A214629 -id:A214629 - cp's OEIS frontend

A260523 Numbers n such that (sum of digits of n) + (product of digits of n) is semiprime.

2, 3, 5, 7, 14, 17, 24, 28, 33, 38, 39, 40, 41, 42, 46, 47, 49, 55, 60, 64, 67, 68, 69, 71, 74, 76, 82, 83, 86, 90, 93, 94, 96, 103, 105, 108, 109, 111, 112, 114, 116, 121, 122, 124, 126, 130, 141, 142, 144, 146, 150, 161, 162, 164, 166, 180, 190, 202, 204, 207

Offset: 1

K. D. Bajpai, Jul 28 2015

			a(6) = 17. (1+7) + (1*7) = 8 + 7 = 15 = 3 * 5, which is semiprime.
a(10) = 38. (3+8) + (3*8) = 11 + 24 = 35 = 5 * 7, which is semiprime.

K. D. Bajpai, Table of n, a(n) for n = 1..10000

Cf. A001358, A007953, A007954, A061762, A214629.

with(numtheory):A260523 := proc() local a; a:= (add(d,d=convert(n, base, 10)) + mul(d,d=convert(n, base, 10))  ); if bigomega(a)=2 then RETURN (n); fi; end: seq(A260523 (),n=1..300);

Select[Range[1000], PrimeOmega[(Plus@@IntegerDigits[#]) + (Times@@IntegerDigits[#])] == 2 &]

for(n=1,500,d=digits(n);s=sumdigits(n);p=prod(i=1,#d,d[i]);if(bigomega(s+p)==2,print1(n,", "))) \\ Derek Orr, Aug 27 2015

A261127 Triangular numbers t such that (sum of digits of t) + (product of digits of t) is a triangular number.

Original entry on oeis.org

0, 3, 10, 105, 120, 136, 190, 210, 300, 406, 703, 780, 820, 1081, 1128, 1431, 1540, 1653, 1770, 1891, 1953, 2080, 2211, 2628, 2701, 2850, 3003, 3160, 3403, 3570, 4560, 4656, 5050, 5460, 7021, 7260, 7503, 8646, 8911, 9453, 10011, 10153, 11026, 12403, 14028, 15400

Offset: 1

K. D. Bajpai, Aug 09 2015

All the terms in this sequence are triangular, and hence 0 or 1 (mod 3).

			a(6) = 136 = 16 * (16+1) / 2, that is triangular number. (1+3+6) + (1*3*6) = 28, which is 7th triangular number.
a(15) = 1128 = 47 * (47+1) / 2, that is triangular number. (1+1+2+8) + (1*1*2*8) = 28, which is 7th triangular number.

K. D. Bajpai, Table of n, a(n) for n = 1..10000

Cf. A000217, A007953, A007954, A061762, A214629.

[n*(n+1) div 2: n in [0..100] | IsSquare(8*k+1) where k is (&+Intseq(n*(n+1) div 2) + &*Intseq(n*(n+1) div 2))];

with(numtheory): A261127:= proc() local a,k,t;t:=n*(n+1)/2; a:= (add(d,d=convert(t, base, 10)) + mul(d,d=convert(t, base, 10)));k:=(-1 + sqrt(8*a + 1))/2; if k=floor(k) then RETURN (t); fi; end: seq(A261127 (),n=0..300);

A261127 = {}; Do[t = n*(n + 1)/2; k = Plus @@ IntegerDigits[t] + Times @@ IntegerDigits[t]; If[IntegerQ[( -1 + Sqrt[8*k + 1])/2], AppendTo[A261127, t]], {n,0,1000}]; A261127

for(n =0, 500, t = n*(n+1)/2; k = (sumdigits(t)); d = digits (t); p = prod(i = 1, #d, d[i]); s = k+p; if(ispolygonal(s,3), print1(t, ", ")));

A344032 a(n) is the least prime that begins a sequence of at least n distinct primes under iteration of A061762.

Original entry on oeis.org

2, 11, 23, 53, 12451, 36779999

Offset: 1

J. M. Bergot and Robert Israel, May 07 2021

			12451 is prime and A061762(12451) = 1*2*4*5*1+1+2+4+5+1 = 53.
53 is prime and A061762(53) = 5*3+5+3 = 23.
23 is prime and A061762(23) = 2*3+2+3 = 11.
11 is prime and A061762(11) = 1*1+1+1 = 3.
3 is prime and A061762(3) = 3+3 = 6 is not prime.
Thus 12451 begins a sequence of 5 distinct primes under the iteration of A061762.  Since 12451 is the least such prime, a(5) = 12451.

Cf. A061762, A214629.

f:= proc(n) local L;
   L:= convert(n,base,10);
   convert(L,`+`)+convert(L,`*`)
end proc:
g:= proc(n) local S,v;
  S:= {n}:
  v:= n;
  do
    v:= f(v);
    if member(v,S) or not isprime(v) then return nops(S) fi;
    S:= S union {v}
  od
end proc:
R:= NULL: p:= 1: m:= 0:
while m < 5 do
  p:= nextprime(p);
  v:= g(p);
  if v > m then R:= R, p$(v-m); m:= v fi
od:
R;

cp's OEIS Frontend

A260523 Numbers n such that (sum of digits of n) + (product of digits of n) is semiprime.

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A261127 Triangular numbers t such that (sum of digits of t) + (product of digits of t) is a triangular number.

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Magma

Maple

Mathematica

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A344032 a(n) is the least prime that begins a sequence of at least n distinct primes under iteration of A061762.

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Author

Keywords

Examples

Crossrefs

Programs

Maple