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.

User: Shyam Sunder Gupta

Shyam Sunder Gupta's wiki page.

Shyam Sunder Gupta has authored 656 sequences. Here are the ten most recent ones:

A387245 Numbers k such that (the concatenation of the first k positive even numbers) + 1 is prime.

Original entry on oeis.org

1, 5, 9, 24, 1400, 8915
Offset: 1

Views

Author

Shyam Sunder Gupta, Aug 24 2025

Keywords

Comments

a(6) = 8915 corresponds to a 39024-digit probable prime. a(7) > 15000.
The corresponding primes are a subsequence of A210734. - Michael S. Branicky, Aug 24 2025

Examples

			5 is a term since 246810 + 1 = 246811 is prime.

Crossrefs

Cf. A019520, A069048, A210734.

Formula

{k | A019520(k) + 1 is prime}. - Michael S. Branicky, Aug 24 2025

A383939 Numbers k such that k and the k-th triangular number T(k) = k*(k+1)/2 have only even digits.

Original entry on oeis.org

0, 28, 40, 64, 400, 2828, 4000, 4064, 6428, 22840, 24028, 40000, 202428, 240028, 400000, 2040040, 2400028, 4000000, 6422840, 6428064, 6646624, 20044064, 20202080, 20400040, 20406080, 24000028, 40000000, 66400064, 200042428, 204000040, 228406080, 240000028
Offset: 1

Views

Author

Shyam Sunder Gupta, Aug 18 2025

Keywords

Comments

The sequence also contains the infinite subsequence 4*10^n for n >= 1.

Examples

			64 is a term since it and T(64) = 2080 both have only even digits.

Links

  • Shyam Sunder Gupta, Triangular Numbers, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 3, 82-125.

Crossrefs

Cf. A000217, A014263, A117978, A347475.

Programs

  • Mathematica
    q[k_] := And @@ (AllTrue[IntegerDigits[#], EvenQ] & /@ {k, k*(k+1)/2}); Select[Range[0, 4*10^6], q] (* Amiram Eldar, Aug 18 2025 *)

A383821 3-automorphic numbers: positive integers k such that 3k^2 ends with k.

Original entry on oeis.org

2, 5, 7, 67, 75, 92, 667, 792, 875, 6667, 6875, 9792, 66667, 69792, 96875, 296875, 369792, 666667, 2369792, 4296875, 6666667, 62369792, 66666667, 262369792, 404296875, 666666667, 6666666667, 7262369792, 9404296875, 27262369792, 39404296875, 66666666667, 639404296875
Offset: 1

Views

Author

Shyam Sunder Gupta, May 11 2025

Keywords

Comments

All 3-automorphic numbers end in 2, 5, or 7 only.
From Michael S. Branicky, May 11 2025: (Start)
Terms of successively larger digits can be created by prepending digits on the left of previous terms; for each length, only 3 positive such "seeds" are valid (some may have leading zeros and thus do not contribute terms at that length).
Infinite since 6..67, with i 6's and then a 7 is a term for all i >= 0.
a(2774) has 1001 digits. (End)

Examples

			67 is in the sequence because 3*67^2 = 13467 which ends with 67.

Links

Crossrefs

Cf. A003226, A033428.
Essentially the union of A030985, A030986, and A067275.

Programs

  • Mathematica
    Select[Range[10^7],IntegerDigits[#]==Take[IntegerDigits[3#^2],-IntegerLength[#]]&] (* James C. McMahon, May 16 2025 *)

Extensions

More terms from Michael S. Branicky, May 11 2025

A383646 Smallest number that takes n steps to reach 1 under iteration of sum-of-cubes-of-digits map.

Original entry on oeis.org

1, 10, 112, 1189, 778, 13477, 2388889999999999999999
Offset: 0

Views

Author

Shyam Sunder Gupta, May 11 2025

Keywords

Comments

These could also be called the smallest cubic happy numbers of height n.
Subsequent terms are too large to display in full.
a(7) = 1127 * 10^3276941015089163237 - 1 (1126 followed by 3276941015089163237 nines).
a(8) = 35678 * 10^((a(7) - 1054)/729) - 1.

Examples

			a(2) = 112, because 112 takes 2 steps to reach 1 (112 --> 1 + 1 + 8 = 10 --> 1 + 0 = 1).

Links

Crossrefs

Cf. A007770, A001273, A035504.

A383640 Internal digits of k^3 include digits of k as substring, k does not end in 0.

Original entry on oeis.org

56, 782, 5111, 8089, 8216, 9553, 11768, 14357, 18229, 53257, 64164, 65137, 72556, 98442, 213405, 271516, 830686, 941976, 1969394, 2420681, 2751442, 4150015, 5354867, 7045156, 9590417, 9699457, 10333214, 13427757, 21955652, 31213974, 32743132, 35272742
Offset: 1

Views

Author

Shyam Sunder Gupta, May 03 2025

Keywords

Examples

			56 is in the sequence as 56^3 = 175616 contains 56 in its decimal expansion.

