A152603 -id:A152603 - cp's OEIS frontend

A158652 Any two consecutive digits in the sequence sum up to a prime.

1, 2, 3, 4, 7, 41, 43, 47, 49, 83, 85, 89, 202, 302, 303, 411, 412, 502, 503, 830, 2020, 2021, 2023, 2025, 2029, 2030, 2032, 3020, 3021, 4111, 4112, 5020, 5021, 6111, 6112, 9202, 9203, 20202, 30202, 30203, 41111, 41112, 50202, 50203, 83020, 202020

Offset: 1

Eric Angelini, Mar 23 2009

a(1)=1 and a(n) is always the smallest integer > a(n-1) not leading to a contradiction. Terms computed by M. F. Hasler.
From M. F. Hasler, Mar 24 2009: (Start)
After the initial 2,3,4,7,41,43,47,49,83,85,89, the following pattern repeats:
202,302,303,
411,412,502,503,830,
2020,2021,2023,2025,2029,2030,2032,3020,3021,
4111,4112,5020,5021,6111,6112,9202,9203,
with each time two extra digits (either 02 or 11):
20202,30202,30203,
41111,41112,50202,50203,83020,
202020,202021,202023,202025,202029,202030,202032,302020,302021,
411111,411112,502020,502021,611111,611112,920202,920203,
and so on. (End)

Robert G. Wilson v, Table of n, a(n) for n=1..100
Eric Angelini, Chiffres consecutifs dans quelques suites
Eric Angelini, Chiffres consecutifs dans quelques suites [Cached copy, with permission]

Cf. A152603-A152610.

f[s_List] := Block[{k = s[[ -1]] + 1, ls = Mod[ s[[ -1]], 10]}, While[ Union@ PrimeQ[ Plus @@@ Partition[ Join[{ls}, IntegerDigits@ k], 2, 1]] != {True}, k++ ]; Append[s, k]]; Nest[f, {1}, 45] (* Robert G. Wilson v, Apr 05 2009 *)

from itertools import count, islice
allowed = {"0":"2357", "1":"1246", "2":"01359", "3":"0248", "4":"1379", "5":"0268", "6":"157", "7":"046", "8":"359", "9":"248"}
def cgen(seed, digits, geq="0"): # numbers satisfying the condition
    if digits == 1:
        yield from (c for c in allowed[seed] if c >= geq); return
    for f in (c for c in allowed[seed] if c >= geq):
        yield from (f + r for r in cgen(f, digits-1))
def nextc(k): # next element of cgen greater than k
    s = str(k)
    for d in count(len(s)):
        geq = s[0] if d == len(s) else "1"
        for c in map(int, cgen(s[-1], d, geq=geq)):
            if c > k: return c
def agen():
    an = 1
    for n in count(1): yield an; an = nextc(an)
print(list(islice(agen(), 40))) # Michael S. Branicky, Jul 12 2022

A152604 a(1) = 1; thereafter a(n) is always the smallest integer > a(n-1) not leading to a contradiction, such that any four consecutive digits in the sequence sum up to a prime.

Original entry on oeis.org

1, 2, 3, 5, 7, 8, 9, 51, 83, 110, 111, 211, 301, 310, 311, 631, 703, 710, 911, 2111, 2113, 2117, 2119, 2153, 2155, 2159, 2171, 2173, 2177, 2179, 2513, 2515, 2519, 2531, 2533, 2537, 2539, 2573, 2575, 2579, 8513, 8515, 8519, 8573, 8579, 8591

Offset: 1

N. J. A. Sloane, Sep 23 2009

Computed by Jean-Marc Falcoz.

From a(69)=1100110 onward starts a repeating pattern of length 54. - M. F. Hasler, Oct 16 2009

Eric Angelini, Chiffres consecutifs dans quelques suites
E. Angelini, Chiffres consecutifs dans quelques suites [Cached copy, with permission]

Cf. A158652, A152603..A152609.

A152604(n,show_all=0)={my(a);for(i=1,n,if( i<8,a=i+(i>3)+(i>4), my(l3d=if(a>99,a%1000,[789,951,183][i-7])); while( a++, my(t=a+l3d*10^#Str(a));forstep(d=#Str(a)-1,0,-1, isprime(sum(j=d,d+3,t\10^j%10)) & next; a+=10^d-a%10^d-1; next(2)); break)); show_all&print1(a", "));a} \\ M. F. Hasler, Oct 16 2009

A152605 a(1) = 1; thereafter a(n) is always the smallest integer > a(n-1) not leading to a contradiction, such that any five consecutive digits in the sequence sum up to a prime.

