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 1-6 of 6 results.

A091629 Product of digits associated with A091628(n). Essentially the same as A007283.

Original entry on oeis.org

6, 12, 24, 48, 96, 192, 384, 768, 1536, 3072, 6144, 12288, 24576, 49152, 98304, 196608, 393216, 786432, 1572864, 3145728, 6291456, 12582912, 25165824, 50331648, 100663296, 201326592, 402653184, 805306368, 1610612736, 3221225472
Offset: 1

Views

Author

Enoch Haga, Jan 24 2004

Keywords

Comments

Sequence arising in Farideh Firoozbakht's solution to Prime Puzzle 251 - 23 is the only pointer prime (A089823) not containing digit "1".
The monotonic increasing value of successive product of digits strongly suggests that in successive n the digit 1 must be present.

Links

Crossrefs

Sequences of the form (2*m+1)*2^n: A000079 (m=0), A007283 (m=1), A020714 (m=2), A005009 (m=3), A005010 (m=4), A005015 (m=5), A005029 (m=6), A110286 (m=7), A110287 (m=8), A110288 (m=9), A175805 (m=10), A248646 (m=11), A164161 (m=12), A175806 (m=13), A257548 (m=15).
Cf. A089823, A091628, A091630, A091631, A091632.
Similar to A003945, A042950, A058764, A087009, A081808.

Programs

Formula

a(n) = 3 * 2^n = product of digits of A091628(n).
From Philippe Deléham, Nov 23 2008: (Start)
a(n) = 6*2^(n-1).
a(n) = 2*a(n-1), with a(1) = 6.
G.f.: 6*x/(1-2*x). (End)
E.g.f.: 3*(exp(2*x) - 1). - G. C. Greubel, Jan 05 2023

Extensions

Edited and extended by Ray Chandler, Feb 07 2004

A091630 Numbers n + product of digits associated with A091628.

Original entry on oeis.org

29, 235, 2247, 22271, 222319, 2222415, 22222607, 222222991, 2222223759, 22222225295, 222222228367, 2222222234511, 22222222246799, 222222222271375, 2222222222320527, 22222222222418831, 222222222222615439
Offset: 1

Views

Author

Enoch Haga, Jan 24 2004

Keywords

Comments

Sequence arising in Farideh Firoozbakht's solution to Prime Puzzle 251 - 23 is the only pointer prime (A089823) not containing the digit "1".
The monotonically increasing value of successive product of digits (A091629) strongly suggests that in successive n the digit 1 must be present.

Examples

			a(1) = 23 + 6 = 29.

Links

Crossrefs

Cf. A089823, A091628, A091629, A091631, A091632.

Formula

a(n) = A091628(n) + A091629(n).
From Chai Wah Wu, Feb 12 2021: (Start)
a(n) = 13*a(n-1) - 32*a(n-2) + 20*a(n-3) for n > 3.
G.f.: x*(-120*x^2 + 142*x - 29)/((x - 1)*(2*x - 1)*(10*x - 1)). (End)

Extensions

Edited and extended by Ray Chandler, Feb 07 2004

A091631 Next prime associated with A091628.

Original entry on oeis.org

29, 227, 2237, 22229, 222247, 2222239, 22222253, 222222227, 2222222243, 22222222273, 222222222301, 2222222222243, 22222222222229, 222222222222227, 2222222222222281, 22222222222222301, 222222222222222281
Offset: 1

Views

Author

Enoch Haga, Jan 24 2004

Keywords

Comments

Sequence arising in Farideh Firoozbakht's solution to Prime Puzzle 251 - 23 is the only pointer prime (A089823) not containing the digit "1".
The monotonically increasing value of successive product of digits (A091629) strongly suggests that in successive n the digit 1 must be present.

Examples

			a(1) = nextprime(23+1) = 29.

Links

Crossrefs

Cf. A007918, A089823, A091628, A091629, A091630, A091632.

Programs

  • PARI
    a(n) = nextprime((10^(n+1) - 1)/9*2 + 2); \\ Michel Marcus, Mar 18 2018

Formula

a(n) = A007918(A091628(n)+1).

