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 1-8 of 8 results.

A348488 Positive numbers whose square starts and ends with exactly one 4.

Original entry on oeis.org

2, 22, 68, 202, 208, 218, 222, 642, 648, 652, 658, 672, 678, 682, 692, 698, 702, 2002, 2008, 2018, 2022, 2028, 2032, 2042, 2048, 2052, 2058, 2068, 2072, 2078, 2082, 2092, 2122, 2128, 2132, 2142, 2148, 2152, 2158, 2168, 2172, 2178, 2182, 2192, 2198, 2202, 2208, 2218, 2222, 2228
Offset: 1

Views

Author

Bernard Schott, Oct 24 2021

Keywords

Comments

When a square ends with 4 (A273375), this square may end with precisely one 4, two 4's or three 4's (A328886).
This sequence is infinite as each 2*(10^m + 1), m >= 1 or 2*(10^m + 4), m >= 2 is a term.
Numbers 2, 22, 222, ..., 2*(10^k - 1) / 9, (k >= 1), as well as numbers 2228, 22228, ..., 2*(10^k + 52) / 9, (k >= 4) are terms and have no digits 0. - Marius A. Burtea, Oct 24 2021

Examples

			22 is a term since 22^2 = 484.
638 is not a term since 638^2 = 407044.
668 is not a term since 668^2 = 446224.

Crossrefs

Cf. A045858, A273375 (squares ending with 4), A017317, A328886 (squares ending with three 4).
Cf. A002276 \ {0} (a subsequence).
Subsequence of A305719.
Similar to: A348487 (k=1), this sequence (k=4), A348489 (k=5), A348490 (k=6).

Programs

  • Magma
    [2] cat [n:n in [4..2300]|Intseq(n*n)[1] eq 4 and Intseq(n*n)[#Intseq(n*n)] eq 4 and Intseq(n*n)[-1+#Intseq(n*n)] ne 4 and Intseq(n*n)[2] ne 4]; // Marius A. Burtea, Oct 24 2021
  • Mathematica
    Join[{2}, Select[Range[10, 2000], (d = IntegerDigits[#^2])[[1]] == d[[-1]] == 4 && d[[-2]] != 4 && d[[2]] != 4 &]] (* Amiram Eldar, Oct 24 2021 *)
  • PARI
    isok(k) = my(d=digits(sqr(k))); (d[1]==4) && (d[#d]==4) && if (#d>2, (d[2]!=4) && (d[#d-1]!=4), 1); \\ Michel Marcus, Oct 24 2021
  • Python
    from itertools import count, takewhile
def ok(n):
  s = str(n*n); return len(s.rstrip("4")) == len(s.lstrip("4")) == len(s)-1
def aupto(N):
  r = takewhile(lambda x: x<=N, (10*i+d for i in count(0) for d in [2, 8]))
  return [k for k in r if ok(k)]
print(aupto(2228)) # Michael S. Branicky, Oct 24 2021

A116499 Numbers k such that the base-10 representation of k^2 has the structure dmmd, where d is a single digit and m is a string of digits.

Original entry on oeis.org

63565788, 777035126, 785900449264170834, 824735400720481514, 41969556121016094021, 7841042904081599033746826, 775474811976872476373043200886, 403578381989727694739321538082071713672671, 820657465670154825723396137907157085701046
Offset: 1

Views

Author

Giovanni Resta, Feb 18 2006

Keywords

Examples

			785900449264170834^2 = 6[17639516153625555][17639516153625555]6.

Links

Crossrefs

Cf. A116500, A116501, A305719.

A348487 Positive numbers whose square starts and ends with exactly one 1.

Original entry on oeis.org

1, 11, 39, 41, 101, 111, 119, 121, 129, 131, 139, 141, 319, 321, 329, 331, 349, 351, 359, 361, 369, 371, 379, 381, 389, 391, 399, 401, 409, 411, 419, 421, 429, 431, 439, 441, 1001, 1009, 1011, 1019, 1021, 1029, 1031, 1039, 1041, 1099, 1101, 1109, 1111, 1119, 1121, 1129, 1131, 1139
Offset: 1

Views

Author

Bernard Schott, Oct 21 2021

Keywords

Comments

When a square ends with 1, this square ends with exactly one 1.
Sequences A000533 and A253213 show that there are an infinity of terms. The square of their terms, for n >= 3, starts and ends with exactly one 1. Also, the numbers 119, 1119, 11119, ..., ((10^k + 71) / 9)^2, (k >= 3) are terms. The squares ((10^k + 71) / 9)^2, have the last digit 1 and because 12*10^(2*k - 3) < ((10^k + 71) / 9)^2 <13*10^(2*k - 3), for k >= 3, the squares ((10^k + 71) / 9)^2, k >= 4, start with 12. - Marius A. Burtea, Oct 21 2021

Examples

			39 is a term since 39^2 = 1521.
109 is not a term since 109^2 = 11881.
119 is a term since 119^2 = 14161.

