A156207 -id:A156207 - cp's OEIS frontend

A156208 Primes appearing as the products of digits and positions in A156207(i) in the order of appearance.

2, 3, 5, 7, 3, 5, 7, 11, 13, 17, 19, 2, 3, 5, 7, 11, 13, 17, 19, 5, 7, 11, 13, 17, 19, 23, 7, 11, 13, 17, 19, 23, 11, 13, 17, 19, 23, 7, 13, 19, 3, 5, 11, 17, 23, 29, 7, 13, 19, 31, 11, 17, 23, 29, 13, 19, 31, 37, 17, 23, 29, 41, 19, 31, 37, 43, 2, 5, 11, 17, 23, 29, 7, 13, 19, 31, 11

Offset: 1

Cino Hilliard, Feb 05 2009, Feb 08 2009

A156207 read without the 1's and without the composites. - R. J. Mathar, Sep 07 2016

			For n=19 we have 1*1 + 2*9 = 19 prime and the sequence.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A156207.

f:= proc(n) local L,i,a;
  L:= convert(n,base,10);
  a:= add(L[-i]*i,i=1..nops(L));
  if isprime(a) then a else NULL fi
end proc:
map(f, [$1..1000]); # Robert Israel, Sep 07 2016

g1(n) = for(j=1,n,if(isprime(g(j)),print1(g(j)",")))
g(n) = v=Vec((Str(n)));sum(x=1,length(v),x*eval(v[x]))

Given a number n with digits d1d2d3...dm, a(n) = d1*1+d2*2+d3*3+...+dm*m.

If a(n) is prime, list it.

Definition clarified. - R. J. Mathar, Sep 07 2016

A117241 Numbers divisible by the sum of k times the k-th digit.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 19, 20, 30, 38, 40, 50, 57, 60, 70, 76, 80, 90, 95, 100, 104, 120, 190, 200, 207, 208, 231, 240, 252, 300, 310, 360, 380, 400, 403, 414, 430, 432, 462, 465, 480, 500, 506, 528, 570, 600, 620, 625, 629, 693, 700, 702, 714, 754, 760, 800, 805, 806

Offset: 1

Luc Stevens (lms022(AT)yahoo.com), Apr 22 2006

			95 is in the sequence because 1*9 + 2*5 = 19 and 95 is divisible by 19.
1232 is in the sequence because 1*1 + 2*2 + 3*3 + 4*2 = 22 and 1232 is divisible by 22.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A156207.

Select[Range[1000], Divisible[#, Plus @@ ((d = IntegerDigits[#]) * Range[Length[d]])] &] (* Amiram Eldar, Feb 08 2021 *)

Missing terms inserted by Amiram Eldar, Feb 08 2021

A156294 Sum of products of the digits of prime numbers and the position of the digits in the prime numbers.

Original entry on oeis.org

2, 3, 5, 7, 3, 7, 15, 19, 8, 20, 5, 17, 6, 10, 18, 11, 23, 8, 20, 9, 13, 25, 14, 26, 23, 4, 10, 22, 28, 12, 26, 10, 28, 34, 36, 14, 32, 22, 34, 24, 42, 20, 22, 28, 40, 46, 7, 15, 27, 33, 17, 35, 13, 15, 33, 23, 41, 19, 37, 21, 27, 29, 24, 8, 14, 26, 12, 30, 32, 38, 22, 40, 36, 26, 44

Offset: 1

Cino Hilliard, Feb 07 2009

			For n = 5, prime(5) = 11 and A156207(11) = 1*1 + 1*2 = 3. So, a(5) = 3. - _Indranil Ghosh_, Feb 11 2017

Indranil Ghosh, Table of n, a(n) for n = 1..50000

Table[Total[IntegerDigits[#] Range@ IntegerLength@ #] &@ Prime@ n, {n, 75}] (* Michael De Vlieger, Feb 11 2017 *)

g(x) = local(v,j);v=Vec((Str(x))); sum(j=1, length(v), j*eval(v[j]))
g1(n) = local(j);forprime(j=1,n,print1(g(j)","))

a(n) = A156207(prime(n)). - R. J. Mathar, Sep 10 2016

A381135 Numbers of the form d_1 d_2 d_3 ... where the sum of their digits multiplied by their digit positions is equal to their number of digits.

Original entry on oeis.org

1, 20, 110, 300, 1010, 2100, 4000, 10010, 12000, 20100, 31000, 50000, 100010, 111000, 200100, 220000, 301000, 410000, 600000, 1000010, 1020000, 1101000, 1300000, 2000100, 2110000, 3001000, 3200000, 4010000, 5100000, 7000000, 10000010, 10110000, 11001000, 12100000

Offset: 1

Leo Crabbe, Feb 14 2025

The digits for these numbers are 1-indexed.

Numbers k such that A156207(k) = A055642(k).

			301000 is a member of this sequence because it has 3*1 + 1*3 = 6 digits.

Michael S. Branicky, Table of n, a(n) for n = 1..10000
Michael S. Branicky, Python program for OEIS A381135

Cf. A055642, A156207.

isok(k) = my(d=digits(k)); sum(i=1, #d, i*d[i]) == #d; \\ Michel Marcus, Feb 17 2025

def ok(n): return sum(int(d)*(i+1) for i, d in enumerate(str(n))) == len(str(n))

# see linked program for a different algorithm
from itertools import count, islice
from sympy.utilities.iterables import partitions
def agen(): # generator of terms
    for d in count(1): yield from sorted(int("".join(str(p[k]) if k in p else "0" for k in range(1, d+1))) for p in partitions(d, m=d, k=d) if max(p.values()) < 10 if 1 in p and p[1] != 0)
print(list(islice(agen(), 34))) # Michael S. Branicky, Feb 19 2025

More terms from Michel Marcus, Feb 17 2025

cp's OEIS Frontend

A156208 Primes appearing as the products of digits and positions in A156207(i) in the order of appearance.

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A117241 Numbers divisible by the sum of k times the k-th digit.

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A156294 Sum of products of the digits of prime numbers and the position of the digits in the prime numbers.

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A381135 Numbers of the form d_1 d_2 d_3 ... where the sum of their digits multiplied by their digit positions is equal to their number of digits.

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