Original entry on oeis.org

1, 2, 3, 4, 7, 12, 30, 51, 83, 231, 232, 312, 323, 327, 413, 414, 530, 541, 701, 811, 812, 1101, 2110, 3011, 6301, 7030, 7103, 8110, 9011, 21011, 21013, 21017, 21019, 21053, 21055, 21059, 21071, 21073, 21077, 21079, 21413, 21415, 21419

Offset: 1

N. J. A. Sloane, Sep 23 2009

Computed by Jean-Marc Falcoz.

From a(116)=6100011 on, there starts a pattern of 75 terms which then repeats indefinitely (with duplication of a substring of 5 digits in the middle of each term). - M. F. Hasler, Oct 16 2009

Eric Angelini, Chiffres consecutifs dans quelques suites
E. Angelini, Chiffres consecutifs dans quelques suites [Cached copy, with permission]

Cf. A158652, A152603, A152604, A152606, A152607, A152608, A152609.

A152605(n,show_all=0,s=[1, 2, 3, 4, 7, 12, 30, 51, 83, 231, 232, 312, 323, 327, 413, 414, 530, 541, 701, 811, 812, 1101])={ my(a); for(i=1,n, if(i<=#s,a=s[i], my(ld=a%10^4); while(a++,my(t=a+ld*10^#Str(a));forstep(d=#Str(a)-1,0,-1,isprime(sum(j=d,d+4,t\10^j%10))&next;a+=10^d-a%10^d-1;next(2));break));show_all&print1(a", "));a } \\ M. F. Hasler, Oct 16 2009

A342215 Two successive terms always share a common "digit pattern" (see the Comments section). The successive "common digit patterns", concatenated, reproduce the successive terms of the sequence, concatenated.

Original entry on oeis.org

1, 10, 100, 101, 1201, 301, 12, 20, 104, 13, 30, 102, 14, 21, 2, 200, 103, 410, 341, 3, 203, 105, 210, 421, 1242, 112, 204, 50, 106, 310, 34, 41, 107, 230, 43, 114, 31, 23, 205, 303, 113, 108, 305, 25, 121, 109, 40, 24, 123, 15, 120, 42, 1142, 211, 26, 207, 140, 45, 250, 110, 610, 36, 131, 160, 302, 134, 4

Offset: 1

Eric Angelini and Carole Dubois, Mar 05 2021

A "common pattern" shared by two successive integers A and B is a string of digits present in both A and B. For example, if A = 1 and B = 10 the common pattern is "1"; if A = 2021 and B = 302 the common pattern is "02".
We allow the successive terms A and B to share more than one pattern, but only in the case of a single shared longer string of digits - longer than the other possible strings; as A = 2021 and B = 231 share both the strings "2" and "1", which are of the same length, B cannot follow A in the sequence. As A = 2021 and B = 2031 share both the strings "20" and "1" and as the string "20" is longer than the string "1", B could follow A in the sequence (the "common pattern" would be "20" here).
This "common pattern" idea was imagined to inspire people having almost no mathematical skills - only two eyes (or one single eye) and a pencil.
Caveat: to reduce the computing time, no term > 10000 was tested.
Given the doubts about this sequence, please do NOT add a b-file. N. J. A. Sloane, Mar 14 2021

			The first ten terms are 1, 10, 100, 101, 1201, 301, 12, 20, 104, 13.
The "common patterns" are 1  10   10   01    01    1   2   0    1 and their concatenation is 1101001011201 - which is exactly the start of the concatenation of the sequence's terms.

Carole Dubois, Conjectured table of n, a(n) for n = 1..789

Cf. A152603

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A158652 Any two consecutive digits in the sequence sum up to a prime.

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A152604 a(1) = 1; thereafter a(n) is always the smallest integer > a(n-1) not leading to a contradiction, such that any four consecutive digits in the sequence sum up to a prime.

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A152605 a(1) = 1; thereafter a(n) is always the smallest integer > a(n-1) not leading to a contradiction, such that any five consecutive digits in the sequence sum up to a prime.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A342215 Two successive terms always share a common "digit pattern" (see the Comments section). The successive "common digit patterns", concatenated, reproduce the successive terms of the sequence, concatenated.

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Author

Keywords

Comments

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Crossrefs