A253824 -id:A253824 - cp's OEIS frontend

A253825 Numbers m = concat(s,t) such that m = (sigma(s)-s) * (sigma(t)-t), where sigma(x)-x is the sum of the aliquot parts of x.

6396, 20680, 124416, 567816, 1719480, 7593432, 10538040, 36382320, 107277800, 123251968, 166601760, 327844840, 933363000, 1286859804, 2524125184, 3398418000, 4561432920, 4566915540, 4911440776, 7097433536, 16913792670, 20565608940, 21099997800, 27639552000

Offset: 1

Paolo P. Lava, Jan 15 2015

			6396 = concat(63,96) -> sigma(63)-63 = 41, sigma(96)-96 = 156 and 41*156 = 6396.
20680 = concat(20,680) -> sigma(20)-20 = 22, sigma(680)-680 = 940 and 22*940 = 20680.
124416 = concat(12,4416) -> sigma(12)-12 = 16, sigma(4416)-4416 = 7776 and 16*7776 = 124416.
567816 = concat(567,816) -> sigma(567)-567 = 410, sigma(816)-816 = 1416 and 401*1416 = 567816.

Max Alekseyev, Table of n, a(n) for n = 1..42 (first 27 terms from Hiroaki Yamanouchi)

Cf. A001065, A000203, A253824.

with(numtheory): P:=proc(q) local s, t, k, n;
for n from 1 to q do for k from 1 to ilog10(n) do s:=n mod 10^k; t:=trunc(n/10^k); if s*t>0 then if (sigma(s)-s)*(sigma(t)-t)=n
then print(n); break; fi; fi; od; od; end: P(10^6);

fQ[n_] := Block[{idn = IntegerDigits@ n, lng = Floor@ Log10@ n}, MemberQ[ Table[s = FromDigits@ Take[idn, {1, i}]; t = FromDigits@ Take[idn, {i + 1, lng + 1}]; (DivisorSigma[1, s] - s) (DivisorSigma[1, t] - t), {i, lng}], n]]; k = 1; lst = {}; While[k < 100000001, If[fQ@ k, AppendTo[lst, k]; Print@ k]; k++] (* Robert G. Wilson v, Jan 26 2015 *)

isok(n) = {len = #Str(n); for (k=1, len-1, na = n\10^k; nb = n % 10^k; if (nb && (n == (sigma(na)-na)*(sigma(nb)-nb)), return (1)););} \\ Michel Marcus, Jan 15 2015

a(8)-a(9) from Robert G. Wilson v, Jan 26 2015

a(10)-a(24) from Hiroaki Yamanouchi, Sep 26 2015

A258813 Numbers k with the property that it is possible to write the base 2 expansion of k as concat(a_2,b_2), with a_2>0 and b_2>0 such that, converting a_2 and b_2 to base 10 as a and b, we have sigma(a) + sigma (b) = sigma(k) - k.

Original entry on oeis.org

9, 15, 27, 39, 51, 77, 143, 207, 329, 377, 473, 611, 903, 1241, 1243, 1273, 1437, 1591, 2117, 2303, 2975, 4189, 8401, 8657, 11993, 13849, 15611, 16771, 18239, 18599, 19359, 25331, 28877, 37291, 41747, 41807, 61549, 67037, 72601, 82169, 83411, 83711, 87449, 99329

Offset: 1

Paolo P. Lava, Jun 11 2015

			9 in base 2 is 1001. If we take 1001 = concat(10,01) then 10 and 01 converted to base 10 are 2 and 1. Finally sigma(2) + sigma(1) = sigma(9) - 9 = 4.
180953 in base 2 is 101100001011011001. If we take 101100001011011001 = concat(101100001011,011001) then 101100001011 and 011001 converted to base 10 are 2827 and 25. Finally sigma(2827) + sigma(25) = sigma(180953) - 180953 = 3127.