Links

Crossrefs

Cf. A052210, A046837.

Programs

  • Mathematica
    Select[Range[5,10^6],Mod[#,10]>0&&SequenceCount[Rest[Drop[IntegerDigits[#^3],-1]],IntegerDigits[#]]>0&] (* James C. McMahon, May 09 2025 *)

A383638 Right-truncatable happy numbers: every prefix is a happy number and no digits are zero.

Original entry on oeis.org

1, 7, 13, 19, 79, 133, 139, 192, 193, 793, 1332, 1333, 1335, 1337, 1339, 1393, 1929, 1933, 7937, 7938, 13323, 13332, 13334, 13339, 13393, 13933, 19293, 19295, 19296, 19333, 79372, 79384, 79386, 133236, 133326, 133399, 133939, 139339, 192934, 192951, 192954
Offset: 1

Views

Author

Shyam Sunder Gupta, May 03 2025

Keywords

Comments

Happy numbers in which repeatedly deleting the least significant digit gives a happy number at every step until a single-digit happy number remains. Last term is a(170) = 1933396239639.

Links

Crossrefs

Cf. A007770.

A383639 Left-truncatable happy numbers: every suffix is a happy number and no digits are zero.

Original entry on oeis.org

1, 7, 31, 91, 97, 291, 331, 391, 397, 931, 2331, 3331, 3391, 3931, 5331, 7331, 7397, 8397, 9291, 9331, 23331, 27397, 32331, 33391, 33931, 39291, 39331, 43331, 48397, 59291, 68397, 69291, 93331, 127397, 159291, 427397, 439291, 459291, 469291, 623331, 632331
Offset: 1

Views

Author

Shyam Sunder Gupta, May 03 2025

Keywords

Comments

Happy numbers in which repeatedly deleting the most significant digit gives a happy number at every step until a single-digit happy number remains. Last term is a(170) = 9639326993331.

Links

Crossrefs

Cf. A007770.

A383291 Successively larger gaps in happy numbers start at this happy number.

Original entry on oeis.org

1, 32, 49, 109, 139, 409, 496, 566, 3392, 4287, 5364, 358962, 488444, 4488044, 59299951, 59999665, 88889733, 488849933, 569199933, 5888999662, 3888899909932
Offset: 1

Views

Author

Shyam Sunder Gupta, Apr 22 2025

Keywords

Examples

			1 and 7 are successive happy numbers and so the first gap is 6. 32 and 44 are successive happy numbers and this is the first gap larger than 6, so 32 is in the sequence as the succeeding term to 1.

Links

  • Shyam Sunder Gupta, Happy Numbers, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 7, 209-222.

Crossrefs

Cf. A007770.

Extensions

a(21) from Amiram Eldar, Apr 26 2025

A383290 Number of happy primes <= 10^n.

Original entry on oeis.org

1, 7, 35, 200, 1465, 11144, 91323, 812371, 7408754, 67982202, 621496655
Offset: 1

Views

Author

Shyam Sunder Gupta, Apr 22 2025

Keywords

Comments

The proportion of happy primes to total primes seems to follow the same pattern as the proportion of happy numbers to total numbers, at least up to 10^10.

Links

  • Shyam Sunder Gupta, Happy Numbers, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 7, 209-222.

Crossrefs

Cf. A007770, A035497, A068571.

Extensions

a(11) from Amiram Eldar, Apr 23 2025

A381046 Happy repdigit numbers.

Original entry on oeis.org

1, 7, 44, 888, 5555, 88888, 1111111, 2222222, 22222222, 77777777, 1111111111, 7777777777, 22222222222, 44444444444, 444444444444, 1111111111111, 4444444444444, 7777777777777, 999999999999999, 7777777777777777, 22222222222222222, 77777777777777777
Offset: 1

Views

Author

Shyam Sunder Gupta, Apr 14 2025

Keywords

Examples

			888 is a term since it is a repdigit number consisting of digit 8 only and is also a happy number (64+64+64 = 192, and 1+81+4 = 86, and 64+36 = 100, and 1+0+0 = 1).

Links

  • Shyam Sunder Gupta, Happy Numbers, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 7, 209-222.

Crossrefs

Intersection of A010785 and A007770.

Programs

  • Mathematica
    happyQ[n_] := NestWhile[Total[IntegerDigits[#]^2] &, n, UnsameQ, All] == 1; Select[Union[Table[k*(10^n - 1)/9, {k, 1, 9}, {n, 17}] // Flatten], happyQ] (* Amiram Eldar, Apr 14 2025 *)

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

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