Crossrefs

Cf. A045855, A090771, A253213, A273372 (squares ending with 1), A017281, A017377.
Cf. A000533, A253213 for n >= 2 (subsequences).
Subsequence of A305719.

Programs

  • Magma
    [1] cat [n:n in [2..1200]|Intseq(n*n)[1] eq 1 and Intseq(n*n)[#Intseq(n*n)] eq 1 and Intseq(n*n)[-1+#Intseq(n*n)] ne 1]; // Marius A. Burtea, Oct 21 2021
  • Mathematica
    Join[{1}, Select[Range[11, 1200], (d = IntegerDigits[#^2])[[1]] == d[[-1]] == 1 && d[[2]] != 1 &]] (* Amiram Eldar, Oct 21 2021 *)
  • PARI
    isok(k) = my(d=digits(sqr(k))); (d[1]==1) && (d[#d]==1) && if (#d>2, (d[2]!=1) && (d[#d-1]!=1), 1); \\ Michel Marcus, Oct 21 2021
  • Python
    from itertools import count, takewhile
def ok(n):
  s = str(n*n); return len(s.rstrip("1")) == len(s.lstrip("1")) == len(s)-1
def aupto(N):
  r = takewhile(lambda x: x<=N, (10*i+d for i in count(0) for d in [1, 9]))
  return [k for k in r if ok(k)]
print(aupto(1140)) # Michael S. Branicky, Oct 21 2021

A348489 Positive numbers whose square starts and ends with exactly one 5.

Original entry on oeis.org

75, 225, 715, 725, 735, 755, 765, 2245, 2255, 2265, 2275, 2285, 2295, 2305, 2315, 2325, 2335, 2345, 2375, 2385, 2395, 2405, 2415, 2425, 2435, 2445, 7075, 7085, 7095, 7105, 7115, 7125, 7135, 7145, 7155, 7165, 7175, 7185, 7195, 7205, 7215, 7225, 7235, 7245
Offset: 1

Views

Author

Bernard Schott, Oct 25 2021

Keywords

Comments

When a square ends with 5, it ends with 25.
From Marius A. Burtea, Oct 25 2021: (Start)
Numbers 75, 765, 7665, 76665, ..., (23*10^k -5) / 3, k >= 1, are terms and have no digits 0, because their squares are 5625, 585225, 58752225, 5877522225, 587775222225, 58777752222225, ...
Also 75, 735, 7335, 73335, ..., (22*10^n+5) / 3, k >= 1, are terms and have no digits 0, because their squares are 5625, 540225, 53802225, 5378022225, 537780222225, 53777802222225, ... (End)

Examples

			75^2 = 5625, hence 75 is a term.
235^2 = 55225, hence 235 is not a term.

Crossrefs

Cf. A045859, A017330 (squares ending with 5).
Similar to: A348487 (k=1), A348488 (k=4), this sequence (k=5), A348490 (k=6), A348491 (k=9).
Subsequence of A305719.

Programs

  • Magma
    [n:n in [4..7500]|Intseq(n*n)[1] eq 5 and Intseq(n*n)[#Intseq(n*n)] eq 5 and Intseq(n*n)[-1+#Intseq(n*n)] ne 5 ]; // Marius A. Burtea, Oct 25 2021
  • Mathematica
    Select[5 * Range[2, 1500], (d = IntegerDigits[#^2])[[1]] == d[[-1]] == 5 && d[[2]] != 5 &] (* Amiram Eldar, Oct 25 2021 *)
  • PARI
    isok(k) = my(d=digits(sqr(k))); (d[1]==5) && (d[#d]==5) && if (#d>2, (d[2]!=5) && (d[#d-1]!=5), 1); \\ Michel Marcus, Oct 25 2021
  • Python
    from itertools import count, takewhile
def ok(n):
  s = str(n*n); return len(s.rstrip("5")) == len(s.lstrip("5")) == len(s)-1
def aupto(N):
  r = takewhile(lambda x: x<=N, (10*i+5 for i in count(0)))
  return [k for k in r if ok(k)]
print(aupto(7245)) # Michael S. Branicky, Oct 26 2021

A348490 Positive numbers whose square starts and ends with exactly one 6.

