k]] == n &@ IntegerDigits[n^k]]] (* Michael De Vlieger, Jul 08 2016, after Harvey P. Dale at A037045 *)
Like 4-white numbers but with blocks of 4 starting at left.
+10
3
0, 1, 1000, 11110, 14638, 15628, 17170, 18217, 19305, 19999, 21649, 22320, 25234, 29041, 30195, 31428
EXAMPLE
14638^4 = 45912080296849936 and 4591 + 2080+ 2968 + 4993 + 6 = 14638.
MAPLE
P:=proc(q, h) local a, b, c, d, n; print(0); for n from 1 to q do
a:=n^h; d:=ilog10(n^h)+1; c:=d-h*trunc(d/h); b:=0;
while a>0 do b:=b+(a mod 10^c); a:=trunc(a/10^c); c:=h; od;
if n=b then print(n); fi; od; end: P(10^15, 4);
Like 5-white numbers but with blocks of 5 starting at left.
+10
3
1, 10000, 73440, 95120, 218510, 221220, 222220, 242900, 245610, 289970, 344070
EXAMPLE
73440^5 = 2136305413264402022400000 and 21363 + 05413 + 26440 + 20224 + 00000 = 73440.
MAPLE
P:=proc(q, h) local a, b, c, d, n; print(0); for n from 1 to q do
a:=n^h; d:=ilog10(n^h)+1; c:=d-h*trunc(d/h); b:=0;
while a>0 do b:=b+(a mod 10^c); a:=trunc(a/10^c); c:=h; od;
if n=b then print(n); fi; od; end: P(10^15, 5);
Like 7-white numbers but with blocks of 7 starting at left.
+10
3
0, 1, 1000000, 20585070, 25104356, 25975583, 27483737, 27940490, 27941490, 28133416, 29069509, 32345773, 32482961, 32581773, 33332330, 34310934, 34676272, 35530163, 35707886, 36067139, 41716867, 42163087, 42568703, 44444440, 47745130
EXAMPLE
20585070^7 = 1566269305839650871270449961448347855098390430000000 and 1566269 + 3058396 + 5087127 + 0449961 + 4483478 + 5509839 + 0430000 + 000 = 20585070.
MAPLE
P:=proc(q, h) local a, b, c, d, n; print(0); for n from 1 to q do
a:=n^h; d:=ilog10(n^h)+1; c:=d-h*trunc(d/h); b:=0;
while a>0 do b:=b+(a mod 10^c); a:=trunc(a/10^c); c:=h; od;
if n=b then print(n); fi; od; end: P(10^15, 7);
Like 6-white numbers but with blocks of 6 starting at left.
+10
2
0, 1, 100000, 1705330, 1818180, 1941030, 2046807, 2227770, 2285010, 2414880, 2598400, 2694600, 2727270, 2728270, 2758239, 2760940, 2857140, 2890810, 2979315, 3040660, 3085911, 3317050, 3541014, 3636460, 4543174
EXAMPLE
1705330^6 = 24595213291709423201966052256969000000 and 245952 + 132917 + 094232 + 019660 + 522569 + 690000 + 00 = 1705330.
MAPLE
P:=proc(q, h) local a, b, c, d, n; print(0); for n from 1 to q do
a:=n^h; d:=ilog10(n^h)+1; c:=d-h*trunc(d/h); b:=0;
while a>0 do b:=b+(a mod 10^c); a:=trunc(a/10^c); c:=h; od;
if n=b then print(n); fi; od; end: P(10^15, 7);
MATHEMATICA
Select[Range[0, 4544000], Total[FromDigits/@Partition[IntegerDigits[#^6], UpTo[6]]]==#&] (* Harvey P. Dale, Dec 25 2023 *)
a(n) is the smallest k that is equal to the sum of the digits of k*(k+1)*...*(k+n-1) in base 10^n, or -1 if such a number does not exist.
+10
0
1, 33, -1, 10692, 74016, 1153845, 19999998, 373722624, 3025660311, 39999999996, -1
COMMENTS
Partition the product of n consecutive integers, from k to k+n-1, into blocks of n digits starting from right. Sequence lists, for each n, the first number k of the least product whose sum of blocks is equal to k, or -1 if such a number does not exist.
a(12) <= 6*10^12 - 6, a(13) <= 4*10^13 - 4, a(14) <= 6*10^14 - 6, a(15) <= 8*10^15 - 8, a(16) <= 6*10^16 - 6, a(17) <= 8*10^17 - 8, a(18) <= 8*10^18 - 8. - Giovanni Resta, May 25 2019
EXAMPLE
a(2) = 33 because 33*34 = 1122 and 11 + 22 = 33.
a(4) = 10692 because 10692*10693*10694*10695 = 13076137707585480 and 5480 + 758 + 1377 + 3076 + 1 = 10692.
MAPLE
P:=proc(q) local a, b, c, j, k, n, x; c:=1; for n from 1 to q do x:=0:
for k from c to q do a:=mul(j, j=k..k+n-1); b:=0; while a>0 do
b:=b+(a mod 10^n); a:=trunc(a/10^n); od; if k>b then x:=x+1;
else if k<b then x:=0: fi; fi; if k=b then lprint(n, k); c:=n: break;
fi; if x=100 then lprint(n, -1); break; fi; od; od; end: P(10^15);
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