Paolo P. Lava, Table of n, a(n) for n = 1..80

Cf. A253824, A253825, A258843, A258844.

with(numtheory): P:=proc(q) local a,b,c,j,k,n;
for n from 1 to q do c:=convert(n,binary,decimal);
j:=0; for k from 1 to ilog10(c) do
a:=convert(trunc(c/10^k),decimal,binary);
b:=convert((c mod 10^k),decimal,binary);
if a*b>0 then if sigma(a)+sigma(b)=sigma(n)-n then print(n);
break; fi; fi; od; od; end: P(10^9);

isok(n) = {b = binary(n); if (#b > 1, for (k=1, #b-1, vba = Vecrev(vector(k, i, b[i])); vbb = Vecrev(vector(#b-k, i, b[k+i])); da = sum(i=1, #vba, vba[i]*2^(i-1)); db = sum(i=1, #vbb, vbb[i]*2^(i-1)); if (da && db && (sigma(da)+sigma(db) == sigma(n)-n), return(1));););} \\ Michel Marcus, Jun 13 2015

A159000 Numbers m such that m = s|t = phi(s)*sigma(t) for some numbers s and t, where "|" denotes concatenation.

Original entry on oeis.org

3360, 19440, 35712, 55800, 120960, 395808, 451584, 548640, 628992, 695520, 763344, 3008768, 3749760, 5602320, 17557344, 46902240, 55031040, 119627904, 162496048, 193933440, 243855360, 249793920, 374473800, 377677440, 548402400

Offset: 1

Farideh Firoozbakht, Jun 04 2009

A159001(n) gives the first part s of a(n).

			3008768 = phi(3008)*sigma(768) so 3008768 is in the sequence.

Max Alekseyev, Table of n, a(n) for n = 1..51 (first 33 terms from Giovanni Resta)

Cf. A159001, A253824, A253825, A260144.

ntnQ[n_]:=MemberQ[(EulerPhi[#[[1]]]DivisorSigma[1,#[[2]]])==n&/@ Table[FromDigits/@TakeDrop[IntegerDigits[n],i],{i,IntegerLength[ n]-1}],True]; Select[Range[55*10^7], ntnQ] (* The program uses the TakeDrop function from Mathematica version 10 *) (* The program takes a long time to run *) (* Harvey P. Dale, Jan 01 2016 *)

isA159000(n)={my(m);for(i=1,#Str(n)-1,m=n%10^i;if(m,m=divrem(n,sigma(m));if(m[2]==0&eulerphi(n\10^i)==m[1],return(i))));0} /* Charles R Greathouse IV, Apr 28 2010 */

Corrected (a(3), a(4), and a(11) missing), extended past a(12), and edited by Charles R Greathouse IV, Apr 28 2010

A257897 Numbers n such that n = concat(a,b) and n | a^b + b^a , with a>0 and b>0.

Original entry on oeis.org

63, 103, 128, 147, 155, 212, 272, 292, 351, 452, 486, 497, 525, 527, 537, 584, 607, 624, 648, 729, 979, 999, 1024, 1296, 1323, 1359, 1533, 1541, 1575, 1809, 1872, 2048, 2050, 2107, 2187, 2448, 2512, 2537, 2564, 2763, 2793, 2886, 3072, 3357, 3927, 4096, 4263, 4284

Offset: 1

Paolo P. Lava, May 12 2015

We can have different solutions for the same number. E.g.: 2048 divides both (20^48 + 48^20) and (204^8 + 8^204). The same occurs for 4096, 4263, 16807, 32768, 96957, 156672, 186624, 252081, 262144, 270729, 352947, 390624 … The first number with 3 different concatenations is 186624 that divides (18^6624 + 6624^18), (186^624 + 624^186) and (1866^24 + 24^1866).

			6^3 + 3^6 = 945 and 945 / 63 = 15;
10^3 + 3^10 = 60049 and 60049 / 103 = 583;
12^8 + 8^12 = 69149458432 and 69149458432 / 128 = 540230144; etc.

