A097050 -id:A097050 - cp's OEIS frontend

A063108 a(1) = 1; thereafter a(n+1) = a(n) + product of nonzero digits of a(n).

1, 2, 4, 8, 16, 22, 26, 38, 62, 74, 102, 104, 108, 116, 122, 126, 138, 162, 174, 202, 206, 218, 234, 258, 338, 410, 414, 430, 442, 474, 586, 826, 922, 958, 1318, 1342, 1366, 1474, 1586, 1826, 1922, 1958, 2318, 2366, 2582, 2742, 2854, 3174, 3258, 3498, 4362

Offset: 1

Paul A. Loomis, Aug 08 2001

Conjecture: no matter what the starting term is, the sequence eventually joins this one. This should be true in any base - base 2, for example, is trivial.

A063114 iterated, beginning with 1. - Reinhard Zumkeller, Jan 15 2012

			a(2) = 1 + 1 = 2; a(3) = 4; a(6) = 16 + 1*6 = 22; a(22) = 206 + 2*6 = 218.

T. D. Noe, Table of n, a(n) for n = 1..10000
P. A. Loomis, An Interesting Family of Iterated Sequences
P. A. Loomis, An Introduction to Digit Product Sequences, J. Rec. Math., 32 (2003-2004), 147-151.
P. A. Loomis, An Introduction to Digit Product Sequences, J. Rec. Math., 32 (2003-2004), 147-151. [Annotated archived copy]
Index entries for Colombian or self numbers and related sequences

Cf. A063112, A063113, A063114, A097050, A051801, A096355, A230102, A232485, A232486, A232487, A232488.

a063108_list = iterate a063114 1  -- Reinhard Zumkeller, Jan 15 2012

with transforms;
f:=proc(n) option remember; if n=1 then 1
else f(n-1)+digprod(f(n-1)); fi; end;
[seq(f(n),n=1..20)];
# N. J. A. Sloane, Oct 12 2013

f[ n_Integer ] := Block[{s = Sort[ IntegerDigits[ n ]]}, While[ s[[ 1 ]] == 0, s = Drop[ s, 1 ]]; n + Times @@ s]; NestList[ f, 1, 65 ]
nxt[n_]:=n+Times@@Select[IntegerDigits[n],#>0&]; NestList[nxt,1,50] (* Harvey P. Dale, Oct 10 2012 *)

lista(n)={ my(a=vector(n)); a[1]=1; for(i=1, #a-1, a[i+1] = a[i] + vecprod(select(x->x, digits(a[i])))); a } \\ Harry J. Smith, Aug 18 2009

A crude heuristic analysis suggests that a(n) grows roughly like (8/9 * (1-y))^(1/(1-y)) * n^(1/1-y) where y = log_10(4.5), i.e., that a(n) ~ 0.033591*n^2.8836.

More terms from Robert G. Wilson v, Aug 09 2001

A065383 a(n) = smallest prime >= n*(n + 1)/2.

Original entry on oeis.org

2, 2, 3, 7, 11, 17, 23, 29, 37, 47, 59, 67, 79, 97, 107, 127, 137, 157, 173, 191, 211, 233, 257, 277, 307, 331, 353, 379, 409, 439, 467, 499, 541, 563, 599, 631, 673, 709, 743, 787, 821, 863, 907, 947, 991, 1039, 1087, 1129, 1181, 1229, 1277, 1327, 1381, 1433

Offset: 0

Reinhard Zumkeller, Nov 05 2001

nonn

Besides 7, terms exclude the greater of the twin primes (A006512). - Bill McEachen, Dec 01 2022

Harry J. Smith, Table of n, a(n) for n = 0..1000

See A097050 for another version.
Cf. A065382, A007491, A064384, A000217.
Cf. A000217.

a065383 n = head $ dropWhile (< a000217 n) a000040_list
-- Reinhard Zumkeller, Aug 03 2012

PrimeNext[n_]:=Module[{k=n},While[ !PrimeQ[k],k++ ];k];f[n_]:=n*(n+1)/2;lst={};Do[AppendTo[lst,PrimeNext[f[n]]],{n,6!}];lst (* Vladimir Joseph Stephan Orlovsky, Feb 26 2010 *)
NextPrime/@(Accumulate[Range[0,60]]-1) (* Harvey P. Dale, Jul 31 2012 *)

{ for (n=0, 1000, write("b065383.txt", n, " ", nextprime(n*(n + 1)/2)) ) } \\ Harry J. Smith, Oct 17 2009

Edited by N. J. A. Sloane, Nov 21 2008

A077128 Smallest number greater than the previous term which is relatively prime to each of the group of the next n numbers.

