Antonio Roldán - cp's OEIS frontend

A352721 Perfect cubes whose decimal digits appear in nonincreasing order.

0, 1, 8, 64, 1000, 8000, 64000, 1000000, 8000000, 64000000, 1000000000, 8000000000, 64000000000, 1000000000000, 8000000000000, 64000000000000, 1000000000000000, 8000000000000000, 64000000000000000, 1000000000000000000, 8000000000000000000, 64000000000000000000

Offset: 1

Antonio Roldán, Mar 30 2022

			64 is in the sequence because it is a perfect cube (64 = 4^3) whose digits appear in nonincreasing order.

Cf. A004647, A028820, A028864, A234848, A273045.

Intersection of A000578 and A009996.

Select[Range[0, 4*10^6]^3, Max@ Differences[IntegerDigits[#]] <= 0 &] (* Amiram Eldar, Mar 30 2022 *)

ok(n) = digits(n) == vecsort(digits(n),,4) && ispower(n,3)

a(n) = A004647(n-1)^3.

A337784 Smaller of two consecutive oblong numbers which are anagrams of each other.

Original entry on oeis.org

23256, 530712, 809100, 11692980, 17812620, 20245500, 22834062, 23527350, 29154600, 83768256, 182236500, 189847062, 506227500, 600127506, 992218500, 1363566402, 1640209500, 2175895962, 2422657620, 2477899062, 2520190602, 3041687952, 3764129256, 4760103042

Offset: 1

Antonio Roldán, Sep 21 2020

All terms are multiples of 9.

The indices of these oblong numbers are 152, 728, 899, 3419, 4220, 4499, 4778, 4850, 5399, 9152, 13499, 13778, 22499, 24497, 31499, 36926, 40499, 46646, 49220, 49778, 50201, 55151, 61352, 68993.

			530712 is in the sequence because it is an oblong number, 530712 = 728 * 729, and the next oblong number, 532170 = 729 * 730, is an anagram of 530712.

Subsequence of A008591.

Cf. A002378, A069567, A227692, A228133, A228135.

s = {}; o1 = 1; d1 = Sort @ IntegerDigits[o1]; Do[o2 = n*(n + 1); d2 = Sort @ IntegerDigits[o2]; If[d2 == d1, AppendTo[s, o1]]; o1 = o2; d1 = d2, {n, 2, 70000}]; s (* Amiram Eldar, Sep 21 2020 *)

ok(k) = {my(b, m=0); if(issquare(4*k + 1), b=truncate(sqrt(4*k + 1) - 1)/2; if(vecsort(digits(k)) == vecsort(digits((b + 1)*(b + 2))), m = 1)); m}

A333821 Numbers k that can be represented in the form k = p^3 - q^3 - r^3, where p, q, r are positive integers satisfying p = q + r.

Original entry on oeis.org

6, 18, 36, 48, 60, 90, 126, 144, 162, 168, 210, 216, 252, 270, 288, 330, 360, 378, 384, 396, 468, 480, 486, 540, 546, 594, 630, 720, 750, 792, 816, 858, 918, 924, 972, 990, 1008, 1026, 1140, 1152, 1170, 1260, 1296, 1344, 1386, 1404, 1518, 1530, 1560, 1620, 1638, 1656, 1680, 1728, 1800

Offset: 1

Antonio Roldán, Apr 06 2020

nonn

An alternative representation of k is k = 3*q*r*(q+r), with q, r positive integers, then k is a multiple of 6.

			60 is in the sequence because 60 = 5^3 - 4^3 - 1^3, with 5 = 4 + 1.

Cf. A000578, A003072, A024975, A078129, A255018, A275154.

ok(n) = {my(i=1, a=0, m=0, j); if(n%6==0, while(a<=n&&m==0, j=1; while(j

a(n) = 6 * A121741(n).

A329590 Odd numbers k that cannot be expressed as k = p+q+r, with p prime and (q, r) a pair of twin primes.