Original entry on oeis.org

26, 246, 254, 256, 264, 776, 784, 786, 794, 796, 804, 806, 824, 826, 834, 836, 2454, 2456, 2464, 2466, 2474, 2476, 2484, 2486, 2494, 2496, 2504, 2506, 2514, 2516, 2524, 2526, 2534, 2536, 2544, 2546, 2554, 2556, 2564, 2566, 2594, 2596, 2604, 2606, 2614, 2616, 2624, 2626, 2634, 2636, 2644, 7746
Offset: 1

Views

Author

Bernard Schott, Oct 29 2021

Keywords

Comments

When a square ends with 6, it ends with only one 6.
From Marius A. Burtea, Oct 30 2021 : (Start)
The sequence is infinite because the numbers 806, 8006, 80006, ..., 8*10^k + 6, k >= 2, are terms with squares 649636, 64096036, 6400960036, 640009600036, ..., 64*10^(2*k) + 96*10^k + 36, k >= 2.
Numbers 796, 7996, 79996, 799996, 7999996, 79999996, ..., 10^k*8 - 4, k >= 2, are terms and have no digits 0, because their squares are 633616, 63936016, 6399360016, 639993600016, 63999936000016, 6399999360000016, ....
Also 794, 7994, 79994, 799994, ..., (8*10^k - 6), k >= 2, are terms and have no digits 0, because their squares are 630436, 63904036, 6399040036, 639990400036, 63999904000036, 6399999040000036, ... (End)

Examples

			26^2 = 676, hence 26 is a term.
814^2 = 662596, hence 814 is not a term.

Crossrefs

Cf. A045789, A045860, A273373 (squares ending with 6).
Similar to: A348487 (k=1), A348488 (k=4), A348489 (k=5), this sequence (k=6), A348491 (k=9).
Subsequence of A305719.

Programs

  • Magma
    [n:n in [4..7500]|Intseq(n*n)[1] eq 6 and Intseq(n*n)[#Intseq(n*n)] eq 6 and Intseq(n*n)[-1+#Intseq(n*n)] ne 6 ]; // Marius A. Burtea, Oct 30 2021
  • Mathematica
    Select[Range[10, 7750], (d = IntegerDigits[#^2])[[1]] == d[[-1]] == 6 && d[[2]] != 6 &] (* Amiram Eldar, Oct 30 2021 *)
  • PARI
    isok(k) = my(d=digits(sqr(k))); (d[1]==6) && (d[#d]==6) && if (#d>2, (d[2]!=6) && (d[#d-1]!=6), 1); \\ Michel Marcus, Oct 30 2021
  • Python
    from itertools import count, takewhile
def ok(n):
  s = str(n*n); return len(s.rstrip("6")) == len(s.lstrip("6")) == len(s)-1
def aupto(N):
  r = takewhile(lambda x: x<=N, (10*i+d for i in count(0) for d in [4, 6]))
  return [k for k in r if ok(k)]
print(aupto(2644)) # Michael S. Branicky, Oct 29 2021

A348491 Positive numbers whose square starts and ends with exactly one 9.

Original entry on oeis.org

3, 97, 303, 307, 313, 953, 957, 963, 967, 973, 977, 983, 987, 993, 3003, 3007, 3013, 3017, 3023, 3027, 3033, 3037, 3043, 3047, 3053, 3057, 3063, 3067, 3073, 3077, 3083, 3087, 3093, 3097, 3103, 3107, 3113, 3117, 3123, 3127, 3133, 3137, 3143, 9487, 9493, 9497, 9503, 9507, 9513, 9517
Offset: 1

