Displaying 101-110 of 113 results found.
Sum of squares of digits of prime factors of n, with multiplicity.
+10
0
0, 4, 9, 8, 25, 13, 49, 12, 18, 29, 2, 17, 10, 53, 34, 16, 50, 22, 82, 33, 58, 6, 13, 21, 50, 14, 27, 57, 85, 38, 10, 20, 11, 54, 74, 26, 58, 86, 19, 37, 17, 62, 25, 10, 43, 17, 65, 25, 98, 54, 59, 18, 34, 31, 27, 61, 91, 89, 106, 42
COMMENTS
Differs from A067666 if any prime factor exceeds 1 digit. Fixed points include 16, 27. See also: A067666 Sum of squares of prime factors of n (counted with multiplicity). See also: A003132 Sum of squares of digits of n. See also: A118503 Sum of digits of prime factors of n, with multiplicity.
FORMULA
a(n) = SUM[i=1..k] (e_i)* A003132(p_i) where prime decomposition of n = (p_1)^(e_1) * (p_2)^(e_2) * ... * (p_k)^(e_k).
EXAMPLE
a(22) = 6 because 22 = 2 * 11 and the sum of squares of digits of prime factors is 2^2 + 1^2 + 1^2.
a(121) = 4 because 121 = 11^2 = 11 * 11, so 1^2 + 1^2 + 1^2 + 1^2 = 4.
MATHEMATICA
Join[{0}, Table[Total[Flatten[IntegerDigits/@(Flatten[Table[#[[1]], #[[2]]]&/@ FactorInteger[ n]])]^2], {n, 2, 60}]] (* Harvey P. Dale, Nov 17 2022 *)
Numbers n such that the sum of the squares of the digits of n^n is a square.
+10
0
0, 1, 2, 8, 10, 100, 123, 209, 312, 1000, 1668, 2191, 2268, 4767, 9338, 10000, 11004, 12248, 12322, 15926, 17951, 18202, 19764, 21807, 29509, 42647, 43072, 44750, 54237, 56634, 70383, 74032, 85325, 90906, 95261, 100000
EXAMPLE
8 is in the sequence because 8^8 = 16777216 and 1^2+6^2+7^2+7^2+7^2+2^2+1^2+6^2
= 225 = 15^2.
MAPLE
with(numtheory): digits:=200:nn:=5000:for n from 0 to nn do:l:=length(n^n):n0:=n^n:s:=0:for
m from 1 to l do:q:=n0:u:=irem(q, 10):v:=iquo(q, 10):n0:=v :s:=s+u^2:od:if sqrt(s)=
floor(sqrt(s))then printf(`%d, `, n):else fi:od:
MATHEMATICA
Join[{0}, Select[Range[100000], IntegerQ[Sqrt[Total[IntegerDigits[ #^#]^2]]]&]] (* Harvey P. Dale, Sep 25 2018 *)
PROG
(PARI) isok(n) = my(d = digits(n^n)); issquare (sum(i=1, #d, d[i]^2)); \\ Michel Marcus, Jan 15 2014
EXTENSIONS
Offset corrected and more terms added, Michel Marcus, Jan 15 2014
Smallest prime p such that the sum of the squares of the digits of p equals n (or 0 if no such prime exists).
+10
0
0, 11, 0, 2, 101111, 211, 2111, 101111111, 3, 13, 113, 112121, 23, 21221, 1123, 11213, 41, 1223, 313, 10133, 241, 233, 112223, 21313, 5, 431, 151, 1151, 13313, 251, 2333, 11251, 2243, 53, 1433, 1153, 61, 523, 1523, 11161, 443, 541, 353, 33413, 2621, 163, 1163, 13523, 7, 17
COMMENTS
a(3) = 0 because the numbers of the form 10..010..01 are divisible by 3. Conjecture : except for the numbers 1 and 3, for every possible square digit sum there exists a prime.
EXAMPLE
a(13) = 23 because 2^2 + 3^2 = 13, and 23 is the least such prime.