Original entry on oeis.org

1, 3, 5, 7, 9, 33, 57, 93, 99, 129, 141, 153, 177, 183, 195, 213, 225, 243, 255, 261, 267, 273, 297, 309, 327, 333, 351, 369, 393, 411, 423, 435, 453, 477, 489, 501, 513, 519, 525, 537, 561, 573, 591, 597, 603, 633, 645, 657, 663, 675, 687, 693, 705, 711, 723

Offset: 1

Antonio Roldán, Feb 13 2020

nonn

			33 can be expressed as the sum of three primes in 9 different ways:
33 = 11 + 11 + 11 = 13 + 13 + 7 = 17 + 11 + 5 = 17 + 13 + 3 = 19 + 7 + 7 = 19 + 11 + 3 = 23 + 5 + 5 = 23 + 7 + 3 = 29 + 2 + 2;
there is no pair of twin primes in the addends, so 33 is a term.

Cf. A001359, A034961, A054735, A068307, A124868.

for(n = 0, 500, m = 2*n+1; v = 0; forprime(i = 3, m-8, j = (m-i)/2; if(isprime(j-1) && isprime(j+1), v = 1)); if(v == 0, print1(m,", ")))

isok(k) = {if (! (k % 2), return (0)); forprime(p=3, k, if (isprime((k-p)\2-1) && isprime((k-p)\2+1), return(0));); return (1);} \\ Michel Marcus, Feb 16 2020

A332008 Numbers k such that phi(k) and phi(k+1) are perfect powers (A001597).

Original entry on oeis.org

1, 15, 16, 63, 101, 125, 255, 256, 272, 504, 512, 513, 629, 679, 1358, 1359, 1728, 1970, 2047, 2222, 2509, 2834, 3458, 3705, 4094, 4095, 4400, 4577, 4616, 4913, 5403, 6817, 7295, 7956, 8729, 11667, 11672, 16132, 16384, 16523, 17507, 23085, 24198, 24564, 24624, 25220, 25601, 27216, 27384, 28564, 29444

Offset: 1

Antonio Roldán, Feb 04 2020

nonn

			phi(101) = 10^2, and phi(102) = 2^5.
phi(3458) = 6^4, and phi(3459) = 48^2.

Cf. A000010, A001597, A166955, A281016.

[1] cat [k:k in [3..30000]|IsPower(EulerPhi(k))  and IsPower(EulerPhi(k+1))]; // Marius A. Burtea, Feb 05 2020

perfectPowerQ[1] = True; perfectPowerQ[n_] := GCD @@ FactorInteger[n][[;; , 2]] > 1; Select[Range[30000], And @@ perfectPowerQ /@ EulerPhi[# + {0, 1}] &] (* Amiram Eldar, Feb 04 2020 *)

v=[1]; for(i = 2, 30000, if(ispower(eulerphi(i)), if(ispower(eulerphi(i+1)), v = concat(v, i)))); v

A309884 Numbers k such that A003132(k^3) = A003132(k), where A003132(n) is the sum of the squares of the digits of n.

Original entry on oeis.org

0, 1, 10, 74, 100, 740, 1000, 3488, 7400, 10000, 23658, 30868, 34880, 47508, 48517, 52187, 58947, 59468, 67685, 68058, 74000, 76814, 78368, 78845, 84878, 100000, 108478, 145877, 149217, 163871, 179685, 186884, 188647, 218977, 219878, 236580, 238758, 248967, 278638, 292597, 308680

Offset: 1

Antonio Roldán, Aug 21 2019

If k is in the sequence, then so are k*10^r, with r >= 1.

			74^3 = 405224, A003132(74) = 7^2 + 4^2 = 65, A003132(405224) = 4^2 + 0^2 + 5^2 + 2^2 + 2^2 + 4^2 = 65.

Cf. A003132, A058369, A070276, A174460, A176670, A178213, A307735, A309883.