Views

Author

Bernard Schott, Nov 02 2021

Keywords

Comments

When a square ends with 9, it ends with only one 9.
From Marius A. Burtea, Nov 02 2021 : (Start)
The sequence is infinite because the numbers 303, 3003, 30003, ..., 3*(10^k + 1), k >= 2, are terms with squares 91809, 9018009, 900180009, 90001800009, ... 9*(10^(2*k) + 2*10^k + 1), k >= 2.
Numbers 97, 967, 9667, 96667, 966667, ..., (29*10^n + 1) / 3, k >= 1, are terms and have no digits 0, because their squares are 9409, 935089, 93450889, 9344508889, 934445088889, ...
Also 963, 9663, 96663, 966663, 9666663, 96666663, ... (29*10^k - 11) / 3, k >= 2, are terms and have no digits 0, because their squares are 927369, 93373569, 9343735569, 934437355569, 93444373555569, 9344443735555569, ... (End)

Examples

			97^2 = 9409, hence 97 is a term.
997^2 = 994009, hence  997 is not a term.

Crossrefs

Subsequence of A305719, A063226, and A045863.
Cf. A017377, A045863, A273374 (squares ending with 9).
Similar to: A348487 (k=1), A348488 (k=4), A348489 (k=5), A348490 (k=6), this sequence (k=9).

Programs

  • Magma
    [3] cat [n:n in [4..9600]|Intseq(n*n)[1] eq 9 and Intseq(n*n)[#Intseq(n*n)] eq 9]; // Marius A. Burtea, Nov 02 2021
  • Mathematica
    Join[{3}, Select[Range[10, 10^4], (d = IntegerDigits[#^2])[[1]] == d[[-1]] == 9 && d[[2]] != 9 &]] (* Amiram Eldar, Nov 02 2021 *)
  • PARI
    isok(k) = my(d=digits(sqr(k))); (d[1]==9) && (d[#d]==9) && if (#d>2, (d[2]!=9) && (d[#d-1]!=9), 1); \\ Michel Marcus, Nov 03 2021
  • PARI
    list(lim)=my(v=List([3])); for(d=2, 2*#digits(lim\=1), my(s=sqrtint(9*10^(d-1)-1)+1); s+=[3,2,1,0,3,2,1,0,5,4][s%10+1]; forstep(n=s, min(sqrtint(10^d-10^(d-2)-1), lim), if(s%10==3, [4,6], [6,4]), listput(v, n))); Vec(v) \\ Charles R Greathouse IV, Nov 03 2021
  • Python
    from itertools import count, takewhile
def ok(n):
  s = str(n*n); return len(s.rstrip("9")) == len(s.lstrip("9")) == len(s)-1
def aupto(N):
  r = takewhile(lambda x: x<=N, (10*i+d for i in count(0) for d in [3, 7]))
  return [k for k in r if ok(k)]
print(aupto(9517)) # Michael S. Branicky, Nov 02 2021

A348832 Positive numbers whose square starts and ends with exactly 444.

Original entry on oeis.org

666462, 666538, 666962, 667038, 2107462, 2107538, 2107962, 2108038, 2108462, 2108538, 2108962, 2109038, 2109462, 6663462, 6663538, 6663962, 6664038, 6664462, 6664538, 6664962, 6665038, 6665462, 6665538, 6665962, 6666038, 6667462, 6667538, 6667962, 6668038, 6668462, 6668538, 6668962
Offset: 1

Views

Author

Bernard Schott, Nov 09 2021

Keywords

Comments

The 1st problem of British Mathematical Olympiad (BMO) in 1995 (see link) asked to find all positive integers whose squares end in three 4’s (A039685); this sequence is the subsequence of these integers whose squares also start in precisely three 4's (no four or more 4's). Two such infinite subsequences are proposed below.
When a square starts and ends with digits ddd, then ddd is necessarily 444.
The first 3 digits of terms are either 210, 666 or 667, while the last 3 digits are either 038, 462, 538 or 962 (see examples).
From Marius A. Burtea, Nov 09 2021 : (Start)
The sequence is infinite because the numbers 667038, 6670038, 66700038, 667000038, ..., 667*10^k + 38, k >= 3, are terms because are square 444939693444, 44489406921444, 4448895069201444, 444889050692001444, 44488900506920001444, ...
Also, 6663462, 66633462, 666333462, 6663333462, ..., (1999*10^k + 386) / 3, k >= 4, are terms and have no digits 0, because their squares are 44401725825444, 4440018258105444, 444000282580905444, 44400012825808905444,
4440001128258088905444, ... (End)

Examples

			666462 is a term since 666462^2 = 444171597444.
21038 is not a term since 21038^2 = 442597444.

