A067275 -id:A067275 - cp's OEIS frontend

A105971 Number of distinct prime divisors of 66...667 (with n 6's).

1, 1, 2, 2, 2, 1, 2, 1, 1, 3, 1, 3, 3, 4, 3, 3, 3, 2, 4, 1, 2, 2, 1, 2, 5, 2, 4, 4, 3, 4, 3, 3, 3, 3, 4, 3, 4, 3, 4, 2, 1, 3, 5, 4, 4, 6, 6, 5, 4, 2, 4, 4, 4, 6, 5, 4, 4, 3, 3, 5, 5, 3, 1, 4, 5, 1, 5, 5, 5, 4, 3, 2, 4, 4, 4, 4, 5, 4, 5, 6, 2, 7, 6, 4, 7, 5, 5, 4, 5, 2, 5, 4, 3, 6, 4, 6, 6, 4, 3, 5

Offset: 0

Parthasarathy Nambi, Apr 28 2005

			The number of distinct prime divisors of 67 is 1 (prime).
The number of distinct prime divisors of 667 is 2.
The number of distinct prime divisors of 6667 is 2.

Amiram Eldar, Table of n, a(n) for n = 0..206

Cf. A001221, A067275, A104889, A104524.

Table[PrimeNu[(6*10^(n + 1) + 3)/9], {n, 0, 30}] (* Amiram Eldar, Jan 27 2020 *)
Table[PrimeNu[FromDigits[PadLeft[{7},n,6]]],{n,100}] (* Harvey P. Dale, Apr 05 2023 *)

a(n) = A001221(A067275(n+2)). - Amiram Eldar, Jan 27 2020

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

Shyam Sunder Gupta, May 11 2025

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)

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

Michael S. Branicky, Table of n, a(n) for n = 1..2773
Shyam Sunder Gupta, Elegance of Squares, Cubes, and Higher Powers, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 2, 29-81.

Cf. A003226, A033428.

Essentially the union of A030985, A030986, and A067275.

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

More terms from Michael S. Branicky, May 11 2025

A073554 Number of Fibonacci numbers F(k), k <= 10^n, which end in 7.

Original entry on oeis.org

0, 14, 134, 1334, 13334, 133334, 1333334, 13333334, 133333334, 1333333334, 13333333334, 133333333334, 1333333333334, 13333333333334, 133333333333334, 1333333333333334, 13333333333333334, 133333333333333334, 1333333333333333334, 13333333333333333334, 133333333333333333334, 1333333333333333333334, 13333333333333333333334

Offset: 1

Shyam Sunder Gupta, Aug 15 2002

			a(2) = 14 because there are 14 Fibonacci numbers up to 10^2 which end in 7.

Index entries for linear recurrences with constant coefficients, signature (11,-10).

Cf. A073548 (end in 2), A073549 (6), A073550 (1), A073551 (3), (A073552 (4)), A073553 (5), this sequence (7), A073555 (8), A073556 (9).

Cf. A000045, A067275.

Join[{0},Table[10 FromDigits[PadRight[{1},n,3]]+4,{n,30}]] (* Harvey P. Dale, Mar 29 2023 *)

If n>1 then a(n) = (2*10^n + 10)/15. - Robert Gerbicz, Sep 06 2002
a(n) = A073550(n) for n >= 3. - Georg Fischer, Oct 13 2022
From Elmo R. Oliveira, Jul 22 2025: (Start)
G.f.: 2*x^2*(7 - 10*x)/((1-x)*(1-10*x)).
E.g.f.: 2*(-6 - 15*x + 5*exp(x) + exp(10*x))/15.
a(n) = 2*A067275(n) for n >= 2.
a(n) = 11*a(n-1) - 10*a(n-2) for n > 3. (End)

More terms from Robert Gerbicz, Sep 06 2002

A179782 Numbers n such that the decimal representation of n is contained as substring in that of the n-th pentagonal number.

Original entry on oeis.org

0, 1, 5, 7, 25, 67, 482, 551, 625, 667, 2937, 6667, 9284, 9376, 9649, 48179, 49900, 55712, 66667, 89517, 90625, 161579, 631206, 666667, 890625, 1348613, 2089517, 3863187, 4999000, 6666667, 7109376, 7477735, 8575619, 10721030, 12890625

Offset: 1

Jonathan Vos Post, Jul 27 2010

This is to pentagonal numbers A000326 as A119238 is to triangular numbers A000217 and as A018834 is to squares A000290. All numbers of the form (10^n-1)/3*2+1 are contained in this list {1, 7, 67, 667, 6667, 66667, 666667, 6666667, 66666667, ...} Alois P. Heinz. Extension: Values 8-25 by Claudio Meller, 26-37 by D. S. McNeil.

			The 5th pentagonal number, 35, which contains 5.
The 7th pentagonal number, 70, which contains 7.
The 25th pentagonal number, 925, which contains 25.
The 67th pentagonal number, 6700, which contains 67.
The 482nd pentagonal number, 348245, which contains 482.
The 667th pentagonal number, 667000, which contains 667.

Cf. A000217, A000290, A000326, A018834, A067275, A119238.

[n for n in range(10**4) if str(n) in str((3*n**2-n)//2)]

Code clarified by D. S. McNeil, Aug 08 2010

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A105971 Number of distinct prime divisors of 66...667 (with n 6's).

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

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A073554 Number of Fibonacci numbers F(k), k <= 10^n, which end in 7.

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A179782 Numbers n such that the decimal representation of n is contained as substring in that of the n-th pentagonal number.

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