Paolo P. Lava, Table of n, a(n) for n = 1..400

Cf. A253824, A253825, A255725, A255726.

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 a>0 and b>0 then if type((a^b+b^a)/n,integer)
then print(n); break; fi; fi; od; od; end: P(10^9);

tst[n_]:=Catch@Block[{a,b}, Do[a=Floor[n/10^k]; b=Mod[n,10^k]; If[Mod[ PowerMod[a, b, n] + PowerMod[b, a, n], n]==0, Throw@True], {k, IntegerLength[n]-1}]; False]; Select[Range@1000, tst] (* Giovanni Resta, May 12 2015 *)

A260144 Numbers k such that there exist two numbers a and b where k = a.b = sigma(a)*phi(b) ("." means concatenation).

Original entry on oeis.org

1568, 14049280, 140492800, 368089904, 506300928, 1404928000, 14049280000, 124856008704, 140492800000, 1404928000000

Offset: 1

Paolo P. Lava and Giovanni Resta, Jul 17 2015

The sequence is infinite because, for m>0, 14049280*10^m = sigma(1404) * phi(9280*10^m).

			1568 is in the sequence because 1568 = sigma(156) * phi(8).

Cf. A000010, A000203, A159000, A253824.

a(7)-a(10) from Max Alekseyev, May 26 2025

A251605 Numbers k = concat(s,t) such that k = antisigma(s) + antisigma(t), where antisigma(x) = A024816(x).

Original entry on oeis.org

226, 1046, 3881, 20206, 1970256, 57619059, 62109258

Offset: 1

Paolo P. Lava, Feb 02 2015

			226 = concat(22,6); 22*23/2 - sigma(22) = 253 - 36 = 217; 6*7/2 - sigma(6) = 21 - 12 = 9. Finally 217 + 9 = 226.

Cf. A000203, A024816, A253824, A253825.

with(numtheory): P:=proc(q) local s, t, k; global n; for n from 1 to q do for k from 1 to ilog10(n) do s:=n mod 10^k; t:=trunc(n/10^k); if s>0 then if s*(s+1)/2-sigma(s)+t*(t+1)/2-sigma(t)=n
then print(n); break; fi; fi; od; od; end: P(10^7);

Incorrect term 240 removed by and a(6)-a(7) from Jinyuan Wang, Feb 16 2021

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

Paolo P. Lava, May 26 2015

			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.

Giovanni Resta, Table of n, a(n) for n = 1..50

Cf. A000010, A239562, A249766, A251605, A251860, A253824, A253825, A255725, A255726, A257897.

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);

a(18) inserted by Giovanni Resta, May 27 2015

A259670 Numbers n with the property that it is possible to write the base 2 expansion of n as concat(a_2,b_2), with a_2>0 and b_2>0 such that, converting a_2 and b_2 to base 10 as a and b, we have antisigma(a) + antisigma(b) = n.

Original entry on oeis.org

50, 77, 179, 346, 347, 550, 1758, 1909, 9205, 27884, 30660, 37354, 52019, 88052, 107974, 131590, 164413, 232447, 295682, 326133, 328491, 1494561, 1541005, 1541851

Offset: 1

Paolo P. Lava, Jul 03 2015

			50 in base 2 is 110010. If we take 110010 = concat(1100,10) then 1100 and 10 converted to base 10 are 12 and 2. Finally 12*13/2 - sigma(12) + 2*3/2 - sigma(2) = 78 - 28 + 3 - 3 = 50.
179 in base 2 is 1001101. If we take 1001101 = concat(11100,1) then 11100 and 1 converted to base 10 are 5 and 19. Finally 5*6/2 - sigma(5) + 19*20/2 - sigma(19) = 15 - 6 + 190 - 20 = 179.