References

  • A. Gardiner, The Mathematical Olympiad Handbook: An Introduction to Problem Solving, Oxford University Press, 1997, reprinted 2011, Pb 1 pp. 55 and 95-96 (1995)

Links

  • British Mathematical Olympiad 1975, Problem 1.

Crossrefs

Cf. A017317, A328886.
Subsequence of A039685, A045858, A273375, A305719, A346892.
Similar to: A348488 (d=4), A348831 (dd=44), this sequence (ddd=444).

Programs

  • Magma
    fd:=func; fs:=func; [n:n in [1..6700000]|fd(n) and fs(n)]; // Marius A. Burtea, Nov 09 2021
  • Mathematica
    Select[Range[100, 7*10^6], (d = IntegerDigits[#^2])[[1 ;; 3]] == d[[-3 ;; -1]] == {4, 4, 4} && d[[-4]] != 4 && d[[4]] != 4 &] (* Amiram Eldar, Nov 09 2021 *)
  • Python
    from itertools import count, takewhile
def ok(n):
  s = str(n*n); return len(s.rstrip("4")) == len(s.lstrip("4")) == len(s)-3
def aupto(N):
  ends = [38, 462, 538, 962]
  r = takewhile(lambda x: x<=N, (1000*i+d for i in count(0) for d in ends))
  return [k for k in r if ok(k)]
print(aupto(6668962)) # Michael S. Branicky, Nov 09 2021

A348831 Positive numbers whose square starts and ends with exactly 44, and no 444.

Original entry on oeis.org

212, 2112, 6638, 6662, 6688, 20988, 21012, 21062, 21112, 21138, 21162, 21188, 21212, 66338, 66362, 66388, 66412, 66438, 66488, 66512, 66562, 66588, 66612, 66712, 66738, 66762, 66788, 66812, 66838, 66862, 66888, 66912, 66938, 66988, 67012, 67062, 209762, 209788
Offset: 1

Views

Author

Bernard Schott, Nov 08 2021

Keywords

Comments

When a square starts and ends with digits dd, then dd is necessarily 44.
The last 2 digits of terms are either 12, 38, 62 or 88.
From Marius A. Burtea, Nov 09 2021 : (Start)
The sequence is infinite because the numbers 212, 2112, 21112, ..., (19*10^k + 8) / 9, k >= 3, are terms because the remainder when dividing by 1000 is 544 and 445*10^(2*k - 2) < ((19*10^k + 8) / 9)^2 < 447*10^(2*k - 2), k >= 3.
Also 6638, 66338, 663338, 6633338, 66333338, 663333338, 6633333338, ..., (199*10^k + 14) / 3, k >= 2, are terms and have no digits 0, because their squares are: 44063044, 4400730244, 4400730244, 440017302244, 44001173022244, 4400111730222244, 440011117302222244, ... (End)

Examples

			212 is a term since 212^2 = 44944.
662 is not a term since 662^2 = 438244.
668 is not a term since 668^2 = 446224.
2108 is not a term since 2108^2 = 4443664.
21038 is not a term since 21038^2 = 442597444.
21088 is not a term since 21088^2 = 444703744.

Crossrefs

Cf. A017317.
Subsequence of A045858, A273375, A305719 and A346774.
Similar to: A348488 (d=4), this sequence (dd=44), A348832 (ddd=444).

Programs

  • Magma
    fd:=func; fs:=func; [n:n in [1..210000]|fd(n) and fs(n)]; // Marius A. Burtea, Nov 08 2021
  • Mathematica
    Select[Range[10, 300000], (d = IntegerDigits[#^2])[[1 ;; 2]] ==  d[[-2 ;; -1]] == {4, 4} && d[[-3]] != 4 && d[[3]] != 4 &] (* Amiram Eldar, Nov 08 2021 *)
  • Python
    from itertools import count, takewhile
def ok(n):
  s = str(n*n); return len(s.rstrip("4")) == len(s.lstrip("4")) == len(s)-2
def aupto(N):
  ends = [12, 38, 62, 88]
  r = takewhile(lambda x: x<=N, (100*i+d for i in count(0) for d in ends))
  return [k for k in r if ok(k)]
print(aupto(209788)) # Michael S. Branicky, Nov 08 2021
Showing 1-8 of 8 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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