cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A306212 Numbers that are the sum of squares of three distinct positive integers in arithmetic progression.

Original entry on oeis.org

14, 29, 35, 50, 56, 66, 77, 83, 93, 107, 110, 116, 126, 140, 149, 155, 158, 165, 179, 194, 197, 200, 210, 219, 224, 242, 245, 251, 261, 264, 275, 290, 293, 302, 308, 315, 318, 332, 341, 350, 365, 371, 372, 381, 395, 398, 413, 428, 434, 435, 440, 450, 461, 462, 464, 482
Offset: 1

Views

Author

Antonio Roldán, Jan 29 2019

Keywords

Examples

			35 = 1^2 + 3^2 + 5^2, with 3 - 1 = 5 - 3 = 2;
371 = 1^2 + 9^2 + 17^2, with 9 - 1 = 17 - 9 = 8. Also 371 = 9^2 + 11^2 + 13^2, with 11 - 9 = 13 - 11 = 2.

Links

Crossrefs

Cf. A000290, A000378, A000408, A085317, A120328, A292313, A306213, A306214.

Programs

  • Maple
    N:= 1000: # for terms <= N
S:= {seq(seq(3*a^2+2*b^2, b=1..min(a-1, floor(sqrt((N-3*a^2)/2)))),a=1..floor(sqrt(N/3)))}:
sort(convert(S,list)); # Robert Israel, Jun 08 2020
  • PARI
    for(n=3, 600, k=sqrt(n/3); a=2; v=0; while(a<=k&&v==0, b=(n-3*a^2)/2; if(b==truncate(b)&&issquare(b), d=sqrt(b); if(d>=1&&d<=a-1, v=1; print1(n,", "))); a+=1))
  • PARI
    w=List(); for(n=3, 600, k=sqrt(n/3); for(a=2, k, for(c=1, a-1, v=(a-c)^2+a^2+(a+c)^2; if(v==n, listput(w,n))))); print(vecsort(Vec(w),,8))

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

Views

Author

Antonio Roldán, Jan 29 2019

Keywords

Comments

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

Examples

			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.

Links

Crossrefs

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

Programs

  • Maple
    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
  • PARI
    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))

A359030 Positive numbers that are the sum of cubes of three distinct integers in arithmetic progression.

Original entry on oeis.org

9, 27, 36, 57, 72, 99, 132, 153, 216, 219, 243, 288, 297, 324, 369, 387, 405, 408, 456, 489, 495, 531, 576, 603, 612, 645, 684, 729, 792, 855, 867, 963, 972, 996, 1017, 1056, 1071, 1125, 1179, 1197, 1224, 1233, 1353, 1368, 1407, 1455, 1476, 1539, 1548, 1584, 1701, 1728, 1737, 1752, 1845, 1881
Offset: 1

Views

Author

Robert Israel, Dec 15 2022

Keywords

Comments

Numbers that can be represented in at least one way as 3*a*(a^2 + 2*b^2) for positive integers a and b.
In contrast to A306213, the arithmetic progression need not consist only of positive numbers.

Examples

			a(4) = 57 is a term because 57 = (-2)^3 + 1^3 + 4^3 where (-2, 1, 3) are in arithmetic progression.

Links

Crossrefs

Cf. A306213, A359078.

Programs

  • Maple
    N:= 2000: # for terms <= N
L:= NULL:
for a from 1 while 3*a^3 <= N do
  for b from 1 do
    x:= 3*a*(a^2 + 2*b^2);
    if x > N then break fi;
    L:= L,x
od od:
sort(convert({L},list));

A359055 Numbers that can be represented in more than one way as the sum of cubes of three distinct positive numbers in arithmetic progression.

Original entry on oeis.org

5643, 12384, 31977, 45144, 99072, 123849, 152361, 153792, 255816, 259776, 269739, 274968, 334368, 361152, 477576, 500445, 705375, 792576, 863379, 912339, 928017, 950931, 990792, 1090584, 1218888, 1230336, 1548000, 1629144, 1700424, 1737252, 1799523, 1813512, 1935549, 1941192, 2046528, 2078208
Offset: 1

Views

Author

Robert Israel, Dec 14 2022

Keywords

Comments

Numbers k such that there are at least two pairs of positive numbers (a,d) such that k = a^3 + (a+d)^3 + (a+2d)^3.
The first term that has three such representations is 255816 = 8^3 + 34^3 + 60^3 = 18^3 + 38^3 + 58^3 = 43^3 + 44^3 + 45^3.
346380489216 has four such representations: 1188^3 + 3888^3 + 6588^3, 1728^3 + 4104^3 + 6480^3, 4248^3 + 4824^3 + 5400^3 and 4665^3 + 4864^3 + 5063^3. It might not be the first.

Examples

			a(1) = 5643 is a term because 5643 = 1^3 + (1+8)^3 + (1+2*8)^3 = 6^3 + (6+5)^3 + (6+2*5)^3.

Links

Crossrefs

Cf. A306213.

Programs

  • Maple
    N:= 10^7: # to get terms <= N
S:= {}: S2:= {}:
for a from 1 while a^3 + (a+1)^3 + (a+2)^3 <= N do
  for d from 1 do
    x:= a^3 + (a+d)^3 + (a+2*d)^3;
    if x > N then break fi;
    if member(x,S) then S2:= S2 union {x} fi;
    S:= S union {x}
od od:
sort(convert(S,list));
Showing 1-4 of 4 results.

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