A239563
Numbers n such that there are a, b with abs(sigma(a) - sigma(b)) = sigma(n) - n and a U b = n, where U is decimal concatenation.
Original entry on oeis.org
23, 47, 139, 529, 611, 911, 1109, 1445, 1621, 2003, 2521, 2531, 4007, 4361, 4879, 7169, 9011, 9013, 9481, 13009, 18883, 22081, 24257, 25031, 45349, 49901, 56081, 90011, 98941, 99101, 133705, 160031, 169181, 200003, 202289, 210181, 250031, 307289, 336961, 357101
Offset: 1
For n = 9481 we can consider 9481 = 9 U 481 and sigma(9481) = 10000, sigma(481) = 532, sigma(9) = 13 and 532 - 13 = 519 = 10000 - 9481.
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with(numtheory);
T:=proc(t) local w,x,y; x:=t; y:=0; while x>0 do x:=trunc(x/10); y:=y+1; od; end:
P:=proc(q) local a,b,c,d,i,n;
for n from 1 to q do a:=sigma(n); b:=T(n);
for i from 1 to b-1 do c:=trunc(n/10^i); d:=n-c*10^i;
if abs(sigma(c)-sigma(d))=a-n then print(n); break; fi;
od; od; end: P(10^9);
A239686
Numbers n such that if n = a U b (where U denotes concatenation) then sigma*(a) + sigma*(b) = abs(sigma*(n) - n), where sigma*(n) is the sum of the anti-divisors of n.
Original entry on oeis.org
47, 118, 205, 846, 898, 1219, 4181, 4236, 4701, 4929, 6014, 6516, 13276, 30445, 59956, 61916, 63216, 67314, 72066, 79554, 90674, 106316, 128998, 129179, 136816, 142486, 143396, 180448, 229914, 284894, 357841, 421318, 483286, 486721, 487618, 500218, 642445
Offset: 1
Anti-divisors of 4701 are 2, 6, 7, 17, 79, 119, 1343, 553, 3134 and their sum is 5260. Consider 4701 as 4 U 701. Anti-divisors of 4 is 3 and of 701 are 2, 3, 23, 61, 467 whose sum is 556. At the end we have that 5260 - 4701 = 559 = 3 + 556.
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with(numtheory);
T:=proc(t) local w, x, y; x:=t; y:=0; while x>0 do x:=trunc(x/10); y:=y+1; od; end:
P:=proc(q) local a, b, c, d, f, g, i, j, k,n;
for n from 1 to q do b:=T(n); k:=0; j:=n; while j mod 2<>1 do k:=k+1; j:=j/2; od;
a:=sigma(2*n+1)+sigma(2*n-1)+sigma(n/2^k)*2^(k+1)-6*n-2;
for i from 1 to b-1 do c:=trunc(n/10^i); d:=n-c*10^i; if c>2 and d>2 then
k:=0; j:=c; while j mod 2<>1 do k:=k+1; j:=j/2; od;
f:=sigma(2*c+1)+sigma(2*c-1)+sigma(c/2^k)*2^(k+1)-6*c-2;
k:=0; j:=d; while j mod 2<>1 do k:=k+1; j:=j/2; od;
g:=sigma(2*d+1)+sigma(2*d-1)+sigma(d/2^k)*2^(k+1)-6*d-2;
if f+g=a-n then print(n); break; fi; fi; od; od; end: P(10^9);
A249065
Numbers n such that n = concatenate(a, b) and sigma(a) + sigma(b) = phi(n).
Original entry on oeis.org
12, 420, 966, 2700, 5700, 5940, 7440, 12540, 71100, 76680, 107880, 118680, 121920, 156420, 160650, 161880, 259080, 284580, 339150, 353430, 395850, 498420, 585200, 600780, 701220, 746130, 752100, 771420, 870870, 1052220, 1249680, 1654620, 1684020, 1696080
Offset: 1
For n = 966 we can consider 966 as 96 U 6; sigma(96) + sigma(6) = 252 + 12 = 264 = phi(966).
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with(numtheory); P:=proc(q) local a,b,i,n; for n from 1 to q do
for i from 1 to ilog10(n) do a:=trunc(n/10^i); b:=n-a*10^i;
if sigma(a)+sigma(b)=phi(n) then print(n); break;
fi; od; od; end: P(10^9);
A258319
Numbers n such that n = concat(a,b) and n = phi(n) + phi(a) + phi(b), with a>0 and b>0, where phi(n) is the Euler totient function of n.
Original entry on oeis.org
25, 177, 1177, 2501, 17105, 21337, 22681, 32581, 217009, 409501, 561601, 577501, 861841, 1025821, 1401841, 1738081, 2836465, 8331361, 10284193, 19971901, 20103001, 27835921, 31949921, 34897501, 100763053, 107314217, 111512701, 121806001, 150658561, 155874001
Offset: 1
25 = concat(2,5); phi(25) + phi(2) + phi(5) = 20 + 1 + 4 = 25;
177 = concat(1,77); phi(177) + phi(1) + phi(77) = 116 + 1 + 60 = 177; etc.
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with(numtheory); P:=proc(q) local a, b, i; global n; for n from 1 to q do
for i from 1 to ilog10(n) do a:=trunc(n/10^i); b:=n-a*10^i;
if a>0 and b>0 then if phi(n)+phi(a)+phi(b)=n
then print(n); break; fi; fi; od; od; end: P(10^9);
Showing 1-4 of 4 results.
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