Original entry on oeis.org

2, 5, 7, 11, 17, 23, 29, 37, 47, 59, 67, 79, 97, 107, 127, 137, 157, 173, 191, 211, 233, 257, 277, 307, 331, 353, 379, 409, 439, 467, 499, 541, 563, 599, 631, 673, 709, 743, 787, 821, 863, 907, 947, 991, 1039, 1087, 1129, 1181, 1229, 1277, 1327, 1381, 1433

Offset: 1

Amarnath Murthy, Oct 29 2002

nonn

Conjecture : every member is a prime.

			a(6) = 23 is the smallest number coprime to 16,17,18,19,20 and 21. - _R. J. Mathar_, Sep 02 2008

Cf. A097050. - R. J. Mathar, Sep 02 2008

A000217 := proc(n) n*(n+1)/2 ; end: A077128 := proc(n) option remember ; local ts,a,goodk,k ; if n = 1 then RETURN(2) ; fi; ts := [seq(A000217(n-1)+i,i=1..n)] ; for a from procname(n-1)+1 do goodk := true ; for k in ts do if gcd(a,k) <> 1 then goodk := false; break ; fi; od: if goodk then RETURN(a) ; fi; od: end: for n from 1 to 100 do printf("%d,",A077128(n)) ; od: # R. J. Mathar, Sep 02 2008

Extended beyond a(10) by R. J. Mathar, Sep 02 2008

A181956 Smallest prime greater than n*(n+1)^2/2.

Original entry on oeis.org

2, 3, 11, 29, 53, 97, 149, 227, 331, 457, 607, 797, 1019, 1277, 1579, 1931, 2333, 2767, 3251, 3803, 4421, 5087, 5821, 6637, 7507, 8461, 9479, 10589, 11777, 13063, 14419, 15877, 17431, 19079, 20849, 22691, 24659, 26717, 28901, 31219, 33623, 36187, 38833, 41627

Offset: 0

Gerasimov Sergey, Apr 03 2012

nonn

			a(1)=2 because prime 2 > (0*(0+1)^2/2) = 0, a(2)=3 because prime 3 > (1*(1+1)^2/2) = 2, a(3)=11 because prime 11 > (2*(2+1)^2/2) = 9.

Charles R Greathouse IV, Table of n, a(n) for n = 0..10000

Cf. A006002, A097050.

Table[NextPrime[n*(n+1)^2/2], {n, 0, 50}] (* T. D. Noe, Apr 03 2012 *)

a(n)=nextprime(n*(n+1)^2/2+1) \\ Charles R Greathouse IV, Aug 03 2012

a(n) ~ n^3 / 2. - Charles R Greathouse IV, Aug 03 2012

A147849 a(n) is the smallest triangular number > n-th prime.

Original entry on oeis.org

3, 6, 6, 10, 15, 15, 21, 21, 28, 36, 36, 45, 45, 45, 55, 55, 66, 66, 78, 78, 78, 91, 91, 91, 105, 105, 105, 120, 120, 120, 136, 136, 153, 153, 153, 153, 171, 171, 171, 190, 190, 190, 210, 210, 210, 210, 231, 231, 231, 231, 253, 253, 253, 253, 276, 276, 276, 276

Offset: 1

Zak Seidov, Nov 15 2008

nonn

Cf. A097050 (smallest prime > n-th triangular number).

Cf. A000040, A000217.

a1 = Reap[Do[p = Prime[m]; Do[t = n (n + 1)/2; If[t > p, Sow[t]; Break[]], {n, 200}], {m, 100}]][[2, 1]]

cp's OEIS Frontend

A063108 a(1) = 1; thereafter a(n+1) = a(n) + product of nonzero digits of a(n).

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A065383 a(n) = smallest prime >= n*(n + 1)/2.

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A077128 Smallest number greater than the previous term which is relatively prime to each of the group of the next n numbers.

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Programs

Maple

Extensions

A181956 Smallest prime greater than n*(n+1)^2/2.

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Keywords

Examples

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Programs

Mathematica

PARI

Formula

A147849 a(n) is the smallest triangular number > n-th prime.

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Mathematica