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: Daniel Sterman

Daniel Sterman's wiki page.

Daniel Sterman has authored 5 sequences.

A339073 Number of strings of Hebrew letters with a gematria value equal to n.

Original entry on oeis.org

1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1023, 2045, 4088, 8172, 16336, 32656, 65280, 130496, 260864, 521473, 1042434, 2083846, 4165649, 8327214, 16646264, 33276208, 66519792, 132974368, 265818368, 531376129, 1062231296, 2123421181, 4244760561, 8485359561, 16962400080, 33908170232, 67783096912
Offset: 1

Views

Author

Daniel Sterman, Nov 22 2020

Keywords

Comments

A051596-restricted compositions of n.

Examples

			The four strings with a gematria of 3 are:
אאא (111)
אב (12)
בא (21)
ג (3)
Note: Hebrew is written right-to-left, which is why the order of the digits appears to be reversed.

Links

Crossrefs

Cf. A051596, A000079.

Programs

  • Maple
    g:= 1/(1-add(x^i,i=1..9)-add(x^(10*i),i=1..9)-add(x^(100*i),i=1..4)):
S:= series(g,x,101):
seq(coeff(S,x,n),n=1..100); # Robert Israel, Nov 25 2020
  • Mathematica
    Table[SeriesCoefficient[1/(1 - Sum[x^j, {j, Join[Range[9], 10 Range[9], 100 Range[4]]}]), {x, 0, n}], {n, 100}] (* Jan Mangaldan, Nov 27 2020 *)

Formula

From Doron Zeilberger, Nov 23 2020: (Start)
G.f.: Sum(a(n)*x^n, n=0..infinity) =
1/(1-add(x^i,i=1..9)-add(x^(10*i),i=1..9)-add(x^(100*i),i=1..4))
= 1/(1-x-...-x^9 - x^10- ... -x^90 - x^100-x^200-x^300-x^400).
Asymptotics:
a(n) ~ 0.50221591060212746248115807725009875743325273964521...*(1.9990196005347377028156443471636402056440270173905...)^n
If alpha is the smallest positive root of P:=1-x-...-x^9 - x^10- ... -x^90 - x^100-x^200-x^300-x^400=0
then the above asymptotic formula is exactly -(alpha*P'(alpha))* (1/alpha)^n.
(End)

Extensions

More terms from Robert Israel, Nov 25 2020

A302687 a(1) = 1; a(2) = 2; then a(n) is the smallest number > a(n-1) such that a(n) divides concat(a(1), a(2), ..., a(n-1)).

Original entry on oeis.org

1, 2, 3, 41, 43, 129, 9567001, 21147541, 22662659, 23817877, 24837187, 28850377, 28872229, 37916473, 48749751, 70416307, 439229167, 834385607, 2270365163, 2278377431, 3751789547, 4433933101, 4810754611, 14432263833, 15632412757, 30530543651, 42441819717, 65591903199, 65857498407
Offset: 1

Views

Author

Daniel Sterman, Apr 11 2018

Keywords

Examples

			a(3) = 3, which makes the concatenation of the first three terms: 123. After 3, the next-highest factor of 123 is 41, so a(4) = 41. The concatenation of the first four terms is then 12341. After 41, the next-highest factor of 12341 is 43, so a(5) = 43.

Crossrefs

Compare A240588, in which each term does not need to strictly increase as long as it has not yet appeared in the sequence.
Compare also A171785, in which each term must divide the concatenation of all terms in the sequence including itself.
In A029455, each term divides the concatenation of all smaller positive integers.
In A110740, each term divides the concatenation of all strictly smaller positive integers.

Programs

  • Maple
    A[1]:= 1: A[2]:= 2: C:= 1:
for n from 3 to 20 do
  C:= A[n-1]+C*10^(ilog10(A[n-1])+1);
  A[n]:= min(select(`>`,numtheory:-divisors(C),A[n-1]))
od:
seq(A[i],i=1..20); # Robert Israel, Apr 12 2018

Extensions

a(16)-a(20) from Robert Israel, Apr 12 2018
a(21)-a(29) from Daniel Suteu, Apr 12 2018

A279846 Numbers that are centered k-gonal numbers for three or more values of k.