Extensions

Edited and extended by Ray Chandler, Feb 07 2004

A091632 Excess of n + product of digits over next prime associated with A091628.

Original entry on oeis.org

0, 8, 10, 42, 72, 176, 354, 764, 1516, 3022, 6066, 12268, 24570, 49148, 98246, 196530, 393158, 786406, 1572834, 3145674, 6291440, 12582874, 25165764, 50331634, 100663192, 201326576, 402653180, 805306350, 1610612690, 3221225038
Offset: 1

Views

Author

Enoch Haga, Jan 24 2004

Keywords

Comments

Sequence arising in Farideh Firoozbakht's solution to Prime Puzzle 251 - 23 is the only pointer prime (A089823) not containing the digit "1".
The monotonically increasing value of successive excess (and product of digits (A091629)) strongly suggests that in successive n the digit 1 must be present.

Examples

			a(2) = 235 - 227 = 8.

Links

Crossrefs

Cf. A089823, A091628, A091629, A091630, A091631.

Formula

a(n) = A091630(n) - A091631(n).

Extensions

Edited and extended by Ray Chandler, Feb 07 2004

A089823 Primes p such that the next prime after p can be obtained from p by adding the product of the digits of p.

Original entry on oeis.org

23, 61, 1123, 1231, 1321, 2111, 2131, 11261, 11621, 12113, 13121, 15121, 19121, 21911, 22511, 27211, 61211, 116113, 131231, 312161, 611113, 1111211, 1111213, 1111361, 1112611, 1123151, 1411411, 1612111, 2111411, 2121131, 3112111
Offset: 1

Views

Author

Joseph L. Pe, Jan 09 2004

Keywords

Comments

I call these primes (multiplicative) "pointer primes", in the sense that such primes p "point" to the next prime after p when the product of the digits of p is added to p. 23 is the only pointer prime < 10^7 which does not contain the digit "1". Are there other pointer primes not containing the digit "1"?
See Prime Puzzle 251 link for several arguments that 23 is the only pointer prime not containing digit "1".

Examples

			23 + product of digits of 23 = 29, which is the next prime after 23. Hence 23 belongs to the sequence.

Links

Crossrefs

Cf. A091628, A091629, A091630, A091631, A091632.

Programs

  • Mathematica
    r = {}; Do[p = Prime[i]; q = Prime[i + 1]; If[p + Apply[Times, IntegerDigits[p]] == q, r = Append[r, p]], {i, 1, 10^6}]; r

A093135 Expansion of g.f. (1-8*x)/((1-x)*(1-10*x)).

Original entry on oeis.org

1, 3, 23, 223, 2223, 22223, 222223, 2222223, 22222223, 222222223, 2222222223, 22222222223, 222222222223, 2222222222223, 22222222222223, 222222222222223, 2222222222222223, 22222222222222223, 222222222222222223, 2222222222222222223, 22222222222222222223, 222222222222222222223
Offset: 0

Views

Author

Paul Barry, Mar 24 2004

Keywords

Comments

Second binomial transform of 2*A001045(3*n)/3 + (-1)^n.
Partial sums of A093136.
A convex combination of 10^n and 1.
In general the second binomial transform of k*Jacobsthal(3*n)/3 + (-1)^n is 1, 1+k, 1+11*k, 1+111*k, ... This is the case for k=2.
Essentially the same as A091628 (cf. 2nd formula). - Georg Fischer, Oct 06 2018
a(n) is 3^n represented in bijective base-3 numeration. - Alois P. Heinz, Aug 26 2019

Links

Crossrefs

Cf. A001045, A002279, A047855, A062397, A091628, A093136.

Formula

a(n) = (2*10^n + 7)/9.
a(n) = 10*a(n-1) - 7 (with a(0)=1). - Vincenzo Librandi, Aug 02 2010
From Elmo R. Oliveira, Apr 03 2025: (Start)
E.g.f.: exp(x)*(7 + 2*exp(9*x))/9.
a(n) = 11*a(n-1) - 10*a(n-2).
a(n) = (A062397(n) - A002279(n))/2. (End)

Extensions

More terms from Elmo R. Oliveira, Apr 03 2025
Showing 1-6 of 6 results.

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