MAPLE
with(numtheory):for k from 2 to 100 do: id:=0:for p from 1 to 100000 while(id=0)
do:n:=ithprime(p):l:=length(n):n0:=n:s:=0:for m from 1 to l do:q:=n0:u:=irem(q, 10):v:=iquo(q, 10):n0:=v :s:=s+u^2:od: if s=k then id:=1:printf(`%d, `, n):else fi:od:od:
Primes prime(k) such that the sum of the squares of digits of prime(k) equals the sum of the squares of digits of k.
+10
0
449, 2141, 2213, 3137, 4339, 4349, 4421, 5701, 9623, 13093, 14653, 16447, 16567, 22901, 27091, 36061, 41491, 45403, 45413, 45433, 50417, 52691, 54631, 54941, 55049, 56963, 57089, 58943, 60601, 61043, 61051, 63607, 65831, 66301, 67843, 68543, 72019, 73421
EXAMPLE
449 is the 87th prime and sum of the squares of digits of both is 113;
56963 is the 5778th prime and sum of the squares of digits of both is 187.
MATHEMATICA
Prime[Select[Range[10000], Apply[Plus, IntegerDigits[#]^2] == Apply[Plus, IntegerDigits[Prime[#]]^2] &]]
Primes such that the sum of the squares of their digits equals the number of their digits.
+10
0
11, 102001, 1000121, 1000211, 1002101, 1010201, 1020011, 1020101, 1021001, 1102001, 1120001, 1201001, 2001101, 2100011, 2110001, 100012111, 100101121, 100110121, 100112101, 100121011, 100211101, 101020111, 101100211, 101102101, 101110201, 101210101, 102100111
COMMENTS
A subsequence of 1, 11, 111, 1111, 2000, 10002, 10020, 10200, 11111,... which contains n such that A003132(n) = A055642(n). - R. J. Mathar, Nov 07 2011
EXAMPLE
a(2) = 102001 is in the sequence because 1^2+0^2+2^2+0^2+0^2+1^2 = 6 = length(a(2)).
MATHEMATICA
fQ[n_] := Plus @@ (IntegerDigits[n]^2) == IntegerLength[n]; Select[Prime[Range[100000000]], fQ] (* Robert G. Wilson v, Nov 07 2011 *)
Sum of the 16th powers of the decimal digits of n.
+10
0
0, 1, 65536, 43046721, 4294967296, 152587890625, 2821109907456, 33232930569601, 281474976710656, 1853020188851841, 1, 2, 65537, 43046722, 4294967297, 152587890626, 2821109907457, 33232930569602, 281474976710657, 1853020188851842, 65536, 65537, 131072, 43112257
COMMENTS
This is to exponent 16 as A007953 is to exponent 1, A003132 is to exponent 2, and A055013 is to exponent 4.
EXAMPLE
a(14) = 1^16 + 4^16 = 4294967297 = 641 * 6700417.
Numbers n such that sum of digits of n equals the least prime dividing n and sum of squares of digits of n equals the greatest prime dividing n.
+10
0
133, 803, 2023, 106811, 383177, 1071949, 1342027, 2025343, 2569757, 2911123, 3341831, 3993883, 6285901, 10860071, 11194319, 13270013, 21736109, 21871477, 22159451, 22421587, 26011229, 27600257, 31174391, 32656681, 34880611, 40435193, 41755573, 53738911
EXAMPLE
383177 = 29 * 73 * 181 is in the sequence because 29 = 3+8+3+1+7+7 and 181 = 3^2+8^2+3^2+1^2+7^2+7^2.
MAPLE
with(numtheory):A:= proc(n) add(u, u=convert(n, base, 10)) ; end proc: B:= proc(m) add(v^2, v=convert(m, base, 10)) ; end proc: for i from 2 to 1000000 do:x:=factorset(i):n1:=nops(x):if x[1]=A(i) and x[n1]=B(i) then print(i):else fi:od:
PROG
(PARI) is_ A217690(n)={my(d=digits(n), s=norml2(d), f); (n%s || !isprime(s) || n%(d=sum(i=1, #d, d[i])) || !isprime(d)) & return; !(f=factor(n/(d*s))[, 1]) || (d <= f[1] & s >= f[#f])} \\ Charles R Greathouse IV and M. F. Hasler, Oct 11 2012
Numbers n whose sum of digits, sum of square of digits and sum of cube of digits all have the same digital root.