[0] cat [k:k in [1..310000]| &+[c^2: c in Intseq(k)] eq &+[c^2: c in Intseq(k^3)]]; // Marius A. Burtea, Aug 26 2019

digSum[n_] := Total[IntegerDigits[n]^2]; Select[Range[0, 310000], digSum[#] == digSum[#^3] &] (* Amiram Eldar, Aug 22 2019 *)

for(i = 0, 400000, if(norml2(digits(i^3)) == norml2(digits(i)), print1(i, ", ")))

A309883 Numbers k such that A003132(k^2) = A003132(k), where A003132(n) is the sum of the squares of the digits of n.

Original entry on oeis.org

0, 1, 10, 35, 100, 152, 350, 377, 452, 539, 709, 1000, 1299, 1398, 1439, 1519, 1520, 1569, 1591, 1679, 1965, 2599, 2838, 3332, 3500, 3598, 3770, 4520, 4586, 4754, 4854, 5390, 5501, 5835, 5857, 6388, 6595, 6735, 6861, 6951, 7090, 7349, 7887, 8395, 9795, 10000, 10056, 10159, 10389, 11055, 11091, 12990, 12999

Offset: 1

Antonio Roldán, Aug 21 2019

If k is in the sequence, then so are k*10^r, r >= 1.

			377^2 = 142129, A003132(377) = 3^2 + 7^2 + 7^2 = 107, A003132(142129) = 1^2 + 4^2 + 2^2 + 1^2 + 2^2 + 9^2 = 107.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A003132, A058369, A070276, A174460, A176670, A178213, A307735, A309884.

[0] cat [k:k in [1..13000]| &+[c^2: c in Intseq(k)] eq &+[c^2: c in Intseq(k^2)]]; // Marius A. Burtea, Aug 24 2019

filter:= proc(n) local t;
  add(t^2, t = convert(n,base,10)) = add(t^2, t = convert(n^2,base,10))
end proc:
select(filter, [$0..20000]); # Robert Israel, Apr 30 2023

digSum[n_] := Total[IntegerDigits[n]^2]; Select[Range[0, 13000], digSum[#] == digSum[#^2] &] (* Amiram Eldar, Aug 22 2019 *)

for(i = 0, 30000, if(norml2(digits(i^2)) == norml2(digits(i)), print1(i, ", ")))

def A003132(n):
    s = 0
    while n > 0:
        s, n = s+(n%10)**2, n//10
    return s
n, a = 0, 0
while n < 50:
    if A003132(a) == A003132(a*a):
        n = n+1
        print(n,a)
    a = a+1 # A.H.M. Smeets, Aug 23 2019

A308664 Numbers k such that tau(k) and phi(k) are the legs of a Pythagorean triple.

Original entry on oeis.org

20, 36, 60, 100, 300

Offset: 1

Antonio Roldán, Jul 14 2019

The sequence is finite since for all large enough n, we have tau(n) < n^(1/4) and phi(n) > n^(3/4) while, if x < y are the legs of a Pythagorean triangle, we always have y < x^2/2. - Giovanni Resta, Jul 27 2019

From Resta's inequality it can be deduced that phi(n) <= 2304. Then it's easy to see that the sequence is full. - Max Alekseyev, May 30 2024

			60 is in this sequence because tau(60) = 12 and phi(60) = 16, legs of the Pythagorean triple {12, 16, 20} (12^2 + 16^2 = 20^2).

Cf. A000005, A000010, A020488, A062516, A063469, A063470, A112954.

Select[Range[300], IntegerQ@Sqrt[DivisorSigma[0, #]^2 + EulerPhi[#]^2] &] (* Amiram Eldar, Jul 26 2019 *)

for(i = 1,  2000, a = eulerphi(i); b = numdiv(i); if(issquare(a^2 + b^2), print1(i,", ")))

"full" keyword added by Max Alekseyev, May 30 2024

A307735 Integers k such that if m = k + A003132(k) then k = m - A003132(m).