Cf. A024816, A253824, A253825, A258813, A258843, A258844.

with(numtheory): P:=proc(q) local a,b,c,j,k,n;
for n from 1 to q do c:=convert(n,binary,decimal);
j:=0; for k from 1 to ilog10(c) do
a:=convert(trunc(c/10^k),decimal,binary);
b:=convert((c mod 10^k),decimal,binary);
if a*b>0 then if a*(a+1)/2-sigma(a)+b*(b+1)/2-sigma(b)=n then print(n);
break; fi; fi; od; od; end: P(10^9);

f[n_] := Block[{d = IntegerDigits[n, 2], len = IntegerLength[n, 2], k}, ReplaceAll[Reap[Do[k = {FromDigits[Take[d, i], 2], FromDigits[Take[d, -(len - i)], 2]}; If[! MemberQ[k, 0], Sow@ k], {i, 1, len - 1}]], {} -> {1}][[-1, 1]]]; Select[Range@ 100000, MemberQ[Total /@ (# (# + 1)/2 - DivisorSigma[1, #] &) /@ f@ #, #] &] (* Michael De Vlieger, Jul 03 2015 *)

A260097 Numbers n = concat(s,t) such that n = sigma(s*t), where sigma(x) is the sum of the divisors of x.

Original entry on oeis.org

124, 2352, 2604, 6804, 15240, 63180, 225302, 632400, 1531152, 2537040, 4592588, 7160400, 7603680, 26100144, 26378352, 31492032, 33747840, 49447728, 88385040, 104941200, 162496048, 175600040, 197499456, 403242624, 483741216, 797091840, 2077442640, 2942021520, 4045874976, 4828299840

Offset: 1

Paolo P. Lava, Jul 16 2015

			124 = concat(12,4) and sigma(12*4) = sigma(48) = 124.
2352 = concat(23,52) and sigma(23*52) = sigma(1196) = 2352.

Cf. A000203, A253824, A253825.

with(numtheory); P:=proc(q) local a,b,i,n,p; 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*b)=n then print(n); break; fi; od; od; end: P(10^9);

isok(n) = {len = #Str(n); for (k=1, len-1, na = n\10^k; nb = n % 10^k; if (nb && (n == sigma(na*nb)), return (1)); ); } \\ Michel Marcus, Jul 17 2015

from sympy import divisor_sigma
A260097_list= []
for n in range(11,10**6):
    s = str(n)
    for l in range(1, len(s)):
        m = int(s[:l])*int(s[l:])
        if m > 0 and n == divisor_sigma(m):
            A260097_list.append(n)
            break # Chai Wah Wu, Jul 18 2015

a(14)-a(15) from Chai Wah Wu, Jul 18 2015
a(16)-a(28) from Lars Blomberg, Jul 21 2015
a(29)-a(30) from Giovanni Resta, Jul 24 2015

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A253825 Numbers m = concat(s,t) such that m = (sigma(s)-s) * (sigma(t)-t), where sigma(x)-x is the sum of the aliquot parts of x.

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A258813 Numbers k with the property that it is possible to write the base 2 expansion of k as concat(a_2,b_2), with a_2>0 and b_2>0 such that, converting a_2 and b_2 to base 10 as a and b, we have sigma(a) + sigma (b) = sigma(k) - k.

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A159000 Numbers m such that m = s|t = phi(s)*sigma(t) for some numbers s and t, where "|" denotes concatenation.

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A257897 Numbers n such that n = concat(a,b) and n | a^b + b^a , with a>0 and b>0.

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A260144 Numbers k such that there exist two numbers a and b where k = a.b = sigma(a)*phi(b) ("." means concatenation).

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A251605 Numbers k = concat(s,t) such that k = antisigma(s) + antisigma(t), where antisigma(x) = A024816(x).

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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.

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A259670 Numbers n with the property that it is possible to write the base 2 expansion of n as concat(a_2,b_2), with a_2>0 and b_2>0 such that, converting a_2 and b_2 to base 10 as a and b, we have antisigma(a) + antisigma(b) = n.

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