+10
0
1, 9, 10, 11, 19, 36, 47, 63, 74, 90, 91, 99, 100, 101, 109, 110, 111, 119, 136, 147, 163, 174, 190, 191, 199, 306, 316, 333, 360, 361, 369, 396, 407, 417, 444, 470, 471, 479, 497, 603, 613, 630, 631, 639, 666, 693, 704, 714, 740, 741, 749, 777, 794, 900, 901, 909, 910, 911, 919, 936, 947, 963, 974, 990, 991, 999, 1000, 1001
EXAMPLE
a(6)=36 since (first power) 3+6=9; (second power) 3^2+6^2=9+36=45 and 4+5=9; (third power) 3^3+6^3=27+216=243 and 2+4+3=9.
PROG
(PARI) dr(n) = if (n, (n-1)%9+1); \\ from A010888dsq(n) = if (n, d = digits(n); sum(id = 1, #d, d[id]^2));
dcb(n) = if (n, d = digits(n); sum(id = 1, #d, d[id]^3));
ok(n) = (dr(n) == dr(dcb(n))) && (dr(n) == dr(dsq(n))); \\ Michel Marcus, Sep 24 2013
Numbers n such that A290223(n) = 0.
+10
0
1, 19, 81, 162, 181, 199, 243, 262, 324, 343, 405, 424, 486, 505, 567, 648, 685, 729, 766, 810, 847, 891, 910, 928, 972, 1053, 1072, 1134, 1153, 1215, 1234, 1296, 1315, 1377, 1458, 1495, 1539, 1576, 1620, 1657, 1701, 1720, 1738, 1782, 1801, 1819, 1863, 1944, 1981, 1999, 2025, 2044, 2106, 2125, 2187, 2206, 2268
COMMENTS
This sequence is believed to be infinite.
EXAMPLE
181 is in this sequence because 181 - (1+8+1)^2 = 81. Then 81 - (8+1)^2 = 0.
PROG
(PARI)
a(n)=k=n; c=1; v=List(); listput(v, k); while(c, if(k>=0, k-=sumdigits(k)^2; c+=1; if(k==2||k==3||k==0||k==6||k==9, return(k)); if(vecsearch(Vec(v), k), return(sumdigits(abs(k)))); listput(v, k)); if(k<0, k+=sumdigits(-k)^2; c+=1; if(k==2||k==3||k==0||k==6||k==9, return(k)); if(vecsearch(Vec(v), k), return(sumdigits(abs(k)))); listput(v, k)); c+=1)
for(n=1, 10^4, if(a(n)==0, print1(n, ", ")))
Numbers n such that A290223(n) = 2.
+10
0
2, 23, 62, 77, 119, 194, 287, 398
COMMENTS
This sequence is believed to be finite. a(9) > 10^5, if it exists.
EXAMPLE
62 is in this sequence because 62 - (6+2)^2 = -2. Then -2 + (2)^2 = 2.
PROG
(PARI)
a(n)=k=n; c=1; v=List(); listput(v, k); while(c, if(k>=0, k-=sumdigits(k)^2; c+=1; if(k==2||k==3||k==0||k==6||k==9, return(k)); if(vecsearch(Vec(v), k), return(sumdigits(abs(k)))); listput(v, k)); if(k<0, k+=sumdigits(-k)^2; c+=1; if(k==2||k==3||k==0||k==6||k==9, return(k)); if(vecsearch(Vec(v), k), return(sumdigits(abs(k)))); listput(v, k)); c+=1)
for(n=1, 10^5, if(a(n)==2, print1(n, ", ")))
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