Original entry on oeis.org

0, 9, 205, 212, 217, 366, 457, 663, 1314, 1315, 1348, 1672, 1742, 1792, 1797, 2005, 2012, 2017, 2129, 2201, 2208, 2213, 2216, 2305, 2404, 2405, 2465, 2564, 2565, 2671, 2741, 2748, 2789, 2829, 3114, 3115, 3205, 3303, 3306, 3394, 3436, 3475, 3696, 3819, 4204, 4205, 4245, 4347, 4475, 4542, 4629, 4647, 4688

Offset: 1

Antonio Roldán, Apr 25 2019

A003132(n) is the sum of the squares of the digits of n.

			205 is in the sequence because 205 + 2^2 + 0^2 + 5^2 = 234 and 234 - 2^2 - 3^2 - 4^2 = 205.

Cf. A003132, A108662, A175396, A209303, A223035, A225049, A258881.

sod2[n_] := Total @ (IntegerDigits[n]^2); aQ[n_] := sod2[n + (s=sod2[n])] == s; Select[Range[0, 4700], aQ] (* Amiram Eldar, Jul 03 2019 *)

for(i = 0 , 5000 , a = i + norml2(digits(i)) ; b = a - norml2(digits(a)) ; if(i == b , print1(i , ", ")))

A306214 Numbers that are the sum of fourth powers of three distinct positive integers in arithmetic progression.

Original entry on oeis.org

98, 353, 707, 962, 1568, 2177, 2658, 3107, 4322, 4737, 5648, 7187, 7793, 7938, 9587, 11312, 12657, 13058, 15392, 15938, 17123, 19362, 20657, 23153, 23603, 25088, 28593, 30963, 31202, 32738, 34832, 35747, 40962, 42528, 45233, 45377, 49712, 49763, 54722, 57153, 57267, 61250, 63938, 67667, 69152

Offset: 1

Antonio Roldán, Jan 29 2019

nonn

The remainder of a(n) divided by 16 is less than 5. - Jinyuan Wang, Feb 03 2019

			353 = 2^4 + 3^4 + 4^4, with 3 - 2 = 4 - 3 = 1;
7187 = 1^4 + 5^4 + 9^4, with 5 - 1 = 9 - 5 = 4.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000583, A126657, A133531, A190176, A306212, A306213.

N:= 10^5: # for all terms <= N
Res:= NULL:
for a from 1 to floor((N/3)^(1/4)) do
  for d from 1 do
    v:= a^4 + (a+d)^4 + (a+2*d)^4;
    if v > N then break fi;
    Res:= Res, v
  od
od:
sort(convert({Res},list)); # Robert Israel, Feb 17 2019

for(n=1, 70000, k=(n/3)^(1/4); a=2; v=0; while(a<=k&&v==0, d=sqrt(sqrt(2*n+30*a^4)/2-3*a^2); if(d==truncate(d)&&d>=1&&d<=a-1, v=1; print1(n,", ")); a+=1))

cp's OEIS Frontend

User: Antonio Roldán

A352721 Perfect cubes whose decimal digits appear in nonincreasing order.

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A337784 Smaller of two consecutive oblong numbers which are anagrams of each other.

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A333821 Numbers k that can be represented in the form k = p^3 - q^3 - r^3, where p, q, r are positive integers satisfying p = q + r.

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A329590 Odd numbers k that cannot be expressed as k = p+q+r, with p prime and (q, r) a pair of twin primes.

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A332008 Numbers k such that phi(k) and phi(k+1) are perfect powers (A001597).

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A309884 Numbers k such that A003132(k^3) = A003132(k), where A003132(n) is the sum of the squares of the digits of n.

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A309883 Numbers k such that A003132(k^2) = A003132(k), where A003132(n) is the sum of the squares of the digits of n.

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A308664 Numbers k such that tau(k) and phi(k) are the legs of a Pythagorean triple.

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