Original entry on oeis.org

1, 31, 37, 43, 46, 61, 67, 73, 79, 85, 91, 106, 109, 121, 127, 133, 136, 145, 151, 157, 166, 169, 181, 199, 211, 217, 226, 232, 235, 241, 253, 265, 271, 274, 277, 289, 295, 301, 307, 313, 316, 325, 331, 337, 343, 361, 379, 391, 397, 406, 409, 421, 433, 451, 463, 469, 481, 496, 505, 511
Offset: 1

Views

Author

Daniel Sterman, Dec 20 2016

Keywords

Comments

Numbers satisfying 1 + n*m*(m+1)/2 for three or more values of (n,m), where n >= 0 m > 1.
Numbers in this sequence appear in A101321 at least four times (because the second column contains every positive integer).

Examples

			109 is in the sequence because 109 is a centered triangular number, a centered 18-gonal number, and a centered 36-gonal number.

Links

Crossrefs

Cf. A062712 (rough equivalent for polygonal numbers).
Cf. A101321 (table of all centered polygonal numbers).
Cf. A275340 (list of nontrivial centered polygonal numbers).

A279831 Numbers that are centered k-gonal numbers for two or more values of k.

Original entry on oeis.org

1, 7, 13, 16, 19, 22, 25, 31, 37, 43, 46, 49, 55, 61, 64, 67, 73, 76, 79, 85, 91, 97, 103, 106, 109, 111, 115, 121, 127, 133, 136, 139, 141, 145, 148, 151, 154, 157, 163, 166, 169, 172, 175, 181, 187, 190, 191, 193, 196, 199, 205, 211, 217, 221, 223, 226, 229, 232, 235, 241, 247, 253, 256
Offset: 1

Views

Author

Daniel Sterman, Dec 20 2016

Keywords

Comments

Numbers satisfying 1 + n*m*(m+1)/2 for two or more values of (n,m), where n>=0 m>1.
Numbers in this sequence appear in A101321 at least three times (because the second column contains every positive integer).

Examples

			19 is in the sequence because 19 is both a centered triangular number and a centered hexagonal number.

Links

Crossrefs

Cf. A090428 (rough equivalent for polygonal numbers).
Cf. A101321 (table of all centered polygonal numbers).
Cf. A275340 (list of nontrivial centered polygonal numbers).

A279830 a(n) = the least integer that is centered polygonal in exactly n ways.

Original entry on oeis.org

4, 7, 37, 31, 91, 181, 211, 421, 631, 1891, 1261, 2521, 6931, 18481, 20791, 13861, 27721, 41581, 83161, 138601, 245701, 235621, 180181, 556921, 360361, 540541, 1670761, 1081081, 1413721, 2702701, 2162161, 6486481, 3063061, 8288281, 13430341, 6846841, 10270261, 6126121
Offset: 1

Views

Author

Daniel Sterman, Dec 20 2016

Keywords

Comments

a(n) has exactly n representations as a centered r-gonal number P(r,m) = 1 + r*m*(m+1)/2, with m > 1, r > 0.
a(n) appears n+1 times in A101321, due to the second column containing every positive integer.
a(n)-1 is the first appearance of n+1 in A007862.

Examples

			a(4)=31, because 31 is a centered triangular number (A005448), a centered pentagonal number (A005891), a centered decagonal number (A062786), and a central polygonal number (A002061). No number less than 31 has 4 representations.

Links

Crossrefs

Cf. A007862 (see alternative definition: the number of ways to represent n+1 as a centered polygonal number).
Cf. A063778 (the equivalent for polygonal numbers).
Subset of A275340 (the list of nontrivial centered polygonal numbers).
Subset of A101321 (centered polygonal numbers read by antidiagonals).

Programs

  • Mathematica
    f[n_] := Length@Select[Divisors[2 n - 2], IntegerQ@Sqrt[1 + 4 #] &] - 1;
Do[If[IntegerQ[A279830[f[i]]], , A279830[f[i]] = i], {i, 10000}];
A279830 /@ Range[13]
(* Davin Park, Dec 28 2016 *)

Extensions

Corrected and extended by Davin Park, Dec 27 2016

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